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In this paper, we study a dynamic pickup and delivery problem with docking constraints. There is a homogeneous fleet of vehicles to serve pickup-and-delivery requests at given locations. The vehicles can be loaded up to their capacity, while unloading has to follow the last-in-first-out (LIFO) rule. The locations have a limited number of docking ports for loading and unloading, which may force the vehicles to wait. The problem is dynamic since the transportation requests arrive real-time, over the day. Accordingly, the routes of the vehicles are to be determined dynamically. The goal is to satisfy all the requests such that a combination of tardiness penalties and traveling costs is minimized. We propose a cost function approximation based solution method. In each decision epoch, we solve the respective optimization problem with a perturbed objective function to ensure the solutions remain adaptable to accommodate new requests. We penalize waiting times and idle vehicles. We propose a variable neighborhood search based method for solving the optimization problems, and we apply two existing local search operators, and we also introduce a new one. We evaluate our method using a widely adopted benchmark dataset, and the results demonstrate that our approach significantly surpasses the current state-of-the-art methods.

Traditional mathematical programming solvers require long computational times to solve constrained minimization problems of complex and large-scale physical systems. Therefore, these problems are often transformed into unconstrained ones, and solved with computationally efficient optimization approaches based on first-order information, such as the gradient descent method. However, for unconstrained problems, balancing the minimization of the objective function with the reduction of constraint violations is challenging. We consider the class of time-dependent minimization problems with increasing (possibly) nonlinear and non-convex objective function and non-decreasing (possibly) nonlinear and non-convex inequality constraints. To efficiently solve them, we propose a penalty-based guardrail algorithm (PGA). This algorithm adapts a standard penalty-based method by dynamically updating the right-hand side of the constraints with a guardrail variable which adds a margin to prevent violations. We evaluate PGA on two novel application domains: a simplified model of a district heating system and an optimization model derived from learned deep neural networks. Our method significantly outperforms mathematical programming solvers and the standard penalty-based method, and achieves better performance and faster convergence than a state-of-the-art algorithm (IPDD) within a specified time limit.

The paper aims to investigate relevant computational issues of deep neural network architectures with an eye to the interaction between the optimization algorithm and the classification performance. In particular, we aim to analyze the behaviour of state-of-the-art optimization algorithms in relationship to their hyperparameters setting in order to detect robustness with respect to the choice of a certain starting point in ending on different local solutions. We conduct extensive computational experiments using nine open-source optimization algorithms to train deep Convolutional Neural Network architectures on an image multi-class classification task. Precisely, we consider several architectures by changing the number of layers and neurons per layer, in order to evaluate the impact of different width and depth structures on the computational optimization performance.

We consider geodesically convex optimization problems involving distances to a finite set of points $A$ in a CAT(0) cubical complex. Examples include the minimum enclosing ball problem, the weighted mean and median problems, and the feasibility and projection problems for intersecting balls with centers in $A$. We propose a decomposition approach relying on standard Euclidean cutting plane algorithms. The cutting planes are readily derivable from efficient algorithms for computing geodesics in the complex.

This paper proposes a data-driven approach to design a feedforward Neural Network (NN) controller with a stability guarantee for systems with unknown dynamics. We first introduce data-driven representations of stability conditions for Neural Feedback Loops (NFLs) with linear plants. These conditions are then formulated into a semidefinite program (SDP). Subsequently, this SDP constraint is integrated into the NN training process resulting in a stable NN controller. We propose an iterative algorithm to solve this problem efficiently. Finally, we illustrate the effectiveness of the proposed method and its superiority compared to model-based methods via numerical examples.

This study experimentally optimizes the mixing of a turbulent jet at $Re=10000$ with the surrounding air by targeted shear layer actuation. The forcing is composed of superposed harmonic signals of different azimuthal wavenumber $m$ generated by eight loudspeakers circumferentially distributed around the nozzle lip. Amplitudes and frequencies of the individual harmonic contributions serve as optimization parameters and the time-averaged centerline velocity downstream of the potential core is used as a metric for mixing optimization. The actuation is optimized through Bayesian optimization. Three search spaces are explored - axisymmetric forcing, $m=0$, superposed axisymmetric and helical forcing, $m \in \{0,1\}$, and axisymmetric actuation combined with two counter-rotating helical modes, $m \in \{-1,0,1\}$. High-speed PIV is employed to analyze the jet response to the optimized forcing. The optimization processes are analyzed by persistent data topology. In the search space of axisymmetric excitation, the routine identifies an actuation at the natural frequency of the flow to be most efficient, with the centerline velocity being decreased by $15\%$. The optimal solutions in both the two-mode and three-mode search space converge to a similar forcing with one axial and one helical mode combined at a frequency ratio of around $2.3$. Spectral analysis of the PIV images reveals that for the identified optimal forcing frequencies, a non-linear interaction between forced and natural structures in the jet flow is triggered, leading to a reduction in centerline velocity of around $35\%$. The topology of the most complex search space from the discrete data reveals four basins of attractions, classified into three forcing patterns including axisymmetric, axisym.-helical, and axisym.-flapping. Two deep basins are related to the optimal axisym.-helical pattern, and the others are shallower.

