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Benoît Legat

Publications and source records attributed to Benoît Legat.

16 recordsLinked to original sources

Fast, Differentiable, GPU-Accelerated Ray Tracing for Multiple Diffraction and Reflection Paths

We present a fast, differentiable, GPU-accelerated optimization method for ray path tracing in environments containing planar reflectors and straight diffraction edges. Based on Fermat's principle, our approach reformulates the path-finding problem as the minimization of total path length, enabling efficient parallel execution on modern GPU architectures. Unlike existing methods that require separate algorithms for reflections and diffractions, our unified formulation maintains consistent problem dimensions across all interaction sequences, making it particularly suitable for vectorized computation. Through implicit differentiation, we achieve efficient gradient computation without differentiating through solver iterations, significantly outperforming traditional automatic differentiation approaches. Numerical simulations demonstrate convergence rates comparable to specialized Newton methods while providing superior scalability for large-scale applications. The method integrates seamlessly with differentiable programming libraries such as JAX and DrJIT, enabling new possibilities in inverse design and optimization for wireless propagation modeling. The source code is openly available at https://github.com/jeertmans/fpt-jax.

eess.SP

A General and Streamlined Differentiable Optimization Framework

Differentiating through constrained optimization problems is increasingly central to learning, control, and large-scale decision-making systems, yet practical integration remains challenging due to solver specialization and interface mismatches. This paper presents a general and streamlined framework-an updated DiffOpt.jl-that unifies modeling and differentiation within the Julia optimization stack. The framework computes forward - and reverse-mode solution and objective sensitivities for smooth, potentially nonconvex programs by differentiating the KKT system under standard regularity assumptions. A first-class, JuMP-native parameter-centric API allows users to declare named parameters and obtain derivatives directly with respect to them - even when a parameter appears in multiple constraints and objectives - eliminating brittle bookkeeping from coefficient-level interfaces. We illustrate these capabilities on convex and nonconvex models, including economic dispatch, mean-variance portfolio selection with conic risk constraints, and nonlinear robot inverse kinematics. Two companion studies further demonstrate impact at scale: gradient-based iterative methods for strategic bidding in energy markets and Sobolev-style training of end-to-end optimization proxies using solver-accurate sensitivities. Together, these results demonstrate that differentiable optimization can be deployed as a routine tool for experimentation, learning, calibration, and design-without deviating from standard JuMP modeling practices and while retaining access to a broad ecosystem of solvers.

cs.LG

Dionysos.jl: a Modular Platform for Smart Symbolic Control

We introduce Dionysos.jl, a modular package for solving optimal control problems for complex dynamical systems using state-of-the-art and experimental techniques from symbolic control, optimization, and learning. More often than not with Cyber-Physical systems, the only sensible way of developing a controller is by discretizing the different variables, thus transforming the control task into a purely combinatorial problem on a finite-state mathematical object, called an abstraction of this system. Although this approach offers a safety-critical framework, the available techniques suffer important scalability issues. In order to render these techniques practical, it is necessary to construct smarter abstractions that differ from classical techniques by partitioning the state-space in a non trivial way.

eess.SY

Certifying ground-state properties of quantum many-body systems

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.

quant-ph

Flexible Differentiable Optimization via Model Transformations

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.

cs.LG

Low-Rank Univariate Sum of Squares Has No Spurious Local Minima

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.

math.OC

JuMP 1.0: Recent improvements to a modeling language for mathematical optimization

JuMP is an algebraic modeling language embedded in the Julia programming language. JuMP allows users to model optimization problems of a variety of kinds, including linear programming, integer programming, conic optimization, semidefinite programming, and nonlinear programming, and handles the low-level details of communicating with solvers. After nearly 10 years in development, JuMP 1.0 was released in March, 2022. In this short communication, we highlight the improvements to JuMP from recent releases up to and including 1.0.

cs.PL

Sequential sum-of-squares programming for analysis of nonlinear systems

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.

math.OC

Geometric control of hybrid systems

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.

math.OC

Geometric control of algebraic systems

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.

math.OC

Abstraction-based branch and bound approach to Q-learning for hybrid optimal control

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.

math.OC

Stability of Planar Switched Systems under Delayed Event Detection

In this paper, we analyse the impact of delayed event detection on the stability of a 2-mode planar hybrid automata. We consider hybrid automata with a unique equilibrium point for all the modes, and we find the maximum delay that preserves stability of that equilibrium point. We also show for the class of hybrid automata treated that the instability of the equilibrium point for the equivalent hybrid automaton with delay in the transitions is equivalent to the existence of a closed orbit in the hybrid state space, a result that is inspired by the Joint Spectral Radius theorem. This leads to an algorithm for computing the maximum stable delay exactly. Other potential applications of our technique include co-simulation, networked control systems and delayed controlled switching with a state feedback control.

math.DS

Piecewise semi-ellipsoidal control invariant sets

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.

math.OC

An entropy-based bound for the computational complexity of a switched system

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.

math.OC

Minimally Constrained Stable Switched Systems and Application to Co-simulation

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.

math.OC

Computing controlled invariant sets for hybrid systems with applications to model-predictive control

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.

math.OC