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10,459 results on '"[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]"'

1. Tropical Linear Regression and Mean Payoff Games: Or, How to Measure the Distance to Equilibria

Author

Marianne Akian, Stéphane Gaubert, Yang Qi, Omar Saadi, TROPICAL (TROPICAL), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
FOS: Computer and information sciences, Computer Science::Computer Science and Game Theory, Optimization and Control (math.OC), Computer Science - Computer Science and Game Theory, General Mathematics, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], MSC Primary 06F20, 14T10, 15A80, 16Y20, 16Y60, Secondary 20N20, 15A24, Mathematics - Optimization and Control, Physics::Atmospheric and Oceanic Physics, Computer Science and Game Theory (cs.GT)
Abstract

We study a tropical linear regression problem consisting in finding the best approximation of a set of points by a tropical hyperplane. We establish a strong duality theorem, showing that the value of this problem coincides with the maximal radius of a Hilbert's ball included in a tropical polyhedron. We also show that this regression problem is polynomial-time equivalent to mean payoff games. We illustrate our results by solving an inverse problem from auction theory. In this setting, a tropical hyperplane represents the set of equilibrium prices. Tropical linear regression allows us to quantify the distance of a market to the set of equilibria, and infer secret preferences of a decision maker.

Published
2023

2. Controllability Properties of Solar Sails

Author

Alesia Herasimenka, Lamberto Dell’Elce, Jean-Baptiste Caillau, Jean-Baptiste Pomet, Université Côte d'Azur (UCA), Mathematics for Control, Transport and Applications (McTAO), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Space and Planetary Science, Control and Systems Engineering, Applied Mathematics, Aerospace Engineering, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Electrical and Electronic Engineering
Abstract

Trajectory design and stationkeeping of solar sails about a celestial body can be formulated as control problems with positivity constraints. Specifically, when reemitted radiation is neglected and the sail is modeled as a flat surface, which are reasonable assumptions for control purposes, the force generated by the solar radiation pressure is contained in a pointed convex cone of revolution with the axis in the sun–satellite direction. Therefore, classical approaches to infer controllability based on the Lie algebra rank condition do not apply to these problems. This study offers a novel condition to decide on controllability of control systems with positivity constraints. This condition is effective because it can be verified by solving an auxiliary convex optimization problem for which reliable numerical methods are available. A crucial ingredient of this approach is the theory of positive trigonometric polynomials. The practical interest of this condition is the assessment of a minimum requirement on the optical properties of the sail, which may be of use for mission design purposes.

Published
2023

3. Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory

Author

Ervedoza, Sylvain, Le Balc’h, Kévin, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020)

Subjects
Quantitative unique continuation, Applied Mathematics, Control, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Elliptic equations, Analysis
Abstract

The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in 2-d and observed from a non-empty open subset ω ⊂ Ω. More precisely, for every real-valued bounded potential V, our main result shows that, when Ω has a finite number of holes, the observability constant of the elliptic operator −∆ + V, with domain H^2 ∩ H^1_0(Ω), is of the form C exp (C |V|^{1/2} \log^{1/2}(|V|)) where C is a positive constant depending only on Ω and ω. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov, in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane. The main difference and additional difficulty is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua's open problem concerning small-time global null-controllability of slightly super-linear heat equations.

Published
2023

4. Index theorems for graph-parametrized optimal control problems

Author

Andrei Agrachev, Stefano Baranzini, Ivan Beschastnyi, Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Universidade de Aveiro, and The work of the third author was supported through the CIDMA Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology ('FCT - Fundacao para a Ciencia e a Tecnologia') within the project UIDP/04106/2020.

Subjects
Mathematics - Differential Geometry, Differential Geometry (math.DG), Optimization and Control (math.OC), Applied Mathematics, FOS: Mathematics, General Physics and Astronomy, Statistical and Nonlinear Physics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Mathematical Physics
Abstract

In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two Hessians related to two different graphs or two different sets of boundary conditions. Several applications such as the iteration formulas or lower bounds for the index are proved., 30 pages, 4 figures

Published
2023

5. Fast and stable schemes for non-linear osmosis filtering

Author

Calatroni, L., Morigi, S., Parisotto, S., Recupero, G. A., Morphologie et Images (MORPHEME), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut de Biologie Valrose (IBV), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Signal, Images et Systèmes (Laboratoire I3S - SIS), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Matematica [Bologna], Alma Mater Studiorum Università di Bologna [Bologna] (UNIBO), Department of Applied Mathematics and Theoretical Physics [Cambridge] (DAMTP), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Computational Mathematics, Computational Theory and Mathematics, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 68U10, 94A08, 49K20, 65M05, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We consider a non-linear variant of the transport-diffusion osmosis model for solving a variety of imaging problems such as shadow/soft-light removal and compact data representation. The non-linear behaviour is encoded in terms of a general scalar function g with suitable properties, which allows to balance the diffusion intensity on the different regions of the image while preventing smoothing artefacts. For the proposed model, conservation properties (intensity and non-negativity) are proved and a variational interpretation is showed for specific choices of g. Upon suitable spatial discretisation, both an explicit and a semi-implicit iterative scheme are considered, for which convergence restrictions and unconditional stability are proved, respectively. To validate the proposed modelling and the computational speed of the numerical schemes considered, we report several results and comparisons for the problem of shadow/light-spot removal and compact data representation, showing that artefact-free and computationally efficient results are obtained in comparison to standard linear and anisotropic models, and state-of-the art approaches., Comment: 25 pages, 14 figures

Published
2023

6. SONC optimization and exact nonnegativity certificates via second-order cone programming

Author

Magron, Victor, Wang, Jie, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), ANR-18-ERC2-0004,COPS,Optimisation garantie pour la vérification des systèmes cyber-physiques(2018), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), European Project: 813211,H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (Main Programme), H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers ,10.3030/813211,POEMA(2019), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, and European Project: 813211,H2020,POEMA(2019)

Subjects
FOS: Computer and information sciences, Computer Science - Symbolic Computation, Algebra and Number Theory, MathematicsofComputing_NUMERICALANALYSIS, Symbolic Computation (cs.SC), sum of binomial squares, exact nonnegativity certificate, Mathematics - Algebraic Geometry, Computational Mathematics, rounding-projection algorithm, Optimization and Control (math.OC), sum of nonnegative circuit polynomials, second-order cone programming, polynomial optimization, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, FOS: Mathematics, second-order conerepresentation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Algebraic Geometry (math.AG), Mathematics - Optimization and Control
Abstract

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables., 29 pages, 7 tables, 6 figures, extended version of the article published in the proceedings of ISSAC 2020. arXiv admin note: text overlap with arXiv:1906.06179

Published
2023

7. A Lagrangian Approach for Aggregative Mean Field Games of Controls with Mixed and Final Constraints

Author

J. Frédéric Bonnans, Justina Gianatti, Laurent Pfeiffer, Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires (COMMANDS), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Universidad Nacional de Rosario [Santa Fe], Centro Franco Argentino de Ciencias de la Información y de Sistemas [Rosario] (CIFASIS), Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires] (CONICET)-Universidad Nacional de Rosario [Santa Fe], Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), and Universidad Nacional de Rosario [Santa Fe]-Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires] (CONICET)

Subjects
Computer Science::Computer Science and Game Theory, aggregative games, Control and Optimization, Optimization and Control (math.OC), Applied Mathematics, Lagrangian equilibria, FOS: Mathematics, TheoryofComputation_GENERAL, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], constrained optimal control, Mathematics - Optimization and Control, Mean field games of controls
Abstract

The objective of this paper is to analyze the existence of equilibria for a class of deterministic mean field games of controls. The interaction between players is due to both a congestion term and a price function which depends on the distributions of the optimal strategies. Moreover, final state and mixed state-control constraints are considered, the dynamics being nonlinear and affine with respect to the control. The existence of equilibria is obtained by Kakutani's theorem, applied to a fixed point formulation of the problem. Finally, uniqueness results are shown under monotonicity assumptions.

Published
2023

8. A PDE-ODE model for traffic control with autonomous vehicles

Author

Liard, Thibault, Stern, Raphael, Delle Monache, Maria Laura, Universidad de Deusto [Bilbao] (DEUSTO), Universidad de Deusto (DEUSTO), Department of Civil and Environmental Engineering [Urbana], University of Illinois at Urbana-Champaign [Urbana], and University of Illinois System-University of Illinois System

Subjects
Statistics and Probability, Traffic control, Applied Mathematics, MathematicsofComputing_NUMERICALANALYSIS, Autonomous vehicles, General Engineering, Computer Science Applications, Computer Science::Robotics, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], PDE-ODE systems, [MATH]Mathematics [math], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We consider a partial differential equation - ordinary differential equation system to describe the dynamics of traffic flow with autonomous vehicles. In the model, the bulk flow of human drivers is represented by a scalar conservation law, while each autonomous vehicle is described by an ordinary differential equation. The coupled PDE-ODE model is introduced, and existence of solutions for this model is shown, along with a proposed algorithm to construct approximate solutions. Next, we propose a control strategy for the speeds of the autonomous vehicles to minimize the average fuel consumption of the entire traffic flow. Existence of solutions for the optimal control problem is proved, and we numerically show that a reduction in average fuel consumption is possible with an AV acting as a moving bottleneck.