In this paper, the moment matching techniques are adopted to obtain reduced-order closed-loop systems with reduced-order controllers that maintain the closed-loop stability and guarantee desired asymptotic performance, after revealing the relationship between the Internal Model Principle used in control design and the time-domain moment matching problem. As a result, the design of a low order controller can be done starting from considering the achieving of asymptotic performance as a moment matching problem, resulting in a reduced order closed-loop system., Comment: 7 pages. Preliminary resukts have been presented in CDC2013

Real-time environmental monitoring using a multi-agent system (MAS) has long been a focal point of cooperative control. It is still a challenging task to provide cost-effective services for potential emergencies in surface environments. This paper explores the transformation of a general surface into a two-dimensional (2D) disk through the construction of a conformal mapping. Multiple agents are strategically deployed within the mapped convex disk, followed by mapping back to the original surface environment. This approach circumvents the complexities associated with handling the difficulties and intricacies of path planning. Technical analysis encompasses the design of distributed control laws and the method to eliminate distortions introduced by the mapping. Moreover, the developed coverage algorithm is applied to a scenario of monitoring surface deformation. Finally, the effectiveness of the proposed algorithm is validated through numerical simulations.

This paper presents optimization models for electric Mobility-as-a-Service systems, whereby electric vehicles not only provide on-demand mobility, but also perform charging and Vehicle-to-Grid (V2G) operations to enhance the fleet operator profitability. Specifically, we formulate the optimal fleet operation problem as a mixed-integer linear program, with the objective combining of operational costs and revenues generated from servicing requests and grid electricity sales. Our cost function explicitly captures battery price and degradation, reflecting their impact on the fleet total cost of ownership due to additional charging and discharging activities. Simulation results for Eindhoven, The Netherlands, show that integrating V2G activities does not compromise the number of travel requests being served. Moreover, we emphasize the significance of accounting for battery degradation, as the costs associated with it can potentially outweigh the revenues stemming from V2G operations.

This paper is concerned with collective variables, or reaction coordinates, that map a discrete-in-time Markov process $X_n$ in $\mathbb{R}^d$ to a (much) smaller dimension $k \ll d$. We define the effective dynamics under a given collective variable map $\xi$ as the best Markovian representation of $X_n$ under $\xi$. The novelty of the paper is that it gives strict criteria for selecting optimal collective variables via the properties of the effective dynamics. In particular, we show that the transition density of the effective dynamics of the optimal collective variable solves a relative entropy minimization problem from certain family of densities to the transition density of $X_n$. We also show that many transfer operator-based data-driven numerical approaches essentially learn quantities of the effective dynamics. Furthermore, we obtain various error estimates for the effective dynamics in approximating dominant timescales / eigenvalues and transition rates of the original process $X_n$ and how optimal collective variables minimize these errors. Our results contribute to the development of theoretical tools for the understanding of complex dynamical systems, e.g. molecular kinetics, on large timescales. These results shed light on the relations among existing data-driven numerical approaches for identifying good collective variables, and they also motivate the development of new methods.

We consider the problem of parameter estimation in dynamic systems described by ordinary differential equations. A review of the existing literature emphasizes the need for deterministic global optimization methods due to the nonconvex nature of these problems. Recent works have focused on expanding the capabilities of specialized deterministic global optimization algorithms to handle more complex problems. Despite advancements, current deterministic methods are limited to problems with a maximum of around five state and five decision variables, prompting ongoing efforts to enhance their applicability to practical problems. Our study seeks to assess the effectiveness of state-of-the-art general-purpose global and local solvers in handling realistic-sized problems efficiently, and evaluating their capabilities to cope with the nonconvex nature of the underlying estimation problems.

We propose a new Q-learning variant, called 2RA Q-learning, that addresses some weaknesses of existing Q-learning methods in a principled manner. One such weakness is an underlying estimation bias which cannot be controlled and often results in poor performance. We propose a distributionally robust estimator for the maximum expected value term, which allows us to precisely control the level of estimation bias introduced. The distributionally robust estimator admits a closed-form solution such that the proposed algorithm has a computational cost per iteration comparable to Watkins' Q-learning. For the tabular case, we show that 2RA Q-learning converges to the optimal policy and analyze its asymptotic mean-squared error. Lastly, we conduct numerical experiments for various settings, which corroborate our theoretical findings and indicate that 2RA Q-learning often performs better than existing methods., Comment: 26 pages, 5 figures

Robust Reinforcement Learning (RRL) is a promising Reinforcement Learning (RL) paradigm aimed at training robust to uncertainty or disturbances models, making them more efficient for real-world applications. Following this paradigm, uncertainty or disturbances are interpreted as actions of a second adversarial agent, and thus, the problem is reduced to seeking the agents' policies robust to any opponent's actions. This paper is the first to propose considering the RRL problems within the positional differential game theory, which helps us to obtain theoretically justified intuition to develop a centralized Q-learning approach. Namely, we prove that under Isaacs's condition (sufficiently general for real-world dynamical systems), the same Q-function can be utilized as an approximate solution of both minimax and maximin Bellman equations. Based on these results, we present the Isaacs Deep Q-Network algorithms and demonstrate their superiority compared to other baseline RRL and Multi-Agent RL algorithms in various environments.

