Searcharxiv⌕ Search

arXiv subjects

Loïc Bourdin

Publications and source records attributed to Loïc Bourdin.

At least 19 recordsLinked to original sources
arXiv2026

A Partitioned Optimization Framework for Structure-Aware Problems

This work tackles a class of optimization problems in which fixing some well-chosen combinations of the variables makes the problem substantially easier to solve. We consider that the variables space may be partitioned into subsets that fix these combinations to given values, so that the restriction of the problem to any of the partition sets admits a tractable solution. Then, we exhibit a reformulation of the problem that consists in searching for the partition set index that minimizes the objective value of the solution to the restricted problem. We name partitioned optimization framework (POf) the formalization of this class of problems and this reformulation process. As we prove in this work, the POf allows solving the original problem by focusing on the reformulated problem: all solutions to the reformulated problem are partition indices for which the solution to the associated restricted problem is also a solution to the original problem. Second, we introduce a derivative-free partitioned optimization method (DFPOm) to efficiently solve problems that fit in the POf. We prove that the reformulated problem is nicely handled by derivative-free optimization (DFO) algorithms with a covering step. Then the DFPOm consists in solving the reformulated problem using such DFO algorithm with a covering step to obtain an optimal partition index, and to return the solution to the associated restricted problem as a solution to the initial problem. Finally, we illustrate how the \pof allows solving some classes of problems. We first focus on an infinite-dimensional case, by solving analytically an optimal control problem that challenges standard methods from the literature. Then, we apply the DFPOm on a class of finite-dimensional problems called composite greybox problems, and we highlight the gain in numerical performance provided by the DFPOm by comparing it to two popular DFO solvers.

math.OC↗
arXiv2024

Shape optimization involving the Tresca friction law in a 2D linear elastic model

The aim of this work is to analyse a shape optimization problem in a mechanical friction context. Precisely we perform a shape sensitivity analysis of a Tresca friction problem, that is, a boundary value problem involving the usual linear elasticity equations together with the (nonsmooth) Tresca friction law on a part of the boundary. We prove that the solution to the Tresca friction problem admits a directional shape derivative which moreover coincides with the solution to a boundary value problem involving tangential Signorini's unilateral conditions. Then an explicit expression of the shape gradient of the Tresca energy functional is provided (which allows us to provide numerical simulations illustrating our theoretical results). Our methodology is not based on any regularization procedure, but rather on the twice epi-differentiability of the (nonsmooth) Tresca friction functional which is analyzed thanks to a change of variables which is well-suited in the two-dimensional case. The obstruction in the higher-dimensional case is discussed.

math.OC↗
arXiv2024

Sensitivity analysis and optimal control for a friction problem in the linear elastic model

This paper investigates, without any regularization procedure, the sensitivity analysis of a mechanical friction problem involving the (nonsmooth) Tresca friction law in the linear elastic model. To this aim a recent methodology based on advanced tools from convex and variational analyses is used. Precisely we express the solution to the so-called Tresca friction problem thanks to the proximal operator associated with the corresponding Tresca friction functional. Then, using an extended version of twice epi-differentiability, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving tangential Signorini's unilateral conditions. Finally our result is used to investigate and numerically solve an optimal control problem associated with the Tresca friction model.

math.OC↗
arXiv2024

Shape optimization for variational inequalities: the scalar Tresca friction problem

This paper investigates, without any regularization or penalization procedure, a shape optimization problem involving a simplified friction phenomena modeled by a scalar Tresca friction law. Precisely, using tools from convex and variational analysis such as proximal operators and the notion of twice epi-differentiability, we prove that the solution to a scalar Tresca friction problem admits a directional derivative with respect to the shape which moreover coincides with the solution to a boundary value problem involving Signorini-type unilateral conditions. Then we explicitly characterize the shape gradient of the corresponding energy functional and we exhibit a descent direction. Finally numerical simulations are performed to solve the corresponding energy minimization problem under a volume constraint which shows the applicability.

math.OC↗
arXiv2024

Sensitivity analysis of a scalar mechanical contact problem with perturbation of the Tresca's friction law

This paper investigates the sensitivity analysis of a scalar mechanical contact problem described by a boundary value problem involving the Tresca's friction law. The sensitivity analysis is performed with respect to right-hand source and boundary terms perturbations. In particular the friction threshold involved in the Tresca's friction law is perturbed, which constitutes the main novelty of the present work with respect to the existing literature. Hence we introduce a parameterized Tresca friction problem and its solution is characterized by using the proximal operator associated with the corresponding perturbed nonsmooth convex Tresca friction functional. Then, by invoking the extended notion of twice epi-differentiability depending on a parameter, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving Signorini unilateral conditions. Finally numerical simulations are provided in order to illustrate our main result.

