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A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalisation quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: energy is minimised over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all what variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin-one-half Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way., Comment: 25 pages, 14 figures, 15 tables

This work focuses on the generic identifiability of dynamical networks with partial excitation and measurement: a set of nodes are interconnected by transfer functions according to a known topology, some nodes are excited, some are measured, and only a part of the transfer functions are known. Our goal is to determine whether the unknown transfer functions can be generically recovered based on the input-output data collected from the excited and measured nodes. We introduce the notion of separable networks, for which global and so-called local identifiability are equivalent. A novel approach yields a necessary and sufficient combinatorial characterization for local identifiability for such graphs, in terms of existence of paths and conditions on their parity. Furthermore, this yields a necessary condition not only for separable networks, but for networks of any topology., Comment: 8 pages, 1 figure, article to appear in IEEE Conference on Decision and Control 2023

Numerous interesting properties in nonlinear systems analysis can be written as polynomial optimization problems with nonconvex sum-of-squares problems. To solve those problems efficiently, we propose a sequential approach of local linearizations leading to tractable, convex sum-of-squares problems. Local convergence is proven under the assumption of strong regularity and the new approach is applied to estimate the region of attraction of a polynomial aircraft model., Comment: Submitted to 2023 American Control Conference

We introduce DiffOpt.jl, a Julia library to differentiate through the solution of optimization problems with respect to arbitrary parameters present in the objective and/or constraints. The library builds upon MathOptInterface, thus leveraging the rich ecosystem of solvers and composing well with modeling languages like JuMP. DiffOpt offers both forward and reverse differentiation modes, enabling multiple use cases from hyperparameter optimization to backpropagation and sensitivity analysis, bridging constrained optimization with end-to-end differentiable programming. DiffOpt is built on two known rules for differentiating quadratic programming and conic programming standard forms. However, thanks ability to differentiate through model transformation, the user is not limited to these forms and can differentiate with respect to the parameters of any model that can be reformulated into these standard forms. This notably includes programs mixing affine conic constraints and convex quadratic constraints or objective function.

We study the problem of decomposing a polynomial $p$ into a sum of $r$ squares by minimizing a quadratically penalized objective $f_p(\mathbf{u}) = \left\lVert \sum_{i=1}^r u_i^2 - p\right\lVert^2$. This objective is nonconvex and is equivalent to the rank-$r$ Burer-Monteiro factorization of a semidefinite program (SDP) encoding the sum of squares decomposition. We show that for all univariate polynomials $p$, if $r \ge 2$ then $f_p(\mathbf{u})$ has no spurious second-order critical points, showing that all local optima are also global optima. This is in contrast to previous work showing that for general SDPs, in addition to genericity conditions, $r$ has to be roughly the square root of the number of constraints (the degree of $p$) for there to be no spurious second-order critical points. Our proof uses tools from computational algebraic geometry and can be interpreted as constructing a certificate using the first- and second-order necessary conditions. We also show that by choosing a norm based on sampling equally-spaced points on the circle, the gradient $\nabla f_p$ can be computed in nearly linear time using fast Fourier transforms. Experimentally we demonstrate that this method has very fast convergence using first-order optimization algorithms such as L-BFGS, with near-linear scaling to million-degree polynomials., Comment: 18 pages, to appear in SIAM Journal on Optimization

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function., Comment: 20 pages, 2 figures, submitted to Nonlinear Analysis: Hybrid Systems (NAHS). arXiv admin note: substantial text overlap with arXiv:2101.06990

This work focuses on the generic identifiability of dynamical networks with partial excitation and measurement: a set of nodes are interconnected by transfer functions according to a known topology, some nodes are excited, some are measured, and only a part of the transfer functions are known. Our goal is to determine whether the unknown transfer functions can be generically recovered based on the input-output data collected from the excited and measured nodes. We propose a decoupled version of generic identifiability that is necessary for generic local identifiability and might be equivalent as no counter-example to sufficiency has been found yet in systematic trials. This new notion can be interpreted as the generic identifiability of a larger network, obtained by duplicating the graph, exciting one copy and measuring the other copy. We establish a necessary condition for decoupled identifiability in terms of vertex-disjoint paths in the larger graph, and a sufficient one., Comment: 8 pages, 5 figures, full version with proofs of article to appear in IEEE Conference on Decision and Control 2021. arXiv admin note: text overlap with arXiv:2010.04538

In this paper, we present a geometric approach for computing the controlled invariant set of a continuous-time control system. While the problem is well studied for in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function. We compare it with the common algebraic approach for polynomial sublevel sets and show that it is significantly more conservative than our method., Comment: 7 pages, 2 figures, submitted to the 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS 2021)