This paper proposes a novel real-time affordable solution to the trajectory tracking control problem for self-driving cars subject to longitudinal and steering angular velocity constraints. To this end, we develop a dual-mode Model Predictive Control (MPC) solution starting from an input-output feedback linearized description of the vehicle kinematics. First, we derive the state-dependent input constraints acting on the linearized model and characterize their worst-case time-invariant inner approximation. Then, a dual-mode MPC is derived to be real-time affordable and ensuring, by design, constraints fulfillment, recursive feasibility, and uniformly ultimate boundedness of the tracking error in an ad-hoc built robust control invariant region. The approach's effectiveness and performance are experimentally validated via laboratory experiments on a Quanser Qcar. The obtained results show that the proposed solution is computationally affordable and with tracking capabilities that outperform two alternative control schemes., Comment: Preprint of a manuscript currently under review for TCTS

In this article, we study a simplified version of a density-dependent first-order mean field game, in which the players face a penalization equal to the population density at their final position. We consider the problem of finding an equilibrium when the initial distribution is a discrete measure. We show that the problem becomes finite-dimensional: the final piecewise smooth density is completely determined by the weights and positions of the initial measure. We establish existence and uniqueness of a solution using classical fixed point theorems. Finally, we show that Newton's method provides an effective way to compute the solution. Our numerical simulations provide an illustration of how density penalization in a mean field game tends to the smoothen the initial distribution.

In this paper, a backward map is introduced for the purposes of analysis of the nonlinear (stochastic) filter stability. The backward map is important because the filter-stability in the sense of $\chisq$-divergence follows from showing a certain variance decay property for the backward map. To show this property requires additional assumptions on the model properties of the hidden Markov model (HMM). The analysis in this paper is based on introducing a Poincar\'e Inequality (PI) for HMMs with white noise observations. In finite state-space settings, PI is related to both the ergodicity of the Markov process as well as the observability of the HMM. It is shown that the Poincar\'e constant is positive if and only if the HMM is detectable.

The lack of cycling infrastructure in urban environments hinders the adoption of cycling as a viable mode for commuting, despite the evident benefits of (e-)bikes as sustainable, efficient, and health-promoting transportation modes. Bike network planning is a tedious process, relying on heuristic computational methods that frequently overlook the broader implications of introducing new cycling infrastructure, in particular the necessity to repurpose car lanes. In this work, we call for optimizing the trade-off between bike and car networks, effectively pushing for Pareto optimality. This shift in perspective gives rise to a novel linear programming formulation towards optimal bike network allocation. Our experiments, conducted using both real-world and synthetic data, testify the effectiveness and superiority of this optimization approach compared to heuristic methods. In particular, the framework provides stakeholders with a range of lane reallocation scenarios, illustrating potential bike network enhancements and their implications for car infrastructure. Crucially, our approach is adaptable to various bikeability and car accessibility evaluation criteria, making our tool a highly flexible and scalable resource for urban planning. This paper presents an advanced decision-support framework that can significantly aid urban planners in making informed decisions on cycling infrastructure development.

Large Language Models (LLMs) have achieved remarkable success across diverse tasks, yet they remain vulnerable to adversarial attacks, notably the well-documented \textit{jailbreak} attack. Recently, the Greedy Coordinate Gradient (GCG) attack has demonstrated efficacy in exploiting this vulnerability by optimizing adversarial prompts through a combination of gradient heuristics and greedy search. However, the efficiency of this attack has become a bottleneck in the attacking process. To mitigate this limitation, in this paper we rethink the generation of adversarial prompts through an optimization lens, aiming to stabilize the optimization process and harness more heuristic insights from previous iterations. Specifically, we introduce the \textbf{M}omentum \textbf{A}ccelerated G\textbf{C}G (\textbf{MAC}) attack, which incorporates a momentum term into the gradient heuristic. Experimental results showcase the notable enhancement achieved by MAP in gradient-based attacks on aligned language models. Our code is available at https://github.com/weizeming/momentum-attack-llm., Comment: ICLR 2024 Workshop on Reliable and Responsible Foundation Models

Recently, there has been an increasing interest in exploring the application of multiobjective optimization (MOO) in machine learning (ML). The interest is driven by the numerous situations in real-life applications where multiple objectives need to be optimized simultaneously. A key aspect of MOO is the existence of a Pareto set, rather than a single optimal solution, which illustrates the inherent trade-offs between objectives. Despite its potential, there is a noticeable lack of satisfactory literature that could serve as an entry-level guide for ML practitioners who want to use MOO. Hence, our goal in this paper is to produce such a resource. We critically review previous studies, particularly those involving MOO in deep learning (using Physics-Informed Neural Networks (PINNs) as a guiding example), and identify misconceptions that highlight the need for a better grasp of MOO principles in ML. Using MOO of PINNs as a case study, we demonstrate the interplay between the data loss and the physics loss terms. We highlight the most common pitfalls one should avoid while using MOO techniques in ML. We begin by establishing the groundwork for MOO, focusing on well-known approaches such as the weighted sum (WS) method, alongside more complex techniques like the multiobjective gradient descent algorithm (MGDA). Additionally, we compare the results obtained from the WS and MGDA with one of the most common evolutionary algorithms, NSGA-II. We emphasize the importance of understanding the specific problem, the objective space, and the selected MOO method, while also noting that neglecting factors such as convergence can result in inaccurate outcomes and, consequently, a non-optimal solution. Our goal is to offer a clear and practical guide for ML practitioners to effectively apply MOO, particularly in the context of DL., Comment: 21 pages, 12 figures