math.OC↗
arXiv2023

Convergence in nonlinear optimal sampled-data control problems

Consider, on the one part, a general nonlinear finite-dimensional optimal control problem and assume that it has a unique solution whose state is denoted by $x^*$. On the other part, consider the sampled-data control version of it. Under appropriate assumptions, we prove that the optimal state of the sampled-data problem converges uniformly to $x^*$ as the norm of the corresponding partition tends to zero. Moreover, applying the Pontryagin maximum principle to both problems, we prove that, if $x^*$ has a unique weak extremal lift with a costate $p$ that is normal, then the costate of the sampled-data problem converges uniformly to $p$. In other words, under a nondegeneracy assumption, control sampling commutes, at the limit of small partitions, with the application of the Pontryagin maximum principle.

math.OC↗
arXiv2022

Erratum, counterexample and an additional revealing poll step for a result of "Analysis of direct searches for discontinuous functions''

This note provides a counterexample to a theorem announced in the last part of the paper ''Analysis of direct searches for discontinuous functions'', Mathematical Programming Vol. 133, pp.~299--325, 2012. The counterexample involves an objective function $f: \mathbb{R} \to \mathbb{R}$ which satisfies all the assumptions required by the theorem but contradicts some of its conclusions. A corollary of this theorem is also affected by this counterexample. The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points $(x_k)_k$ converging to a point $x_*$ where $f$ is discontinuous and whose objective function value $f(x_*)$ is strictly less than $\lim_{k\to\infty} f(x_k)$. Moreover the dDSM generates no trial point in one of the two branches of $f$ near $x_*$. This note also investigates the proof of the theorem to highlight the inexact statements in the original paper. Finally this work concludes with a modification of the dDSM that allows to recover the properties broken by the counterexample.

math.OC↗
arXiv2021

Legendre's necessary condition for fractional Bolza functionals with mixed initial/final constraints

The present work was primarily motivated by our findings in the literature of some flaws within the proof of the second-order Legendre necessary optimality condition for fractional calculus of variations problems. Therefore we were eager to elaborate a correct proof and it turns out that this goal is highly nontrivial, especially when considering final constraints. This paper is the result of our reflections on this subject. Precisely we consider here a constrained minimization problem of a general Bolza functional that depends on a Caputo fractional derivative of order 0 < $α$ $\le$ 1 and on a Riemann-Liouville fractional integral of order $β$ > 0, the constraint set describing general mixed initial/final constraints. The main contribution of our work is to derive corresponding first-and second-order necessary optimality conditions, namely the Euler-Lagrange equation, the transversality conditions and, of course, the Legendre condition. A detailed discussion is provided on the obstructions encountered with the classical strategy, while the new proof that we propose here is based on the Ekeland variational principle. Furthermore we underline that some subsidiary contributions are provided all along the paper. In particular we prove an independent and intrinsic result of fractional calculus stating that it does not exist a nontrivial function which is, together with its Caputo fractional derivative of order 0 < $α$ < 1, compactly supported. Moreover we also discuss some evidences claiming that Riemann-Liouville fractional integrals should be considered in the formulation of fractional calculus of variations problems in order to preserve the existence of solutions.

math.OC↗
arXiv2020

Unified Riccati theory for optimal permanent and sampled-data control problems in finite and infinite time horizons

We revisit and extend the Riccati theory, unifying continuous-time linear-quadratic optimal permanent and sampled-data control problems, in finite and infinite time horizons. In a nutshell, we prove that:-- when the time horizon T tends to $+\infty$, one passes from the Sampled-Data Difference Riccati Equation (SD-DRE) to the Sampled-Data Algebraic Riccati Equation (SD-ARE), and from the Permanent Differential Riccati Equation (P-DRE) to the Permanent Algebraic Riccati Equation (P-ARE);-- when the maximal step of the time partition $Δ$ tends to $0$, one passes from (SD-DRE) to (P-DRE), and from (SD-ARE) to (P-ARE).Our notations and analysis provide a unified framework in order to settle all corresponding results.

math.OC↗
arXiv2018

On a decomposition formula for the proximal operator of the sum of two convex functions

The main result of the present theoretical paper is an original decomposition formula for the proximal operator of the sum of two proper, lower semicontinuous and convex functions $f$ and $g$. For this purpose, we introduce a new operator, called $f$-proximal operator of $g$ and denoted by $\mathrm{prox}^f_g$, that generalizes the classical notion. Then we prove the decomposition formula $\mathrm{prox}_{f+g} = \mathrm{prox}_f \circ \mathrm{prox}^f_g$. After collecting several properties and characterizations of $\mathrm{prox}^f_g$, we prove that it coincides with the fixed points of a generalized version of the classical Douglas-Rachford operator. This relationship is used for the construction of a weakly convergent algorithm that computes numerically this new operator $\mathrm{prox}^f_g$, and thus, from the decomposition formula, allows to compute numerically $\mathrm{prox}_{f+g}$. It turns out that this algorithm was already considered and implemented in previous works, showing that $\mathrm{prox}^f_g$ is already present (in a hidden form) and useful for numerical purposes in the existing literature. However, to the best of our knowledge, it has never been explicitly expressed in a closed formula and neither been deeply studied from a theoretical point of view. The present paper contributes to fill this gap in the literature. Finally we give an illustration of the usefulness of the decomposition formula in the context of sensitivity analysis of linear variational inequalities of second kind in a Hilbert space.