Published
2023

9. Existence and Completeness of Solutions to Extended Projected Dynamical Systems and Sector-Bounded Projection-Based Controllers

Author

W. P. M. H. Heemels, Aneel Tanwani, Eindhoven University of Technology [Eindhoven] (TU/e), Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)

Subjects
Nonsmooth dynamics, Control and Optimization, Control and Systems Engineering, [INFO.INFO-SY]Computer Science [cs]/Systems and Control [cs.SY], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Projection based control
Abstract

Projection-based control (PBC) systems have significant engineering impact and receive considerable scientific attention. To properly describe closed-loop PBC systems, extensions of classical projected dynamical systems are needed, because partial projection operators and irregular constraint sets (sectors) are crucial in PBC. These two features obstruct the application of existing results on existence and completeness of solutions. To establish a rigorous foundation for the analysis and design of PBC, we provide essential existence and completeness properties for this new class of discontinuous systems.

Published
2023

10. Démélange, déconvolution et débruitage conjoints d’un modèle convolutif parcimonieux avec dérive instrumentale, par pénalisation de rapports de normes ou quasi-normes lissées (PENDANTSS)

Author

Zheng, Paul, Chouzenoux, Emilie, Duval, Laurent, Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH), OPtimisation Imagerie et Santé (OPIS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de vision numérique (CVN), Institut National de Recherche en Informatique et en Automatique (Inria)-CentraleSupélec-Université Paris-Saclay-CentraleSupélec-Université Paris-Saclay, IFP Energies nouvelles (IFPEN), and European Project: ERC-2019-STG-850925,MAJORIS(2020)

Subjects
Signal Processing (eess.SP), FOS: Electrical engineering, electronic engineering, information engineering, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Electrical Engineering and Systems Science - Signal Processing, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing
Abstract

Denoising, detrending, deconvolution: usual restoration tasks, traditionally decoupled. Coupled formulations entail complex ill-posed inverse problems. We propose PENDANTSS for joint trend removal and blind deconvolution of sparse peak-like signals. It blends a parsimonious prior with the hypothesis that smooth trend and noise can somewhat be separated by low-pass filtering. We combine the generalized pseudo-norm ratio SOOT/SPOQ sparse penalties $\ell_p/\ell_q$ with the BEADS ternary assisted source separation algorithm. This results in a both convergent and efficient tool, with a novel Trust-Region block alternating variable metric forward-backward approach. It outperforms comparable methods, when applied to typically peaked analytical chemistry signals. Reproducible code is provided: https://github.com/paulzhengfr/PENDANTSS., Comment: in French language

Published
2023

11. Implicit Differentiation for Hyperparameter Tuning the Weighted Graphical Lasso

Author

Pouliquen, Can, Gonçalves, Paulo, Massias, Mathurin, Vayer, Titouan, Optimisation, Connaissances pHysiques, Algorithmes et Modèles (OCKHAM), Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut Rhône-Alpin des systèmes complexes (IXXI), École normale supérieure de Lyon (ENS de Lyon)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Inria Lyon, Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Droit des Affaires et Nouvelles Technologies (DANTE), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ), ANR-19-CHIA-0009,AllegroAssai,Algorithmes, Approximations, Parcimonie et Plongements pour l'IA(2019), Dynamic Networks : Temporal and Structural Capture Approach (DANTE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Institut Rhône-Alpin des systèmes complexes (IXXI), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)

Subjects
FOS: Computer and information sciences, Optimization, Computer Science - Machine Learning, Implicit differentiation, Machine Learning (stat.ML), Hyperparameter Selection, Machine Learning (cs.LG), Graphical Lasso, Optimization and Control (math.OC), Statistics - Machine Learning, FOS: Mathematics, [INFO]Computer Science [cs], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control
Abstract

International audience; We provide a framework and algorithm for tuning the hyperparameters of the Graphical Lasso via a bilevel optimizationproblem solved with a first-order method. In particular, we derive the Jacobian of the Graphical Lasso solution with respect to itsregularization hyperparameters.

Published
2023

12. Co-designing versatile quadruped robots for dynamic and energy-efficient motions

Author

Fadini, Gabriele, Kumar, Shivesh, Kumar, Rohit, Flayols, Thomas, Prete, Andrea, Carpentier, Justin, Souères, Philippe, Équipe Mouvement des Systèmes Anthropomorphes (LAAS-GEPETTO), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Université Toulouse III - Paul Sabatier (UT3), Deutsches Forschungszentrum für Künstliche Intelligenz GmbH = German Research Center for Artificial Intelligence (DFKI), University of Trento [Trento], Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria), Département d'informatique - ENS Paris (DI-ENS), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), M-RoCK (FKZ 01IW21002) project funded by the German Aerospace Center (DLR) with federal funds from the Federal Ministry of Education and Research, ANR-10-EQPX-4401,ROBOTEX 2.0,ANR-10-EQPX-44-01, ANR-21-ESRE-0015,TIRREX,Infrastructure technologique pour la recherche d'excellence en robotique(2021), ANR-19-P3IA-0001,PRAIRIE,PaRis Artificial Intelligence Research InstitutE(2019), and European Project: 101070165,AGIMUS

Subjects
Methods and Tools for Robot System Design Optimization and Optimal Control Simulation and Animation Mechanism Design, Mechanism Design, Methods and Tools for Robot System Design, Simulation and Animation, Optimization and Optimal Control, [INFO.INFO-RB]Computer Science [cs]/Robotics [cs.RO], [PHYS.MECA.GEME]Physics [physics]/Mechanics [physics]/Mechanical engineering [physics.class-ph], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

This paper presents a bi-level optimization framework to concurrently optimize a quadruped hardware and control policies for achieving dynamic cyclic behaviors. The longterm vision to drive the design of dynamic and efficient robots by means of computational techniques is applied to improve the development of a new quadruped prototype. The scale of the robot and its actuators are optimized for energy efficiency considering a complete model of the motor, that includes friction, torque, and bandwidth limitations. This model is used to optimize the power consumption during bounding and backflip tasks and is validated by tracking the output trajectories on the first prototype iteration. The co-design results show an improvement of up to 87% for a single task optimization. It appears that, for jumping forward, robots with longer thighs perform better, while for backflips, longer shanks are better suited. To understand the trade-off between these different choices, a Pareto set is constructed to guide the design of the next prototype.

Published
2023

13. Multiobjective optimization with a quadratic surrogate-assisted CMA-ES

Author

Gharafi, Mohamed, Hansen, Nikolaus, Brockhoff, Dimo, Le Riche, Rodolphe, Randomized Optimisation (RANDOPT ), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Ecole Nationale Supérieure des Mines de St Etienne (ENSM ST-ETIENNE), Centre National de la Recherche Scientifique (CNRS), GECCO, Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes (LIMOS), Ecole Nationale Supérieure des Mines de St Etienne (ENSM ST-ETIENNE)-Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), and Université Clermont Auvergne [2017-2020] (UCA [2017-2020])

Subjects
multiobjective optimization CMA-ES evolution strategies surrogateassisted optimization quadratic metamodel, [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], evolution strategies, multiobjective optimization, surrogateassisted optimization, CMA-ES, quadratic metamodel, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-NE]Computer Science [cs]/Neural and Evolutionary Computing [cs.NE]
Abstract

International audience; We present a surrogate-assisted multiobjective optimization algorithm. The aggregation of the objectives relies on the Uncrowded Hypervolume Improvement (UHVI) which is partly replaced by a linear-quadratic surrogate that is integrated into the CMA-ES algorithm. Surrogating the UHVI poses two challenges. First, the UHVI is a dynamic function, changing with the empirical Pareto set. Second, it is a composite function, defined differently for dominated and nondominated points. The presented algorithm is thought to be used with expensive functions of moderate dimension (up to about 50) with a quadratic surrogate which is updated based on its ranking ability. We report numerical experiments which include tests on the COCO benchmark. The algorithm shows in particular linear convergence on the double sphere function with a convergence rate that is 6-20 times faster than without surrogate assistance.