We propose a novel algorithm that extends the methods of ball smoothing and Gaussian smoothing for noisy derivative-free optimization by accounting for the heterogeneous curvature of the objective function. The algorithm dynamically adapts the shape of the smoothing kernel to approximate the Hessian of the objective function around a local optimum. This approach significantly reduces the error in estimating the gradient from noisy evaluations through sampling. We demonstrate the efficacy of our method through numerical experiments on artificial problems. Additionally, we show improved performance when tuning NP-hard combinatorial optimization solvers compared to existing state-of-the-art heuristic derivative-free and Bayesian optimization methods., Comment: Accepted to ICML2024

This paper presents a systematic literature review and bibliometric analysis focusing on Biased Random-Key Genetic Algorithms (BRKGA). BRKGA is a metaheuristic framework that uses random-key-based chromosomes with biased, uniform, and elitist mating strategies alongside a genetic algorithm. This review encompasses around~250 papers, covering a diverse array of applications ranging from classical combinatorial optimization problems to real-world industrial scenarios, and even non-traditional applications like hyperparameter tuning in machine learning and scenario generation for two-stage problems. In summary, this study offers a comprehensive examination of the BRKGA metaheuristic and its various applications, shedding light on key areas for future research., Comment: 24 pages, 9 figures

We consider generalizations of the classical inverse problem to Bayesien type estimators, where the result is not one optimal parameter but an optimal probability distribution in parameter space. The practical computational tool to compute these distributions is the Metropolis Monte Carlo algorithm. We derive kinetic theories for the Metropolis Monte Carlo method in different scaling regimes. The derived equations yield a different point of view on the classical algorithm. It further inspired modifications to exploit the difference scalings shown on an simulation example of the Lorenz system.

In this paper, we consider the design of data-driven predictive controllers for nonlinear systems from input-output data via linear-in-control input Koopman lifted models. Instead of identifying and simulating a Koopman model to predict future outputs, we design a subspace predictive controller in the Koopman space. This allows us to learn the observables minimizing the multi-step output prediction error of the Koopman subspace predictor, preventing the propagation of prediction errors. To avoid losing feasibility of our predictive control scheme due to prediction errors, we compute a terminal cost and terminal set in the Koopman space and we obtain recursive feasibility guarantees through an interpolated initial state. As a third contribution, we introduce a novel regularization cost yielding input-to-state stability guarantees with respect to the prediction error for the resulting closed-loop system. The performance of the developed Koopman data-driven predictive control methodology is illustrated on a nonlinear benchmark example from the literature.

We consider Cournot mean field games of controls, a model originally developed for the production of an exhaustible resource by a continuum of producers. We prove uniqueness of the solution under general assumptions on the price function. Then, we prove convergence of a learning algorithm which gives existence of a solution to the mean field games system. The learning algorithm is implemented with a suitable finite difference discretization to get a numerical method to the solution. We supplement our theoretical analysis with several numerical examples and illustrate the impacts of model parameters.

A lexicographic maximum of a set $X \subseteq \mathbb{R}^n$ is a vector in $X$ whose smallest component is as large as possible, and subject to that requirement, whose second smallest component is as large as possible, and so on for the third smallest component, etc. Lexicographic maximization has numerous practical and theoretical applications, including fair resource allocation, analyzing the implicit regularization of learning algorithms, and characterizing refinements of game-theoretic equilibria. We prove that a minimizer in $X$ of the exponential loss function $L_c(\mathbf{x}) = \sum_i \exp(-c x_i)$ converges to a lexicographic maximum of $X$ as $c \rightarrow \infty$, provided that $X$ is stable in the sense that a well-known iterative method for finding a lexicographic maximum of $X$ cannot be made to fail simply by reducing the required quality of each iterate by an arbitrarily tiny degree. Our result holds for both near and exact minimizers of the exponential loss, while earlier convergence results made much stronger assumptions about the set $X$ and only held for the exact minimizer. We are aware of no previous results showing a connection between the iterative method for computing a lexicographic maximum and exponential loss minimization. We show that every convex polytope is stable, but that there exist compact, convex sets that are not stable. We also provide the first analysis of the convergence rate of an exponential loss minimizer (near or exact) and discover a curious dichotomy: While the two smallest components of the vector converge to the lexicographically maximum values very quickly (at roughly the rate $\frac{\log n}{c}$), all other components can converge arbitrarily slowly.

We propose a Markov Chain Monte Carlo (MCMC) algorithm based on Gibbs sampling with parallel tempering to solve nonlinear optimal control problems. The algorithm is applicable to nonlinear systems with dynamics that can be approximately represented by a finite dimensional Koopman model, potentially with high dimension. This algorithm exploits linearity of the Koopman representation to achieve significant computational saving for large lifted states. We use a video-game to illustrate the use of the method.