math.OC↗
arXiv2018

Flip procedure in geometric approximation of multiple-component shapes -- Application to multiple-inclusion detection

We are interested in geometric approximation by parameterization of two-dimensional multiple-component shapes, in particular when the number of components is a priori unknown. Starting a standard method based on successive shape deformations with a one-component initial shape in order to approximate a multiple-component target shape usually leads the deformation flow to make the boundary evolve until it surrounds all the components of the target shape. This classical phenomenon tends to create double points on the boundary of the approximated shape. In order to improve the approximation of multiple-component shapes (without any knowledge on the number of components in advance), we use in this paper a piecewise Bézier parameterization and we consider two procedures called intersecting control polygons detection and flip procedure. The first one allows to prevent potential collisions between two parts of the boundary of the approximated shape, and the second one permits to change its topology by dividing a one-component shape into a two-component shape. For an experimental purpose, we include these two processes in a basic geometrical shape optimization algorithm and test it on the classical inverse obstacle problem. This new approach allows to obtain a numerical approximation of the unknown inclusion, detecting both the topology (i.e. the number of connected components) and the shape of the obstacle. Several numerical simulations are performed.

math.OC↗
arXiv2018

Sensitivity analysis of variational inequalities via twice epi-differentiability and proto-differentiability of the proximity operator

In this paper we investigate the sensitivity analysis of parameterized nonlinear variational inequalities of second kind in a Hilbert space. The challenge of the present work is to take into account a perturbation on all the data of the problem. This requires special adjustments in the definitions of the generalized first- and second-order differentiations of the involved operators and functions. Precisely, we extend the notions, introduced and thoroughly studied by R.T. Rockafellar, of twice epi-differentiability and proto-differentiability to the case of a parameterized lower semi-continuous convex function and its subdifferential respectively. The link between these two notions is tied to Attouch's theorem and to the new concept, introduced in this paper, of convergent supporting hyperplanes. The previous tools allow us to derive an exact formula of the proto-derivative of the generalized proximity operator associated to a parameterized variational inequality, and deduce the differentiability of the associated solution with respect to the parameter. Furthermore, the derivative is shown to be the solution of a new variational inequality involving semi- and twice epi-derivatives of the data. An application is given to parameterized convex optimization problems involving the sum of two convex functions (one of them being smooth). The case of smooth convex optimization problems with inequality constraints is discussed in details. This approach seems to be new in the literature and open several perspectives towards theoretical and computational issues in nonlinear optimization.

math.OC↗
arXiv2017

Cauchy-Lipschitz theory for fractional multi-order dynamics -- State-transition matrices, Duhamel formulas and duality theorems

The aim of the present paper is to contribute to the development of the study of Cauchy problems involving Riemann-Liouville and Caputo fractional derivatives. Firstly existence-uniqueness results for solutions of non-linear Cauchy problems with vector fractional multi-order are addressed. A qualitative result about the behavior of local but non-global solutions is also provided. Finally the major aim of this paper is to introduce notions of fractional state-transition matrices and to derive fractional versions of the classical Duhamel formula. We also prove duality theorems relying left state-transition matrices with right state-transition matrices.

math.CA↗
arXiv2016

Linear-quadratic optimal sampled-data control problems: convergence result and Riccati theory

We consider a general linear control system and a general quadratic cost, where the state evolves continuously in time and the control is sampled, i.e., is piecewise constant over a subdivision of the time interval. This is the framework of a linear-quadratic optimal sampled-data control problem. As a first result, we prove that, as the sampling periods tend to zero, the optimal sampled-data controls converge pointwise to the optimal permanent control. Then, we extend the classical Riccati theory to the sampled-data control framework, by developing two different approaches: the first one uses a recently established version of the Pontryagin maximum principle for optimal sampled-data control problems, and the second one uses an adequate version of the dynamic programming principle. In turn, we obtain a closed-loop expression for optimal sampled-data controls of linear-quadratic problems.

math.OC↗
arXiv2016

Note on Pontryagin maximum principle with running state constraints and smooth dynamics -- Proof based on the Ekeland variational principle

In this note our aim is to give a proof of the Pontryagin maximum principle for a general optimal control problem with running state constraints and smooth dynamics. Our proof is based on the classical Ekeland variational principle. The main result (and its proof) of this note are not new and are already well-known in the literature. The aim of the author is only to provide a complete and detailed proof of this classical theorem in the case of smooth dynamics. If you have any remarks or questions, do not hesitate to contact the author at loic.bourdin@unilim.fr.

math.OC↗
arXiv2016

Variational integrators of fractional Lagrangian systems in the framework of discrete embeddings

This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a variational structure called Lagrangian structure. We are specially interested in the conservation at the discrete level of this Lagrangian structure by discrete embeddings. We then replace in this framework the variational integrators developed in [10, Chapter VI.6] and in [12]. Finally, we extend the notion of discrete embeddings and variational integrators to fractional Lagrangian systems.

math.NA↗