Published
2023

14. The Coherent Multi-Representation Problem with Applications in Structural Biology

Author

Mucherino, Antonio, Institut de Recherche en Informatique et Systèmes Aléatoires (IRISA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Bretagne Sud (UBS)-École normale supérieure - Rennes (ENS Rennes)-Institut National de Recherche en Informatique et en Automatique (Inria)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), and ANR-19-CE45-0019,multiBioStruct,Approches geometriques multi-résolution et multi-échelle pour la determination de structures bio-moleculaires(2019)

Subjects
[SDV.BBM.BS]Life Sciences [q-bio]/Biochemistry, Molecular Biology/Structural Biology [q-bio.BM], [INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS], coherent multi-representation problem, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], distance geometry, structural geometry
Abstract

International audience; We introduce the Coherent Multi-representation Problem (CMP), whose solutions allow us to observe simultaneously different geometrical representations for the vertices of a given simple graph. The idea of graph multi-representation extends the common concept of graph embedding, where every vertex can be embedded in a domain that is unique for each of them. In the CMP, the same vertex can instead be represented in multiple ways, and the main aim is to find a general multi-representation where all the involved variables are "coherent" with one another. We prove that the CMP extends a geometrical problem known in the literature as the distance geometry problem, and we show a preliminary computational experiment on a protein-like instance, which is performed with a new Java implementation specifically conceived for graph multi-representations.

Published
2023

15. Optimal control of a solar sail

Author

Herasimenka, Alesia, Dell'Elce, Lamberto, Caillau, Jean-Baptiste, Pomet, Jean-Baptiste, Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Mathematics for Control, Transport and Applications (McTAO), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
solar sailing, multiple shooting, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], SDP relaxation, differential continuation, Optimal control, non-convex constraints
Abstract

Optimal control of a solar sail orbiting around a fixed center of mass is considered. The sail is modelled as a plane surface with two sides having similar optical properties. The control is assumed to be the attitude of the sail and is so represented as an element of the projective plane. Mapping this plane to compute the actual force generated by a given attitude results in a generally non-convex set of admissible control values. A suitable convex relaxation is introduced to study the optimality system associated with maximising the change of the sail orbital parameters in a given direction along one orbit. Building on previous works, a refined analysis of the control structure is given. In order to compute effective solutions, a multiple shooting approach is retained that is well suited for systems with a sequence of switchings between various control subarcs. Three issues are addressed: initialisation of shooting by means a tailored SDP relaxation that takes advantage of good convergence properties of modern convex optimisation algorithms; changes in the control structure that are accommodated by coupling shooting with differential continuation; implicit character of the Hamiltonian maximisation when using Pontrjagin maximum principle that is taken care of by incorporating the associated equation for the dynamical feedback into the shooting procedure. The method is illustrated on an example from JPL for which the sail inclination is optimally changed over one orbital period.

Published
2023

16. Optimal Renewable Energy Curtailment Minimization Control Using a Combined Electromobility and Grid Model

Author

Čičić, Mladen, Vivas, Carlos, Canudas de Wit, Carlos, Rubio, Francisco, Dynamics and Control of Networks (DANCE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Grenoble Alpes (UGA), Universidad de Sevilla / University of Sevilla, IFAC, and European Project: 694209,Advanced Grant (AdG), ERC-2015-AdG ,Scale-FreeBack(2016)

Subjects
Energy curtailment, Grid optimization, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Electromobility, Renewable energy sources, Optimal control
Abstract

International audience; In this paper we propose an integrated power and transportation system control framework, combining the power grid model with a macroscopic electromobility model including charging stations under V2G operation. In this framework, the electrical vehicles (EVs) act as energy storage, but also as an additional virtual power grid link, transporting energy from one point to another. This new holistic model is used as a basis for optimal control design seeking to minimize renewable energy curtailment, while accounting for the structural limitation of the grid and other SoC constraints necessary for the optimal operation of the EVs. The proposed control scheme is shown to eliminate approximately 50% of curtailment compared to uncoordinated EV charging.

Published
2023

17. Modeling the Impact of Route Recommendations in Road Traffic

Author

Toso, Tommaso, Kibangou, Alain, Frasca, Paolo, Dynamics and Control of Networks (DANCE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Grenoble Alpes (UGA), University of Johannesburg [South Africa] (UJ), and IFAC

Subjects
[SPI.OTHER]Engineering Sciences [physics]/Other, Switching systems, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Route recommendations, Modelling and control of road traffic networks, Traffic congestion, Control of networks, Urban mobility
Abstract

International audience; In this work, effects of real-time route recommendations between two alternative routes are analyzed. The aim is to assess the consequences of route planner applications (apps) on road congestion. For this purpose, the considered model takes the form of a state-dependent switching system that describes a supply-demand mechanism, based on established macroscopic traffic flow models. Through a comprehensive stability analysis of the system, with an emphasis on the dependence on the system parameters (such as road capacities, critical densities, traffic demand), it is shown that real-time recommendations can cause congestion in the network and failure to satisfy user demand.

Published
2023

18. Optimal privacy protection of mobility data: a predictive approach

Author

Molina, Emilio, Fiacchini, Mirko, Cerf, Sophie, Robu, Bogdan, GIPSA - Modelling and Optimal Decision for Uncertain Systems (GIPSA-MODUS), GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Grenoble Alpes (UGA), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria), Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Self-adaptation for distributed services and large software systems (SPIRALS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), and Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)

Subjects
Model predictive and optimization-based control, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Predictive control, [MATH]Mathematics [math], Security and privacy
Abstract

International audience; Location data are extensively used to provide geo-personalized contents to mobile devices users.Sharing such personal data is a major threat to privacy, with risks of re-identification or inference of sensitive information. Location data broadcasted to services can be sanitized, i.e., by adding noise to spatial coordinates.Such protection mechanisms from the literature are widely generic, e.g., not specific to a user and mobility properties. In this work, we advocate that taking into account the specificities of location data (temporal correlation, human mobility patterns, etc.) enables to gain in the privacy-utility trade-off.Specifically, using future mobility prediction greatly improves privacy. We present a novel protection mechanism, based on model predictive control (MPC). The sanitized location is optimally computed so that it maximizes privacy while guaranteeing a utility loss constraint, for present and future locations. Our formulation explicitly takes into account non-constant sampling time, due to moments when no location data is broadcasted.We evaluate experimentally our control on real mobility dataset and compare to the state of the art.Results show that with knowledge of user's future mobility over a few of minutes, we can gain up to 10% of privacy compared to state of the art, while preserving data utility.

Published
2023

19. Identifiability and Observability via decoupled variables: Application to a malaria intra-host model

Author

Bichara, Derdei, Iggidr, Abderrahman, Oumoun, Mohamed, Rapaport, Alain, Sallet, Gauthier, Department of Mathematics [Fullerton], California State University [Fullerton] (CSU), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Université Cadi Ayyad [Marrakech] (UCA), Ecole Nationale des Sciences Appliquées [Marrakech] (ENSA), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Agro Montpellier, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
observability, parameters identification, [SDV.MHEP.MI]Life Sciences [q-bio]/Human health and pathology/Infectious diseases, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Parameters estimation, observer, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], identifiability
Abstract

International audience; For a class of systems for which identifiability is difficult to be proved in the original set of coordinates, we show that when there exists a change of variables which decouples some variables i.e., whose dynamics is independent of the parameters, identifiability can then be assessed without having to compute derivatives. Moreover, we show that this allows to consider an observer to reconstruct the state of the system without having to estimate the unknown parameters. The effectiveness of this approach is illustrated on an intra-host model of malaria infection.

Published
2023

20. On the strong local optimality for state-constrained control-affine problems

Author

Chittaro, F, Poggiolini, L, Contrôle et Diagnostic pour l’Environnement (CDE), Laboratoire d'Informatique et Systèmes (LIS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Matematica e Informatica 'Ulisse Dini' [Firenze] (DIMAI UniFI), Università degli Studi di Firenze = University of Florence (UniFI), and This work was completed with the support of Appel à projet 'Enseignants-Chercheurs invités', Université de Toulon, and by Progetto Internazionaliz-zazione, Università degli Studi di Firenze.

Subjects
2010 MSC: 49K15, 49K30, Pontryagin maximum principle, pure state constraints, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Optimal control, sufficient optimality conditions
Abstract

International audience; In this article we establish first and second order sufficient optimality conditions for a class of single-input control-affine problems, in presence of a scalar state constraint. We consider strong-local optimality (that is, the C 0 topology in the state space). The minimum-time and the Mayer problem are addressed. We restrict our analysis to extremals containing a bang arc, a single boundary arc, followed by a finite number of bang arcs. The sufficient conditions are expressed as a strengthened version of the necessary ones, plus the coerciveness of a suitable finite-dimensional quadratic form. The sufficiency of the given conditions is proven via Hamiltonian methods.