Positive semidefinite (PSD) cone is the cone of positive semidefinite matrices, and is the object of interest in semidefinite programming (SDP). A computational efficient approximation of the PSD cone is the $k$-PSD closure, $1 \leq k < n$, cone of $n\times n$ real symmetric matrices such that all of their $k\times k$ principal submatrices are positive semidefinite. For $k=1$, one obtains a polyhedral approximation, while $k=2$ yields a second order conic (SOC) approximation of the PSD cone. These approximations of the PSD cone have been used extensively in real-world applications such as AC Optimal Power Flow (ACOPF) to address computational inefficiencies where SDP relaxations are utilized for convexification the non-convexities. However a theoretical discussion about the geometry of these conic approximations of the PSD cone is rather sparse. In this short communication, we attempt to provide a characterization of some family of generators of the aforementioned conic approximations.

We consider the problem of efficiently solving a system of $n$ non-linear equations in ${\mathbb R}^d$. Addressing Smale's 17th problem stated in 1998, we consider a setting whereby the $n$ equations are random homogeneous polynomials of arbitrary degrees. In the complex case and for $n= d-1$, Beltr\'{a}n and Pardo proved the existence of an efficient randomized algorithm and Lairez recently showed it can be de-randomized to produce a deterministic efficient algorithm. Here we consider the real setting, to which previously developed methods do not apply. We describe an algorithm that efficiently finds solutions (with high probability) for $n= d -O(\sqrt{d\log d})$. If the maximal degree is very large, we also give an algorithm that works up to $n=d-1$., Comment: 44 pages

The present paper deals with the perturbation analysis of set-valued inclusion problems, a problem format whose relevance has recently emerged in such contexts as robust and vector optimization as well as in vector equilibrium theory. The set-valued inclusions here considered are parameterized by variables belonging to a topological space, with and without constraints. By proper techniques of variational analysis, some qualitative global implicit function theorems are established, which ensure global solvability of these problems and continuous dependence on the parameter of the related solutions. Applications to parametric vector optimization are discussed, aimed at deriving sufficient conditions for the existence of ideal efficient solutions that depend continuously on the parameter perturbations.

This paper studies a class of network games with linear-quadratic payoffs and externalities exerted through a strictly concave interaction function. This class of game is motivated by the diminishing marginal effects with peer influences. We analyze the optimal pricing strategy for this class of network game. First, we prove the existence of a unique Nash Equilibrium (NE). Second, we study the optimal pricing strategy of a monopolist selling a divisible good to agents. We show that the optimal pricing strategy, found by solving a bilevel optimization problem, is strictly better when the monopolist knows the network structure as opposed to the best strategy agnostic to network structure. Numerical experiments demonstrate that in most cases, the maximum revenue is achieved with an asymmetric network. These results contrast with the previously studied case of linear interaction function, where a network-independent price is proven optimal with symmetric networks. Lastly, we describe an efficient algorithm to find the optimal pricing strategy., Comment: 7 pages, 2 figures, revisions under IEEE Control Systems Letters

This paper models gas networks as metric graphs, with isothermal Euler equations at the edges, Kirchhoff's law at interior vertices and time-(in)dependent boundary conditions at boundary vertices. For this setup, a generalized $p$-Wasserstein metric in a dynamic formulation is introduced and utilized to derive $p$-Wasserstein gradient flows, specifically focusing on the non-standard case $p = 3$., Comment: 43 pages, 5 figures, submitted to EJAM (Evolution Equations on Graphs: Analysis and Applications)

Optimization over the set of matrices that satisfy $X^\top B X = I_p$, referred to as the generalized Stiefel manifold, appears in many applications involving sampled covariance matrices such as canonical correlation analysis (CCA), independent component analysis (ICA), and the generalized eigenvalue problem (GEVP). Solving these problems is typically done by iterative methods, such as Riemannian approaches, which require a computationally expensive eigenvalue decomposition involving fully formed $B$. We propose a cheap stochastic iterative method that solves the optimization problem while having access only to a random estimate of the feasible set. Our method does not enforce the constraint in every iteration exactly, but instead it produces iterations that converge to a critical point on the generalized Stiefel manifold defined in expectation. The method has lower per-iteration cost, requires only matrix multiplications, and has the same convergence rates as its Riemannian counterparts involving the full matrix $B$. Experiments demonstrate its effectiveness in various machine learning applications involving generalized orthogonality constraints, including CCA, ICA, and GEVP., Comment: 21 pages, 10 figures

We extend the port-Hamiltonian framework defined with respect to a Lagrangian submanifold and a Dirac structure by augmenting the Lagrangian submanifold with the space of external variables. The new pair of conjugated variables is called energy port. We show that in the most general case, the extension describes constrained Hamiltonian systems whose Hamiltonian function depends on inputs., Comment: 6 pages