In this paper, we design a theoretical framework allowing to apply model predictive control on hybrid systems. For this, we develop a theory of approximate dynamic programming by leveraging the concept of alternating simulation. We show how to combine these notions in a branch and bound algorithm that can further refine the Q-functions using Lagrangian duality. We illustrate the approach on a numerical example., Comment: 13 pages, 1 figure, submitted to the 3rd Annual Learning for Dynamics & Control Conference (L4DC 2021)

This work focuses on the identifiability of dynamical networks with partial excitation and measurement: a set of nodes are interconnected by unknown transfer functions according to a known topology, some nodes are subject to external excitation, and some nodes are measured. The goal is to determine which transfer functions in the network can be recovered based on the input-output data collected from the excited and measured nodes. We propose a local version of network identifiability, representing the ability to recover transfer functions which are approximately known, or to recover them up to a discrete ambiguity. We show that local identifiability is a generic property, establish a necessary and sufficient condition in terms of matrix generic ranks, and exploit this condition to develop an algorithm determining, with probability 1, which transfer functions are locally identifiable. Our implementation presents the results graphically, and is publicly available., Comment: 6 pages, 2 pdf figures, to appear in IEEE Conference on Decision and Control 2020

Computing control invariant sets is paramount in many applications. The families of sets commonly used for computations are ellipsoids and polyhedra. However, searching for a control invariant set over the family of ellipsoids is conservative for systems more complex than unconstrained linear time invariant systems. Moreover, even if the control invariant set may be approximated arbitrarily closely by polyhedra, the complexity of the polyhedra may grow rapidly in certain directions. An attractive generalization of these two families are piecewise semi-ellipsoids. We provide in this paper a convex programming approach for computing control invariant sets of this family., Comment: 7 pages, 3 figures, to be published in IEEE Control Systems Letters

We introduce MathOptInterface, an abstract data structure for representing mathematical optimization problems based on combining pre-defined functions and sets. MathOptInterface is significantly more general than existing data structures in the literature, encompassing, for example, a spectrum of problems classes from integer programming with indicator constraints to bilinear semidefinite programming. We also outline an automated rewriting system between equivalent formulations of a constraint. MathOptInterface has been implemented in practice, forming the foundation of a recent rewrite of JuMP, an open-source algebraic modeling language in the Julia language. The regularity of the MathOptInterface representation leads naturally to a general file format for mathematical optimization we call MathOptFormat. In addition, the automated rewriting system provides modeling power to users while making it easy to connect new solvers to JuMP.

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We analyse the accuracy of this method for constrained switched systems, a class of systems that has attracted increasing attention recently. We provide a new guarantee for the upper bound provided by the sum of squares implementation of the method. This guarantee relies on the p-radius of the system and the entropy of the language of allowed switching sequences. We end this paper with a method to reduce the computation of the JSR of low rank matrices to the computation of the constrained JSR of matrices of small dimension., Comment: 8 pages, 4 figures, will appear in IEEE Transactions on Automatic Control. arXiv admin note: substantial text overlap with arXiv:1710.01814

We propose an algorithm to restrict the switching signals of a constrained switched system in order to guarantee its stability, while at the same time attempting to keep the largest possible set of allowed switching signals. Our work is motivated by applications to (co-)simulation, where numerical stability is a hard constraint, but should be attained by restricting as little as possible the allowed behaviours of the simulators. We apply our results to certify the stability of an adaptive co-simulation orchestration algorithm, which selects the optimal switching signal at run-time, as a function of (varying) performance and accuracy requirements., Comment: Technical report complementing the following conference publication: Gomes, Cl\'audio, Beno\^it Legat, Rapha\"el Jungers, and Hans Vangheluwe. "Minimally Constrained Stable Switched Systems and Application to Co-Simulation." In IEEE Conference on Decision and Control. Miami Beach, FL, USA, 2018

In this paper, we develop a method for computing controlled invariant sets using Semidefinite Programming. We apply our method to the controller design problem for switching affine systems with polytopic safe sets. The task is reduced to a semidefinite programming problem by enforcing an invariance relation in the dual space of the geometric problem. The paper ends with an application to safety critical model predictive control., Comment: 7 pages, 7 figures, submitted to ADHS 2018 (under review)

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We investigate dual formulations for this approach and leverage these dual programs for developing new analysis tools for the JSR. We show that the dual of this convex problem searches for the occupations measures of trajectories with high asymptotic growth rate. We both show how to generate a sequence of guaranteed high asymptotic growth rate and how to detect cases where we can provide lower bounds to the JSR. We deduce from it a new guarantee for the upper bound provided by the sum of squares lyapunov program. We end this paper with a method to reduce the computation of the JSR of low rank matrices to the computation of the constrained JSR of matrices of small dimension. All results of this paper are presented for the general case of constrained switched systems, that is, systems for which the switching signal is constrained by an automaton., Comment: 25 pages (31 including abstract), 6 figures, submitted to SICON (under review)