Published
2023

21. Convergence rate of entropy-regularized multi-marginal optimal transport costs

Author

Nenna, Luca, Pegon, Paul, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-11-LABX-0056,LMH,LabEx Mathématique Hadamard(2011)

Subjects
Primary: 49Q22, Secondary: 49N15, 94A17, 49K40, Entropic regularization, Schrödinger problem, Multi-marginal optimal transport, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Functional Analysis (math.FA), Convex analysis, Mathematics - Functional Analysis, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Optimal transport, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, 49Q22, 49N15, 94A17, 49K40, Analysis of PDEs (math.AP)
Abstract

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter $\varepsilon$ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon\log(1/\varepsilon)+O(\varepsilon)$ for some explicit dimensional constants $C$ depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semi-concave costs for a finer estimate, and lower bounds for $\mathcal{C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

Published
2023

22. Local exact controllability to the steady states of a parabolic system with coupled nonlinear boundary conditions

Author

Bhandari, Kuntal, Boyer, Franck, Institute of Mathematics of the Czech Academy of Science (IM / CAS), Czech Academy of Sciences [Prague] (CAS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), and Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.)

Subjects
method of moments, Parabolic systems, fixed-point argument, boundary controllability, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], source term method
Abstract

In this article, we study the boundary local exact controllability to any steady state of a one-dimensional parabolic system with coupled nonlinear boundary conditions by means of only one control. The significant point is that the state components are interacting only at the boundary points in terms of some nonlinear terms. We consider two cases : either the control function is acting through a mixed nonlinear boundary condition on the first component or through a Neumann condition on the second component. The results are slightly different in the two cases. To study this problem, we first consider the associated linearized systems around the given steady state. The method of moments let us to prove its controllability and to obtain a suitable estimate of the control cost of the form $Me^{M/T}$. To this end, we need to develop a precise spectral analysis of a non self-adjoint operator. Thanks to those preliminary results, we can use the source term method developed in [Liu-Takahashi-Tucsnak 2013], followed by the Banach fixed point argument, to obtain the small-time local boundary exact controllability to the steady state for the original system.

Published
2023

23. Moving horizon estimation of human kinematics and muscle forces

Author

Ceglia, Amedeo, Bailly, François, Begon, Mickael, Université de Montréal. Faculté de médecine. Département de pharmacologie et physiologie, École de kinésiologie et des sciences de l'activité physique [Université de Montréal], Department of Medicine, Université de Montréal, Contrôle Artificiel de Mouvements et de Neuroprothèses Intuitives (CAMIN), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Modeling and simulating humans, Physical human-robot interaction, Sensor-based control, [SPI.MECA.BIOM]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Biomechanics [physics.med-ph], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

Human-robot interaction based on real-time kinematics or electromyography (EMG) feedback improves rehabilitation using assist-as-needed strategies. Muscle forces are expected to provide even more comprehensive information than EMG to control these assistive rehabilitation devices. Measuring in vivo muscle force is challenging, leading to the development of numerical methods to estimate them. Due to their high computational cost, forward dynamics-based optimization algorithms were not viable for real-time estimation until recently. To achieve muscle forces estimation in real time, a moving horizon estimator (MHE) algorithm was used to track experimental biosignals. Two participants were equipped with EMG sensors and skin markers that were streamed in real time and used as targets for the MHE. The upper-limb musculoskeletal (MSK) model was composed of 10 degrees-of-freedom actuated by 31 muscles. The MHE relies on a series of overlapping trajectory optimization subproblems of which the following parameters have been adjusted: the fixed duration and the frame to export. We based this adjustment on the estimation delay, the muscle saturation, the joint kinematic mean power frequency, and errors to experimental data. Our algorithm provided consistent estimates of muscle forces and kinematics with visual feedback at 30 Hz with a 110 ms delay. This method is promising to guide rehabilitation and enrich assistive device control laws with personalized force estimations.

Published
2023

24. Réconciliation de paramètres pour la conception d'alternatives socio-techniques cohérentes d'un point de vue biophysique

Author

Mauviel, Olivier, Courtonne, Jean-Yves, Mandil, Guillaume, Sturm, Peter, Sustainability transition, environment, economy and local policy (STEEP), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), and Université Grenoble Alpes (UGA)

Subjects
Material Flow Analysis, Socio-technical Systems, Prospective, Design, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Parameter reconciliation
Abstract

International audience

Published
2023

25. A Generalized Pell's equation for a class of multivariate orthogonal polynomials

Author

Lasserre, Jean-Bernard, Xu, Yuan, Equipe Polynomial OPtimization (LAAS-POP), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Department of Mathematics, University of Oregon [Eugene], University of Oregon [Eugene], and ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Optimization and Control (math.OC), MSC: 44A60 14Q30 42C05 90C25 90C46 32U15 14P10, Probability (math.PR), FOS: Mathematics, 44A60 14Q30 42C05 90C25 90C46 32U15 14P10, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Mathematics - Probability
Abstract

We extend the polynomial Pell's equation satisfied by univariate Chebyshev polynomials on [--1, 1] from one variable to several variables, using orthogonal polynomials on regular domains that include cubes, balls, and simplexes of arbitrary dimension. Moreover, we show that such an equation is strongly connected (i) to a certificate of positivity (from real algebraic geometry) on the domain, as well as (ii) to the Christoffel functions of the equilibrium measure on the domain. In addition, the solution to Pell's equation reflects an extremal property of orthonormal polynomials associated with an entropy-like criterion.

Published
2023

26. Global stabilization of sterile insect technique model by feedback laws

Author

Bidi, Kala Agbo, Almeida, Luis, Coron, Jean-Michel, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)

Subjects
[SDV.EE.SANT]Life Sciences [q-bio]/Ecology, environment/Health, Sterile Insect Technique, Mosquito population control, Dynamical control system, Pest control, Optimization and Control (math.OC), FOS: Mathematics, Lyapunov global stabilization, Vector borne disease, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 93D15, 93D20, 92D25, Feedback design, Mathematics - Optimization and Control, Backstepping feedback
Abstract

The Sterile Insect Technique or SIT is presently one of the most ecological methods for controlling insect pests responsible for disease transmission or crop destruction worldwide. This technique consists of releasing sterile males into the insect pest population. This approach aims at reducing fertility in the population and, consequently, reduce significantly the native insect population after a few generations. In this work, we study the global stabilization of a pest population at extinction equilibrium by the SIT method. We construct explicit feedback laws that stabilize the model and do numerical simulations to show the efficiency of our feedback laws. The different feedback laws are also compared taking into account their possible implementation in field interventions.

Published
2023

27. Exploiting Constant Trace Property in Large-scale Polynomial Optimization

Author

Ngoc Hoang Anh Mai, J. B. Lasserre, Victor Magron, Jie Wang, Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Equipe Polynomial OPtimization (LAAS-POP), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), ANR-18-ERC2-0004,COPS,Optimisation garantie pour la vérification des systèmes cyber-physiques(2018), European Project: 813211,H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions (Main Programme), H2020-EU.1.3.1. - Fostering new skills by means of excellent initial training of researchers ,10.3030/813211,POEMA(2019), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), and European Project: 813211,H2020,POEMA(2019)

Subjects
Optimization and Control (math.OC), Applied Mathematics, FOS: Mathematics, MathematicsofComputing_NUMERICALANALYSIS, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Software
Abstract

We prove that every semidefinite moment relaxation of a polynomial optimization problem (POP) with a ball constraint can be reformulated as a semidefinite program involving a matrix with constant trace property (CTP). As a result such moment relaxations can be solved efficiently by first-order methods that exploit CTP, e.g., the conditional gradient-based augmented Lagrangian method. We also extend this CTP-exploiting framework to large-scale POPs with different sparsity structures. The efficiency and scalability of our framework are illustrated on second-order moment relaxations for various randomly generated quadratically constrained quadratic programs., 43 pages, 6 algorithms, 23 tables

Published
2022

28. Hyperparameter selection for Discrete Mumford–Shah

Author

Lucas, Charles-Gérard, Pascal, Barbara, Pustelnik, Nelly, Abry, Patrice, Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Laboratoire des Sciences du Numérique de Nantes (LS2N), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-École Centrale de Nantes (Nantes Univ - ECN), Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes université - UFR des Sciences et des Techniques (Nantes univ - UFR ST), Nantes Université - pôle Sciences et technologie, Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes Université - pôle Sciences et technologie, and Nantes Université (Nantes Univ)

Subjects
non-convex minimization, contour detection, Image and Video Processing (eess.IV), Electrical Engineering and Systems Science - Image and Video Processing, Optimization and Control (math.OC), Stein Unbiased Risk Estimate, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Signal Processing, FOS: Electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Mumford-Shah functional, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Electrical and Electronic Engineering, Mathematics - Optimization and Control
Abstract

International audience; This work focuses on a parameter-free joint piecewise smooth image denoising and contour detection. Formulated as the minimization of a discrete Mumford-Shah functional and estimated via a theoretically grounded alternating minimization scheme, the bottleneck of such a variational approach lies in the need to fine-tune their hyperparameters, while not having access to ground truth data.To that aim, a Stein-like strategy providing optimal hyperparameters is designed, based on the minimization of an unbiased estimate of the quadratic risk.Efficient and automated minimization of the estimate of the risk crucially relies on an unbiased estimate of the gradient of the risk with respect to hyperparameters. Its practical implementation is performed using a forward differentiation of the alternating scheme minimizing the Mumford-Shah functional, requiring exact differentiation of the proximity operators involved. Intensive numerical experiments are performed on synthetic images with different geometry and noise levels, assessing the accuracy and the robustness of the proposed procedure.The resulting parameter-free piecewise-smooth estimation and contour detection procedure, not requiring prior image processing expertise nor annotated data, can then be applied to real-world images.