The Pareto front of a set of vectors is the subset which is comprised solely of all of the best trade-off points. By interpolating this subset, we obtain the optimal trade-off surface. In this work, we prove a very useful result which states that all Pareto front surfaces can be explicitly parametrised using polar coordinates. In particular, our polar parametrisation result tells us that we can fully characterise any Pareto front surface using the length function, which is a scalar-valued function that returns the projected length along any positive radial direction. Consequently, by exploiting this representation, we show how it is possible to generalise many useful concepts from linear algebra, probability and statistics, and decision theory to function over the space of Pareto front surfaces. Notably, we focus our attention on the stochastic setting where the Pareto front surface itself is a stochastic process. Among other things, we showcase how it is possible to define and estimate many statistical quantities of interest such as the expectation, covariance and quantile of any Pareto front surface distribution. As a motivating example, we investigate how these statistics can be used within a design of experiments setting, where the goal is to both infer and use the Pareto front surface distribution in order to make effective decisions. Besides this, we also illustrate how these Pareto front ideas can be used within the context of extreme value theory. Finally, as a numerical example, we applied some of our new methodology on a real-world air pollution data set., Comment: The code is available at: https://github.com/benmltu/scalarize

Robust Markov Decision Processes (RMDPs) have received significant research interest, offering an alternative to standard Markov Decision Processes (MDPs) that often assume fixed transition probabilities. RMDPs address this by optimizing for the worst-case scenarios within ambiguity sets. While earlier studies on RMDPs have largely centered on risk-neutral reinforcement learning (RL), with the goal of minimizing expected total discounted costs, in this paper, we analyze the robustness of CVaR-based risk-sensitive RL under RMDP. Firstly, we consider predetermined ambiguity sets. Based on the coherency of CVaR, we establish a connection between robustness and risk sensitivity, thus, techniques in risk-sensitive RL can be adopted to solve the proposed problem. Furthermore, motivated by the existence of decision-dependent uncertainty in real-world problems, we study problems with state-action-dependent ambiguity sets. To solve this, we define a new risk measure named NCVaR and build the equivalence of NCVaR optimization and robust CVaR optimization. We further propose value iteration algorithms and validate our approach in simulation experiments.

High-speed railway stations are crucial junctions in high-speed railway networks. Compared to operations on the tracks between stations, trains have more routing possibilities within stations. As a result, track allocation at a station is relatively complicated. In this study, we aim to solve the train platforming problem for a busy high-speed railway station by considering comprehensive track resources and interlocking configurations. A two-level space-time network is constructed to capture infrastructure information at various levels of detail from both macroscopic and microscopic perspectives. Additionally, we propose a nonlinear programming model that minimizes a weighted sum of total travel time and total deviation time for trains at the station. We apply a Two-level Lagrangian Relaxation (2-L LR) to a linearized version of the model and demonstrate how this induces a decomposable train-specific path choice problem at the macroscopic level that is guided by Lagrange multipliers associated with microscopic resource capacity violation. As case studies, the proposed model and solution approach are applied to a small virtual railway station and a high-speed railway hub station located on the busiest high-speed railway line in China. Through a comparison of other approaches that include Logic-based Benders Decomposition (LBBD), we highlight the superiority of the proposed method; on realistic instances, the 2-L LR method finds solution that are, on average, approximately 2% from optimality. Finally, we test algorithm performance at the operational level and obtain near-optimal solutions, with optimality gaps of approximately 1%, in a very short time.

In autonomous mobility-on-demand systems, effectively managing vehicle flows to mitigate induced congestion and ensure efficient operations is imperative for system performance and positive customer experience. Against this background, we study the potential of staggered routing, i.e., purposely delaying trip departures from a system perspective, in order to reduce congestion and ensure efficient operations while still meeting customer time windows. We formalize the underlying planning problem and show how to efficiently model it as a mixed integer linear program. Moreover, we present a matheuristic that allows us to efficiently solve large-scale real-world instances both in an offline full-information setting and its online rolling horizon counterpart. We conduct a numerical study for Manhattan, New York City, focusing on low- and highly-congested scenarios. Our results show that in low-congestion scenarios, staggering trip departures allows mitigating, on average, 94% of the induced congestion in a full information setting. In a rolling horizon setting, our algorithm allows us to reduce 90% of the induced congestion. In high-congestion scenarios, we observe an average reduction of 66% as the full information bound and an average reduction of 56% in our online setting. Surprisingly, we show that these reductions can be reached by shifting trip departures by a maximum of six minutes in both the low and high-congestion scenarios., Comment: 32 pages, 10 figures

Decentralized learning is appealing as it enables the scalable usage of large amounts of distributed data and resources (without resorting to any central entity), while promoting privacy since every user minimizes the direct exposure of their data. Yet, without additional precautions, curious users can still leverage models obtained from their peers to violate privacy. In this paper, we propose Decor, a variant of decentralized SGD with differential privacy (DP) guarantees. Essentially, in Decor, users securely exchange randomness seeds in one communication round to generate pairwise-canceling correlated Gaussian noises, which are injected to protect local models at every communication round. We theoretically and empirically show that, for arbitrary connected graphs, Decor matches the central DP optimal privacy-utility trade-off. We do so under SecLDP, our new relaxation of local DP, which protects all user communications against an external eavesdropper and curious users, assuming that every pair of connected users shares a secret, i.e., an information hidden to all others. The main theoretical challenge is to control the accumulation of non-canceling correlated noise due to network sparsity. We also propose a companion SecLDP privacy accountant for public use., Comment: Accepted as conference paper at ICML 2024