Published
2022

29. Risk-Sensitive Safety Analysis Using Conditional Value-at-Risk

Author

Kevin M. Smith, Claire J. Tomlin, Margaret P. Chapman, Marco Pavone, Insoon Yang, Riccardo Bonalli, Department of Computer Science [University of Toronto] (DCS), University of Toronto, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Tufts University [Medford], Department of Electrical and Computer Engineering [Seoul National University], Seoul National University [Seoul] (SNU), Stanford University, Department of Electrical Engineering and Computer Science [Berkeley] (EECS), University of California [Berkeley], and University of California-University of California

Subjects
0209 industrial biotechnology, Mathematical optimization, Computer science, CVAR, Systems and Control (eess.SY), 02 engineering and technology, State (functional analysis), Optimal control, Electrical Engineering and Systems Science - Systems and Control, Upper and lower bounds, Computer Science Applications, Expected shortfall, 020901 industrial engineering & automation, Control and Systems Engineering, FOS: Electrical engineering, electronic engineering, information engineering, Trajectory, State space, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Markov decision process, Electrical and Electronic Engineering
Abstract

This paper develops a safety analysis method for stochastic systems that is sensitive to the possibility and severity of rare harmful outcomes. We define risk-sensitive safe sets as sub-level sets of the solution to a non-standard optimal control problem, where a random maximum cost is assessed via Conditional Value-at-Risk (CVaR). The objective function represents the maximum extent of constraint violation of the state trajectory, averaged over a given percentage of worst cases. This problem is well-motivated but difficult to solve tractably because the temporal decomposition for CVaR is history-dependent. Our primary theoretical contribution is to derive computationally tractable under-approximations to risk-sensitive safe sets. Our method provides a novel, theoretically guaranteed, parameter-dependent upper bound to the CVaR of a maximum cost without the need to augment the state space. For a fixed parameter value, the solution to only one Markov decision process problem is required to obtain the under-approximations for any family of risk-sensitivity levels. In addition, we propose a second definition for risk-sensitive safe sets and provide a tractable method for their estimation without using a parameter-dependent upper bound. The second definition is expressed in terms of a new coherent risk functional, which is inspired by CVaR. We demonstrate our primary theoretical contribution via numerical examples.

Published
2022

30. Diffusive Limit Approximation of Pure-Jump Optimal Stochastic Control Problems

Author

Marc Abeille, Bruno Bouchard, Lorenzo Croissant, Criteo AI Lab, Criteo [Paris], CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Control and Optimization, Optimization and Control (math.OC), Applied Mathematics, FOS: Mathematics, Diffusive limit, stochastic optimal control, online auctions, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Management Science and Operations Research, Mathematics - Optimization and Control
Abstract

We consider the diffusive limit of a typical pure-jump Markovian control problem as the intensity of the driving Poisson process tends to infinity. We show that the convergence speed is provided by the H\"older constant of the Hessian of the limit problem, and explain how correction terms can be constructed. This provides an alternative efficient method for the numerical approximation of the optimal control of a pure-jump problem in situations with very high intensity of jump. We illustrate this approach in the context of a display advertising auction problem.

Published
2022

31. Geometrically consistent aerodynamic optimization using an isogeometric Discontinuous Galerkin method

Author

Stefano Pezzano, Régis Duvigneau, Mickaël Binois, Analysis and Control of Unsteady Models for Engineering Sciences (ACUMES), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Computational Mathematics, Computational Theory and Mathematics, Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph]
Abstract

International audience; The objective of the current work is to define a design optimization methodology in aerodynamics, in which all numerical components are based on a unique geometrical representation, consistent with Computer-Aided Design (CAD) standards. In particular, the design is parameterized by Non-Uniform Rational B-Splines (NURBS), the computational domain is automatically constructed using rational Bézier elements extracted from NURBS boundaries without any approximation and the resolution of the flow equations relies on an adaptive Discontinuous Galerkin (DG) method based on rational representations. A Bayesian framework is used to optimize NURBS control points, in a single-or multi-objective, constrained, global optimization framework. The resulting methodology is therefore fully CAD-consistent, high-order in space and time, includes local adaption and shock capturing capabilities, and exhibits high parallelization performance. The proposed methods are described in details and their properties are established. Finally, two design optimization problems are provided as illustrations: the shape optimization of an airfoil in transonic regime, for drag reduction with lift constraint, and the multi-objective optimization of the control law of a morphing airfoil in subsonic regime, regarding the time-averaged lift, the minimum instantaneous lift and the energy consumption.

Published
2022

32. Time optimal control for a mobile robot with a communication objective

Author

Jérôme Lohéac, Irinel-Constantin Morarescu, Vineeth Satheeskumar Varma, Centre de Recherche en Automatique de Nancy (CRAN), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, General Computer Science, Key-words: Time optimal control, Pontryagin maximum principle, Applied Mathematics, Modeling and Simulation, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Time optimal control, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], wireless communication, [SPI.AUTO]Engineering Sciences [physics]/Automatic, Theoretical Computer Science
Abstract

In this paper, we design control strategies that minimize the time required by a mobile robot to accomplish a certain task (reach a target) while transmitting/receiving a message. To better illustrate the solution we consider a simple model for the robot dynamics. The message delivery is done over a wireless network, and we account for path-loss, i.e., the transmission rate depends on the distance to the wireless antenna. In this work, we consider only one wireless antenna and disregard any shadowing phenomena. To render the problem interesting from a practical point of view we assume that the robot cannot move with innite velocity. The general problem involves a switching control signal due to the complementarity of the objectives (message transmission can require to approach the antenna situated in the opposite direction of the nal target to reach). Our minimal-time control design is based on the use of Pontryagin maximum principle. A numerical example illustrates the theoretical results.

Published
2022

33. Small perturbations in the type of boundary conditions for an elliptic operator

Author

Bonnetier, Eric, Dapogny, Charles, Vogelius, Michael, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Department of Mathematics - Rutgers School of Arts and Sciences, Rutgers, The State University of New Jersey [New Brunswick] (RU), Rutgers University System (Rutgers)-Rutgers University System (Rutgers), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)

Subjects
Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Applied Mathematics, General Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this ``background'' situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a ``small'' subset $\omega_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a ``small'' subset $\omega_\varepsilon $ of the Dirichlet boundary. The relevant quantity that measures the ``smallness'' of the subset $\omega_\varepsilon $ differs in the two cases: while it is the harmonic capacity of $\omega_\varepsilon $ in the former case, we introduce a notion of ``Neumann capacity'' to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general $\omega_\varepsilon $, under the sole assumption that it is ``small'' in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset $\omega_\varepsilon $ is a vanishing surfacic ball.

Published
2022

34. Analysis of the Impact of Variable Speed Limits on Environmental Sustainability and Traffic Performance in Urban Networks

Author

Bassel Othman, Giovanni De Nunzio, Domenico Di Domenico, Carlos Canudas-de-Wit, IFP Energies nouvelles (IFPEN), TotalEnergies, Dynamics and Control of Networks (DANCE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), and Université Grenoble Alpes (UGA)

Subjects
Variable Speed Limits, Energy efficiency, Mechanical Engineering, Automotive Engineering, Pollutant emissions, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Model Predictive Control, Artificial Neural Networks, Traffic modeling, Computer Science Applications
Abstract

International audience; This work focuses on evaluating the potential of variable speed limits (VSLs) in a synthetic urban network to improve both environmental sustainability and traffic performance. The traffic system is modeled using the microscopic traffic simulator SUMO, and a physical fuel consumption and NOx emission model is used to assess the vehicles' energy efficiency. Speed limits are controlled through a nonlinear model predictive control (NMPC) approach, in which the traffic evolution and fuel consumption are respectively predicted with a macroscopic traffic model, namely the cell transmission model (CTM), and a pre-calibrated artificial neural network (ANN). The results reveal that in transient phases between different levels of congestion, the proposed eco-VSL controller is faster to decongest the network, resulting in an improvement of the environmental sustainability and the traffic performance both in the controlled network, and at its boundary roads.

Published
2022

35. Interior point methods are not worse than Simplex

Author

Xavier Allamigeon, Daniel Dadush, Georg Loho, Bento Natura, Laszlo A. Vegh, Mathematics of Operations Research, TROPICAL (TROPICAL), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centrum voor Wiskunde en Informatica (CWI), Centrum Wiskunde & Informatica (CWI)-Netherlands Organisation for Scientific Research, Faculty of Electrical Engineering, Mathematics and Computer Science [Twente] (EEMCS), University of Twente, Department Mathematics [London] (LSE), London School of Economics and Political Science (LSE), and Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Subjects
FOS: Computer and information sciences, [INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS], Geometry, Time measurement, Computer science, Current measurement, Partitioning algorithms, n/a OA procedure, Optimization and Control (math.OC), Linear programming, Computer Science - Data Structures and Algorithms, FOS: Mathematics, Data Structures and Algorithms (cs.DS), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Upper bound
Abstract

Whereas interior point methods provide polynomial-time linear programming algorithms, the running time bounds depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. We introduce a new polynomial-time path-following interior point method where the number of iterations also admits a combinatorial upper bound $O(2^{n}n^{15}\log n)$ for an n-variable linear program in standard form. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations. The number of iterations of our algorithm is at most $O(n^{15}\log n)$ times the number of segments of any piecewise linear curve in the wide neighborhood of the central path. In particular, it matches the number of iterations of any path following interior point method up to this polynomial factor. The overall exponential upper bound derives from studying the max central path’, a piecewise-linear curve with the number of pieces bounded by the total length of 2n shadow vertex simplex paths. From the existence of a line segment in the wide neighborhood we derive strong implications on the structure of the corresponding segment of the central path. Our algorithm is able to detect this structure from the local geometry at the current iterate, and constructs a step direction that descends along this segment. The bound $O(n^{15}\log n)$ that applies for arbitrarily long line segments is derived from a combinatorial progress measure. Our algorithm falls into the family of layered least squares interior point methods introduced by Vavasis and Ye (Math. Prog. 1996). In contrast to previous layered least squares methods that partition the kernel of the constraint matrix into coordinate subspaces, our method creates layers based on a general subspace providing more flexibility. Our result also implies the same bound on the number of iterations of the trust region interior point method by Lan, Monteiro, and Tsuchiya (SIOPT 2009).

Published
2022

36. Consensus formation in first-order graphon models with time-varying topologies

Author

Bonnet, Benoît, Duteil, Nastassia Pouradier, Sigalotti, Mario, Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Modelling and Analysis for Medical and Biological Applications (MAMBA), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)

Subjects
Consensus Formation, Algebraic Connectivity, MSC2020 Subject Classification : 05C63, 05C90, 37L15, 93A16, Applied Mathematics, Multi-Agent Systems, Functional Analysis (math.FA), Mathematics - Functional Analysis, Optimization and Control (math.OC), Modeling and Simulation, FOS: Mathematics, 05C63, 05C90, 37L15, 93A16, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math], Mathematics - Optimization and Control, Scrambling Coefficient, Graphon Dynamics, Persistence Conditions
Abstract

In this article, we investigate the asymptotic formation of consensus for several classes of time-dependent cooperative graphon dynamics. After motivating the use of this type of macroscopic models to describe multi-agent systems, we adapt the classical notion of scrambling coefficient to this setting, leverage it to establish sufficient conditions ensuring the exponential convergence to consensus with respect to the $L^{\infty}$-norm topology. We then shift our attention to consensus formation expressed in terms of the $L^2$-norm, and prove three different consensus result for symmetric, balanced and strongly connected topologies, which involve a suitable generalisation of the notion of algebraic connectivity to this infinite-dimensional framework. We then show that, just as in the finite-dimensional setting, the notion of algebraic connectivity that we propose encodes information about the connectivity properties of the underlying interaction topology. We finally use the corresponding results to shed some light on the relation between $L^2$- and $L^{\infty}$-consensus formation, and illustrate our contributions by a series of numerical simulations., Comment: 48 pages, 16 figures

Published
2022

37. Vertex order with optimal number of adjacent predecessors

Author

Jérémy Omer and Tangi Migot

Subjects
vertex ordering, distance geometry problem, discretization order, complexity analysis, [math.math-oc]mathematics [math]/optimization and control [math.oc], [info.info-ro]computer science [cs]/operations research [cs.ro], [info.info-cc]computer science [cs]/computational complexity [cs.cc], Mathematics, QA1-939
Abstract

In this paper, we study the complexity of the selection of a graph discretization order with a stepwise linear cost function. Finding such vertex ordering has been proved to be an essential step to solve discretizable distance geometry problems (DDGPs). DDGPs constitute a class of graph realization problems where the vertices can be ordered in such a way that the search space of possible positions becomes discrete, usually represented by a binary tree. In particular, it is useful to find discretization orders that minimize an indicator of the size of the search tree. Our stepwise linear cost function generalizes this situation and allows to discriminate the vertices into three categories depending on the number of adjacent predecessors of each vertex in the order and on two parameters K and U. We provide a complete study of NP-completeness for fixed values of K and U. Our main result is that the problem is NP-complete in general for all values of K and U such that U ≥ K + 1 and U ≥ 2. A consequence of this result is that the minimization of vertices with exactly K adjacent predecessors in a discretization order is also NP-complete.

Published
2020
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38. Exact computation of an error bound for the balanced linear complementarity problem with unique solution

Author

Jean Charles Gilbert, Jean-Pierre Dussault, Université de Sherbrooke (UdeS), Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Dualité forte, Data bitlength, P-matrice, Separable function, Matrix inverse norm, General Mathematics, Norme d'une matrice inverse, Fonction séparable, Balanced linear complementarity problem, Strong duality, Complexity, Longueur en bits de données, Rowwise convex combination of matrices, Problème de complémentarité linéaire équilibré, Matrice diagonale extrême, Combinaison convexe ligne par ligne de matrices, P-matrix, AMS MSC 2020: 15A09, 15A60, 49N15, 65F35, 65Y20, 90C33, 90C46, 90C60, Error bound, Borne d'erreur, Complexité, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Software, Extreme diagonal matrix
Abstract

International audience; This paper considers the balanced form of the standard linear complementarity problem with unique solution and provides a more precise expression of an upper error bound discovered by Chen and Xiang and published in 2006. This expression has at least two advantages. It makes possible the exact computation of the error bound factor and it provides a satisfactory upper estimate of that factor in terms of the data bitlength when the data is formed of rational numbers. Along the way, we show that, when any rowwise convex combination of two square matrices is nonsingular, the infinity norm of the inverse of these rowwise convex combinations is maximized by an extreme diagonal matrix.; Cet article considère la forme équilibrée du problème de complémentarité linéaire standard avec solution unique et présente une expression plus précise de la borne d'erreur supérieure découverte par Chen et Xiang et publiée en 2006. Cette expression a au moins deux avantages. Elle rend possible l'évaluation exacte du facteur de la borne d'erreur et elle permet d'obtenir une estimation supérieure de ce facteur en termes des longueurs en bits des données lorsque celles-ci s'expriment en nombres rationnels. En chemin, nous montrons que, lorsque les combinaisons convexes ligne par ligne de deux matrices carrées sont inversibles, la norme infinie de l'inverse de ces combinaisons convexes est maximale pour une matrice diagonale extrême.

Published
2022

39. Sharp reachability results for the heat equation in one space dimension

Author

KELLAY, Karim, NORMAND, Thomas, TUCSNAK, Marius, Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Applied Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Analysis
Abstract

This paper gives a complete characterization of the reachable space for a system described by the 1D heat equation with L2 (with respect to time) Dirichlet boundary controls at both ends. More precisely, we prove that this space coincides with the sum of two spaces of analytic functions (of Bergman type). These results are then applied to give a complete description of the reachable space via inputs which are n-times differentiable functions of time. Moreover, we establish a connection between the norm in the obtained sum of Bergman spaces and the cost of null controllability in small time. Finally we show that our methods yield new complex analytic results on the sums of Bergman spaces in infinite sectors.