We present a fast trajectory optimization algorithm for the soft capture of uncooperative tumbling space objects. Our algorithm generates safe, dynamically feasible, and minimum-fuel trajectories for a six-degree-of-freedom servicing spacecraft to achieve soft capture (near-zero relative velocity at contact) between predefined locations on the servicer spacecraft and target body. We solve a convex problem by enforcing a convex relaxation of the field-of-view constraint, followed by a sequential convex program correcting the trajectory for collision avoidance. The optimization problems can be solved with a standard second-order cone programming solver, making the algorithm both fast and practical for implementation in flight software. We demonstrate the performance and robustness of our algorithm in simulation over a range of object tumble rates up to 10{\deg}/s., Comment: Accepted to ISpaRo24

This paper presents a convex optimization framework for eco-driving and vehicle energy management problems. We will first show that several types of eco-driving and vehicle energy management problems can be modelled using the same notions of energy storage buffers and energy storage converters that are connected to a power network. It will be shown that these problems can be formulated as optimization problems with linear cost functions and linear dynamics, and nonlinear constraints representing the power converters. We will show that under some mild conditions, the (non-convex) optimization problem has the same (globally) optimal solution as a convex relaxation. This means that the problems can be solved efficiently and that the solution is guaranteed to be globally optimal. Finally, a numerical example of the eco-driving problem is used to illustrate this claim.

The paper is concerned with a zero-sum differential game in the case where a payoff is determined by the exit time, that is, the first time when the system leaves the game domain. Additionally, we assume that a part of domain's boundary is a lifeline where the payoff is infinite. Hereby, the examined problem generalizes the well-known time-optimal problem as well as time-optimal problem with lifeline. The main result of the paper relies on the solution to the Direchlet problem for the Hamilton-Jacobi equation associated with the game with exit time. We prove the existence of the value function for examined problem and construct suboptimal feedback strategies under assumption that the associated Dirichlet problem for the Hamilton-Jacobi equation admits a viscosity/minimax solution. Additionally, we derive a sufficient condition of existence result to this Dirichlet problem., Comment: 22 pages

This paper presents a new algorithm member for accelerating first-order methods for bilevel optimization, namely the \emph{(Perturbed) Restarted Accelerated Fully First-order methods for Bilevel Approximation}, abbreviated as \texttt{(P)RAF${}^2$BA}. The algorithm leverages \emph{fully} first-order oracles and seeks approximate stationary points in nonconvex-strongly-convex bilevel optimization, enhancing oracle complexity for efficient optimization. Theoretical guarantees for finding approximate first-order stationary points and second-order stationary points at the state-of-the-art query complexities are established, showcasing their effectiveness in solving complex optimization tasks. Empirical studies for real-world problems are provided to further validate the outperformance of our proposed algorithms. The significance of \texttt{(P)RAF${}^2$BA} in optimizing nonconvex-strongly-convex bilevel optimization problems is underscored by its state-of-the-art convergence rates and computational efficiency., Comment: arXiv admin note: text overlap with arXiv:2307.00126

This paper presents a particle-based optimization method designed for addressing minimization problems with equality constraints, particularly in cases where the loss function exhibits non-differentiability or non-convexity. The proposed method combines components from consensus-based optimization algorithm with a newly introduced forcing term directed at the constraint set. A rigorous mean-field limit of the particle system is derived, and the convergence of the mean-field limit to the constrained minimizer is established. Additionally, we introduce a stable discretized algorithm and conduct various numerical experiments to demonstrate the performance of the proposed method.

We study how high charging rate demands from electric vehicles (EVs) in a power distribution grid may collectively cause its dynamic instability, and, accordingly, how a price incentivization strategy can be used to steer customers to settle for lesser charging rate demands so that these instabilities can be avoided. We pose the problem as a joint optimization and optimal control formulation. The optimization determines the optimal charging setpoints for EVs to minimize the $\mathcal{H}_2$-norm of the transfer function of the grid model, while the optimal control simultaneously develops a linear quadratic regulator (LQR) based state-feedback control signal for the battery-currents of those EVs to jointly minimize the risk of grid instability. A subsequent algorithm is developed to determine how much customers may be willing to sacrifice their intended charging rate demands in return for financial incentives. Results are derived for both unidirectional and bidirectional charging, and validated using numerical simulations of multiple EV charging stations in the IEEE 33-bus power distribution model.