Published
2022

40. Discrete optimal couple tracking: A control scheme for electrical-tuning of strong coupling piezoelectric energy harvesters

Author

Adrien Morel, Gaël Pillonnet, David Gibus, Adrien Badel, Laboratoire SYstèmes et Matériaux pour la MEcatronique (SYMME), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Commissariat à l'énergie atomique et aux énergies alternatives - Laboratoire d'Electronique et de Technologie de l'Information (CEA-LETI), Direction de Recherche Technologique (CEA) (DRT (CEA)), and Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)

Subjects
Frequency tuning, Nonlinear electronics, Energy harvesting, Mechanical Engineering, Gradient algorithm, [SPI.NRJ]Engineering Sciences [physics]/Electric power, Piezoelectricity, [SPI.MECA.VIBR]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Vibrations [physics.class-ph], Broadband, [SPI.TRON]Engineering Sciences [physics]/Electronics, Discrete Maximum Power Point Tracking, Predictive model, General Materials Science, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Vibration energy harvesting, Impedance matching
Abstract

In order to track the maximum power point of piezoelectric energy harvesters, an effective approach consists in designing interfaces able to electrically tune the harvester dynamics. Such interface should exhibit—at least—two tunable parameters in order to independently optimize the harvester damping and resonant frequency, that is, an electromechanical impedance matching of the real and imaginary parts of the electrical load. In order to optimize and control these two parameters, it is necessary to implement two control loops that simultaneously impact the harvester dynamics. Furthermore, these loops should combine quick convergence time with low-power consumption in order to react sufficiently fast to compensate for any shift in the vibration frequency and to consume only a small portion of the harvested power. In this paper, we propose a control-law based on the successive evaluation of couples of parameters in order to select the optimal one. First, we determine analytically the expression of a set of optimal couples valid for any two-parameters electrical interfaces. Thereafter, we prove that only a few well-chosen couples of parameters are sufficient to maximize the power on a large frequency band. The proposed methodology has been experimentally verified on a tunable interface, the short-circuit synchronous electric charge extraction. Combined with a strongly coupled piezoelectric energy harvester, we have been able to maintain the harvested power close to the maximal achievable power (i.e. >70% of the maximal achievable power) on a 20 Hz frequency band, with only five couples of parameters. Such approach allows to substitute a multi-parameters convergence algorithm by a single-couple of parameters convergence algorithm, greatly simplifying the algorithm and reducing its convergence time.

Published
2022

41. Dynamic Social Learning Under Graph Constraints

Author

Konstantin Avrachenkov, Suhail M. Shah, Sharayu Moharir, Vivek S. Borkar, Network Engineering and Operations (NEO ), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Department of Electrical Engineering [IIT-Bombay] (EE-IIT), Indian Institute of Technology Kanpur (IIT Kanpur), Hong Kong University of Science and Technology (HKUST), This work was supported by the grant `Machine Learning for NetworkAnalytics' from the Indo-French Centre for Promotion of AdvancedScientific Research (Cefipra) and by the grant `Distributed Learning and Controlfor Network Analysis' from Nokia Bell Labs., and The Hong Kong University of Sciences & Technology (HKUST)

Subjects
FOS: Computer and information sciences, Vertex (graph theory), Computer Science - Machine Learning, Control and Optimization, Computer Networks and Communications, annealed dynamics, Stochastic approximation, Machine Learning (cs.LG), [INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI], [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], graphical constraints, FOS: Mathematics, Mathematics - Optimization and Control, Equivalence (measure theory), Empirical process, Mathematics, Discrete mathematics, Markov chain, dynamic choice with reinforcement, Probability (math.PR), vertex reinforced random walk, Recursion (computer science), Random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Optimization and Control (math.OC), Control and Systems Engineering, Signal Processing, Graph (abstract data type), optimal choice, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Probability
Abstract

International audience; We introduce a model of graph-constrained dynamic choice with reinforcement modeled by positively $\alpha$-homogeneous rewards. We show that its empirical process, which can be written as a stochastic approximation recursion with Markov noise, has the same probability law as a certain vertex reinforced random walk. We use this equivalence to show that for $\alpha > 0$, the asymptotic outcome concentrates around the optimum in a certain limiting sense when 'annealed' by letting $\alpha \to \infty$ slowly.

Published
2022

42. Nonatomic aggregative games with infinitely many types

Author

Paulin Jacquot, Cheng Wan, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), TROPICAL (TROPICAL), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie (EDF R&D OSIRIS), EDF R&D (EDF R&D), and EDF (EDF)-EDF (EDF)

Subjects
FOS: Computer and information sciences, TheoryofComputation_MISCELLANEOUS, Computer Science::Computer Science and Game Theory, Information Systems and Management, General Computer Science, media_common.quotation_subject, 0211 other engineering and technologies, Monotonic function, 02 engineering and technology, Management Science and Operations Research, generalized variational inequality, Industrial and Manufacturing Engineering, nonatomic aggregative game, Computer Science - Computer Science and Game Theory, 0502 economics and business, Convergence (routing), FOS: Mathematics, 050207 economics, Mathematics - Optimization and Control, Finite set, Parametric statistics, Mathematics, media_common, Sequence, 021103 operations research, 05 social sciences, ComputingMilieux_PERSONALCOMPUTING, TheoryofComputation_GENERAL, Coupling (probability), Infinity, Optimization and Control (math.OC), variational equilibrium, Modeling and Simulation, Variational inequality, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC], coupling aggregative constraints, Mathematical economics, monotone game, Computer Science and Game Theory (cs.GT)
Abstract

We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of players types. These equilibria are characterized through an infinite-dimensional variational inequality. We show, under monotonicity conditions, a convergence theorem enables to approximate such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of players types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different players types, with actions subject to coupling constraints., arXiv admin note: substantial text overlap with arXiv:1806.06230

Published
2022

43. Convex Optimization of Bioprocesses

Author

Josh A. Taylor, Alain Rapaport, Denis Dochain, UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique, Department of Electrical and Computer Engineering [University of Toronto] (ECE), University of Toronto, Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM), Université Catholique de Louvain = Catholic University of Louvain (UCL), and Rapaport, Alain

Subjects
convex relaxation, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], compartmental system, Computer Science Applications, wastewater treatment, [INFO.INFO-BT] Computer Science [cs]/Biotechnology, Bioprocess, second-order cone programming, Optimization and Control (math.OC), Control and Systems Engineering, FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-BT]Computer Science [cs]/Biotechnology, Electrical and Electronic Engineering, Mathematics - Optimization and Control
Abstract

We optimize a general model of bioprocesses, which is nonconvex due to the microbial growth in the biochemical reactors. We formulate a convex relaxation and give conditions guaranteeing its exactness in both the transient and steady state cases. When the growth kinetics are modeled by the Monod function under constant biomass or the Contois function, the relaxation is a second-order cone program, which can be solved efficiently at large scales. We implement the model on a numerical example based on a wastewater treatment system.

Published
2022

44. Stochastic String Stability of Vehicle Platoons via Cooperative Adaptive Cruise Control With Lossy Communication

Author

Geert Heijenk, Anton A. Stoorvogel, Paolo Frasca, Francesco Acciani, Department of Applied Mathematics [Twente], University of Twente [Netherlands], Dynamics and Control of Networks (DANCE), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Grenoble Alpes (UGA), Digital Society Institute, Design and Analysis of Communication Systems, Hybrid Systems, and University of Twente

Subjects
Stability criteria, Computer science, Autonomous vehicles, Systems and Control (eess.SY), 02 engineering and technology, Vehicle dynamics, Lossy compression, Stability (probability), string stability, [SPI.AUTO]Engineering Sciences [physics]/Automatic, Stochastic processes, Control theory, 0502 economics and business, FOS: Electrical engineering, electronic engineering, information engineering, FOS: Mathematics, 0202 electrical engineering, electronic engineering, information engineering, Propagation losses, Delays, Mathematics - Optimization and Control, Automatic control, 050210 logistics & transportation, Packet loss, Mechanical Engineering, 05 social sciences, String (computer science), 22/2 OA procedure, Variance (accounting), Computer Science Applications, Vehicle platooning, Cooperative Adaptive Cruise Control, Optimization and Control (math.OC), Cruise control, Automotive Engineering, Computer Science - Systems and Control, 020201 artificial intelligence & image processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

This paper is about obtaining stable vehicle platooning by using Cooperative Adaptive Cruise Control when the communication is unreliable and suffers from message losses. We model communication losses as independent random events and we propose an original design for the cooperative controller, which mitigates the effect of losses. This objective is obtained by a switching controller that has a twofold objective: on the one hand, it promotes both plant stability and string stability of the average error dynamics by an $H_infty$ approach, and on the other hand it minimizes the variance around the average. We show by simulations that the proposed controller is able to compensate even for high probability of losses., Comment: 10 pages, 7 figures; submitted to journal

Published
2022

45. Analysis of the Implicit Euler Time-Discretization of Semiexplicit Differential-Algebraic Linear Complementarity Systems

Author

Bernard Brogliato, Modélisation, simulation et commande des systèmes dynamiques non lisses (TRIPOP), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), INRIA Project Lab Modeliscale, Univ. Grenoble Alpes, INRIA, INRIA Project Lab (IPL) ModeliScale, and FUI ModeliScale

Subjects
discrete-time, recession function, Control and Optimization, linear complementarity system, descriptor-variable system, well-posedness, Applied Mathematics, differential-algebraic system, MSC: 65K15, 90C33, 90C90, 93B52, 93D30, 93C30, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Lur'e equations, passive system
Abstract

International audience; This article is largely concerned with the time-discretization of differential-algebraic equations (DAE) with complementarity constraints, which we name differential algebraic linear complementarity systems (DALCS). Specifically, the Euler implicit discretization of DALCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be wellposed under some conditions, then the convergence of the discretized solutions is studied, and the existence of solutions to the continuous-time system is shown as a consequence. Passivity of some operators is pivotal to the analysis. Examples from circuits, mechanics and switching DAE illustrate the applicability of the developments.

Published
2022

46. Convergence of a Constrained Vector Extrapolation Scheme

Author

Mathieu Barré, Adrien Taylor, Alexandre d'Aspremont, Statistical Machine Learning and Parsimony (SIERRA), Département d'informatique - ENS Paris (DI-ENS), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris sciences et lettres (PSL), MB acknowledges support from an AMX fellowship. AT acknowledges support from the European ResearchCouncil (grant SEQUOIA 724063. AA would like to acknowledge support from the ML and Optimisation joint research initiative with the fonds AXA pour la recherche and , a Google focused award, as well as funding by the French government under management of Agence Nationale de la Recherche as part of the 'Investissements d’avenir' program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute)., ANR-19-P3IA-0001,PRAIRIE,PaRis Artificial Intelligence Research InstitutE(2019), and European Project: 724063,ERC-2016-COG,SEQUOIA(2017)

Subjects
ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

International audience; Among extrapolation methods, Anderson acceleration (AA) is a popular technique for speeding up convergence of iterative processes toward their limit points. AA proceeds by extrapolating a better approximation of the limit using a weighted combination of previous iterates. Whereas AA was originally developed in the context of nonlinear integral equations, or to accelerate the convergence of iterative methods for solving linear systems, it is also used to extrapolate the solution of nonlinear systems. Simple additional stabilization strategies can be used in this context to control conditioning issues. In this work, we study a constrained vector extrapolation scheme based on an offline version of AA with fixed window size, for solving nonlinear systems arising in optimization problems, where the stabilization strategy consists in bounding the magnitude of the extrapolation weights. We provide explicit convergence bounds for this method and, as a by-product, upper bounds on a constrained version of the Chebyshev problem on polynomials.

Published
2022

47. Time-Scale Hybrid Continuous-Discrete Sliding Frank-Wolfe Method

Author

Clara, Lage, Nelly, Pustelnik, Jean-Michel, Arbona, Benjamin, Audit, Département de Physique [ENS Lyon], École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon, Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Laboratoire de biologie et modélisation de la cellule (LBMC UMR 5239), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE45-0002,NanoPoRep,Le sequençage nanopore pour une cartographie haute-performance de la réplication de l'ADN(2018), and ANR-19-CE12-0028,HUDROR,Origines de réplication humaines : réconcilier des visions disparates(2019)

Subjects
Time-scale dictionary, Sparse Coding, NanoForkSpeed, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Sliding Frank-Wolfe, [INFO.INFO-BI]Computer Science [cs]/Bioinformatics [q-bio.QM]
Abstract

The continuous sparse coding aims to improve the estimation accuracy by avoiding discretization of the parameter set. In this context, Sliding Frank-Wolfe algorithm has been successfully applied on deconvolution problems when the dictionary involves position shifts. In this article, we treat the problem of dictionary involving positions and scales shifts. We propose a new algorithm inspired on Sliding Frank-Wolfe that considers an hybrid model involving discretized scales and continuous positions. The model proves to be efficient in the context of DNA replication.

Published
2023

48. Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows

Author

Marcotte, Sibylle, Gribonval, Rémi, Peyré, Gabriel, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Optimisation, Connaissances pHysiques, Algorithmes et Modèles (OCKHAM), Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut Rhône-Alpin des systèmes complexes (IXXI), École normale supérieure de Lyon (ENS de Lyon)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Inria Lyon, Institut National de Recherche en Informatique et en Automatique (Inria), ANR-19-CHIA-0009,AllegroAssai,Algorithmes, Approximations, Parcimonie et Plongements pour l'IA(2019), ANR-19-P3IA-0001,PRAIRIE,PaRis Artificial Intelligence Research InstitutE(2019), and European Project

Subjects
FOS: Computer and information sciences, Computer Science - Machine Learning, Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-NE]Computer Science [cs]/Neural and Evolutionary Computing [cs.NE], Mathematics - Optimization and Control, Machine Learning (cs.LG)
Abstract

Understanding the geometric properties of gradient descent dynamics is a key ingredient in deciphering the recent success of very large machine learning models. A striking observation is that trained over-parameterized models retain some properties of the optimization initialization. This "implicit bias" is believed to be responsible for some favorable properties of the trained models and could explain their good generalization properties. The purpose of this article is threefold. First, we rigorously expose the definition and basic properties of "conservation laws", which are maximal sets of independent quantities conserved during gradient flows of a given model (e.g. of a ReLU network with a given architecture) with any training data and any loss. Then we explain how to find the exact number of these quantities by performing finite-dimensional algebraic manipulations on the Lie algebra generated by the Jacobian of the model. Finally, we provide algorithms (implemented in SageMath) to: a) compute a family of polynomial laws; b) compute the number of (not necessarily polynomial) conservation laws. We provide showcase examples that we fully work out theoretically. Besides, applying the two algorithms confirms for a number of ReLU network architectures that all known laws are recovered by the algorithm, and that there are no other laws. Such computational tools pave the way to understanding desirable properties of optimization initialization in large machine learning models.

Published
2023

49. Insights on equidistributed real spectral values in second-order delay systems: perspectives in partial pole placement

Author

Schmoderer, Timothée, Boussaada, Islam, Niculescu, Silviu-Iulian, Bedouhene, Fazia, Dynamical Interconnected Systems in COmplex Environments (DISCO), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Institut Polytechnique des Sciences Avancées (IPSA), Laboratoire de Mathématiques Pures et Appliquées [Tizi-Ouzou] (LMPA), and Université Mouloud Mammeri [Tizi Ouzou] (UMMTO)

Subjects
Partial pole placement, Time-delay differential equation, Stability and stabilization, Exponential decay, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Coexisting-real-root-induced-dominancy, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Spectral method
Abstract

In this work, we show that the coexistence of the maximal number of real spectral values of generic single-delay retarded second-order differential equations guarantees the realness of the rightmost spectral value. From a control theory standpoint, this entails that a delayed proportionalderivative (PD) controller can stabilize a delayed second-order differential equation. By assigning the maximum number of negative roots to the corresponding characteristic function (a quasipolynomial), we establish the conditions for asymptotic stability. If the assigned real spectral values are uniformly distributed, we specify a necessary and sufficient condition for the rightmost root to be negative, thus guaranteeing the exponential decay rate of the system's solutions. We illustrate the proposed design methodology in the delayed PD control of the damped harmonic oscillator. It is worth mentioning that this work represents a natural continuation of [Amrane et al., 2018] and [Bedouhene et al., 2020], addressing the problem of coexisting real spectral values for linear dynamical systems including delays in their models.

Published
2023

50. Sum-of-squares relaxations for polynomial min-max problems over simple sets

Author

Bach, Francis, Statistical Machine Learning and Parsimony (SIERRA), Département d'informatique - ENS Paris (DI-ENS), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria), ANR-19-P3IA-0001,PRAIRIE,PaRis Artificial Intelligence Research InstitutE(2019), and European Project: 724063,ERC-2016-COG,SEQUOIA(2017)

Subjects
Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control
Abstract

We consider min-max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables. When the variables belong to simple sets (e.g., a hypercube, the Euclidean hypersphere, or a ball), we derive a sum-of-squares formulation based on a primal-dual approach. In the simplest setting, we provide a convergence proof when the degree of the relaxation tends to infinity and observe empirically that it can be finitely convergent in several situations. Moreover, our formulation leads to an interesting link with feasibility certificates for polynomial inequalities based on Putinar's Positivstellensatz.

Published
2023

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