It is now possible to conduct large scale perturbation screens with complex readout modalities, such as different molecular profiles or high content cell images. While these open the way for systematic dissection of causal cell circuits, integrated such data across screens to maximize our ability to predict circuits poses substantial computational challenges, which have not been addressed. Here, we extend two Gromov-Wasserstein Optimal Transport methods to incorporate the perturbation label for cross-modality alignment. The obtained alignment is then employed to train a predictive model that estimates cellular responses to perturbations observed with only one measurement modality. We validate our method for the tasks of cross-modality alignment and cross-modality prediction in a recent multi-modal single-cell perturbation dataset. Our approach opens the way to unified causal models of cell biology., Comment: 16 pages, 4 figures, correspondence to Aviv Regev and Romain Lopez

Buildings account for 40 % of global energy consumption. A considerable portion of building energy consumption stems from heating, ventilation, and air conditioning (HVAC), and thus implementing smart, energy-efficient HVAC systems has the potential to significantly impact the course of climate change. In recent years, model-free reinforcement learning algorithms have been increasingly assessed for this purpose due to their ability to learn and adapt purely from experience. They have been shown to outperform classical controllers in terms of energy cost and consumption, as well as thermal comfort. However, their weakness lies in their relatively poor data efficiency, requiring long periods of training to reach acceptable policies, making them inapplicable to real-world controllers directly. Hence, common research goals are to improve the learning speed, as well as to improve their ability to generalize, in order to facilitate transfer learning to unseen building environments. In this paper, we take a federated learning approach to training the reinforcement learning controller of an HVAC system. A global control policy is learned by aggregating local policies trained on multiple data centers located in different climate zones. The goal of the policy is to simultaneously minimize energy consumption and maximize thermal comfort. The federated optimization strategy indirectly increases both the rate at which experience data is collected and the variation in the data. We demonstrate through experimental evaluation that these effects lead to a faster learning speed, as well as greater generalization capabilities in the federated policy compared to any individually trained policy.

Hyperspectral Imaging (HSI) serves as an important technique in remote sensing. However, high dimensionality and data volume typically pose significant computational challenges. Band selection is essential for reducing spectral redundancy in hyperspectral imagery while retaining intrinsic critical information. In this work, we propose a novel hyperspectral band selection model by decomposing the data into a low-rank and smooth component and a sparse one. In particular, we develop a generalized 3D total variation (G3DTV) by applying the $\ell_1^p$-norm to derivatives to preserve spatial-spectral smoothness. By employing the alternating direction method of multipliers (ADMM), we derive an efficient algorithm, where the tensor low-rankness is implied by the tensor CUR decomposition. We demonstrate the effectiveness of the proposed approach through comparisons with various other state-of-the-art band selection techniques using two benchmark real-world datasets. In addition, we provide practical guidelines for parameter selection in both noise-free and noisy scenarios.

Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. Given a set of points $[\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n] = \mathbf{X} \in \mathbb{R}^{d \times n}$ and a vector $\mathbf{y} \in \mathbb{R}^d$, the goal is to find coefficients $\mathbf{w} \in \mathbb{R}^n$ so that $\mathbf{X} \mathbf{w} \approx \mathbf{y}$, subject to some desired structure on $\mathbf{w}$. In this work we seek $\mathbf{w}$ that forms a local reconstruction of $\mathbf{y}$ by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of $\mathbf{X}$ that are close to $\mathbf{y}$ when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of $\mathbf{X}$ have a unique Delaunay triangulation, the optimal coefficients' number of non-zero entries is upper bounded by $d+1$, thereby providing local sparse solutions when $d \ll n$. Under the same condition we also show that for any $\mathbf{y}$ contained in the convex hull of $\mathbf{X}$ there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing $\mathbf{y}$. This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of $\mathbf{X}$. We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex., Comment: 26 pages, 8 figures

Neural collapse (NC) is a simple and symmetric phenomenon for deep neural networks (DNNs) at the terminal phase of training, where the last-layer features collapse to their class means and form a simplex equiangular tight frame aligning with the classifier vectors. However, the relationship of the last-layer features to the data and intermediate layers during training remains unexplored. To this end, we characterize the geometry of intermediate layers of ResNet and propose a novel conjecture, progressive feedforward collapse (PFC), claiming the degree of collapse increases during the forward propagation of DNNs. We derive a transparent model for the well-trained ResNet according to that ResNet with weight decay approximates the geodesic curve in Wasserstein space at the terminal phase. The metrics of PFC indeed monotonically decrease across depth on various datasets. We propose a new surrogate model, multilayer unconstrained feature model (MUFM), connecting intermediate layers by an optimal transport regularizer. The optimal solution of MUFM is inconsistent with NC but is more concentrated relative to the input data. Overall, this study extends NC to PFC to model the collapse phenomenon of intermediate layers and its dependence on the input data, shedding light on the theoretical understanding of ResNet in classification problems., Comment: 14 pages, 5 figures

The search for global minima is a critical challenge across multiple fields including engineering, finance, and artificial intelligence, particularly with non-convex functions that feature multiple local optima, complicating optimization efforts. We introduce the Quantum Global Minimum Finder (QGMF), an innovative quantum computing approach that efficiently identifies global minima. QGMF combines binary search techniques to shift the objective function to a suitable position and then employs Variational Quantum Search to precisely locate the global minimum within this targeted subspace. Designed with a low-depth circuit architecture, QGMF is optimized for Noisy Intermediate-Scale Quantum (NISQ) devices, utilizing the logarithmic benefits of binary search to enhance scalability and efficiency. This work demonstrates the impact of QGMF in advancing the capabilities of quantum computing to overcome complex non-convex optimization challenges effectively., Comment: 12 pages, 3 figures