arXiv subjects

Emilio Vilches

Publications and source records attributed to Emilio Vilches.

At least 19 recordsLinked to original sources

arXiv2026

Volterra Sweeping Processes with Multivalued Perturbations under Compactness Conditions

We study integro-differential sweeping processes of Volterra type in a separable Hilbert space, in which the velocity is governed by the normal cone to a prox-regular moving set and is driven by an outer set-valued perturbation together with a history-dependent integral term that endows the dynamics with memory. The perturbation is assumed measurable, with closed convex values, of linear growth, and upper semicontinuous from the strong to the weak topology. We study the existence of absolutely continuous solutions under either of two alternative compactness hypotheses: ball-compactness of the moving sets, or a measure-of-noncompactness condition on the perturbation. After a reduction of the constrained dynamics to an unconstrained differential inclusion and uniform a priori bounds on the state and its velocity, existence follows from a fixed-point theorem for set-valued maps with contractible values. The memory of the process makes this contractibility delicate, and we obtain it through a continuation argument that propagates the history of the dynamics. As applications, we solve a quasistatic frictionless viscoelastic contact problem with long memory and a spatially distributed bioeconomic fishery model with ecological memory, both featuring uncertain set-valued forcing and a moving constraint set that is not ball-compact.

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arXiv2026

Sharp bounds for stochastic proximal and projection estimators via radial dominance

We study stochastic barycentric estimators for proximal points and metric projections obtained by exponentially reweighting Gaussian perturbations. Our main result is an abstract comparison theorem for probability measures with densities proportional to an exponential weight, under a radial dominance condition relative to a prescribed profile. This yields an explicit upper bound for the norm of the associated barycenter in terms of a one-dimensional comparison measure. We also provide tractable sufficient conditions for radial dominance, including strong convexity, addition of nonnegative convex terms, and star-shaped constraints. As a consequence, we obtain a refined convergence rate for stochastic proximal estimators of weakly convex functions, together with asymptotic sharpness of the constant. The same framework yields a corresponding rate for stochastic projection estimators onto closed convex sets. We further establish basic structural properties of the barycentric approximation operator, such as smoothness and cocoercivity. Numerical experiments illustrate the predicted rate, the dimensional scaling of the constant, and its asymptotic sharpness.

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arXiv2026

Integral Formulation and the Br\'ezis-Ekeland-Nayroles-Type Principle for Prox-Regular Sweeping Processes

We study sweeping processes in a Hilbert space driven by time-dependent uniformly prox-regular sets, allowing the moving constraint to exhibit discontinuities of bounded variation. We introduce a new integral formulation for bounded-variation trajectories, given by a global variational inequality tested against continuous admissible trajectories, and we compare it with the standard differential-measure formulation, in which the differential measure of the trajectory is constrained by the proximal normal cone. In the prox-regular (generally nonconvex) framework, the variational inequality necessarily includes a quadratic correction term reflecting the hypomonotonicity of proximal normal cones. Under mild regularity assumptions on the moving set, including lower semicontinuity in time, uniform prox-regularity of the values, and a selection-extension property guaranteeing a rich class of test trajectories (satisfied, for instance, in the convex case and for bounded prox-regular sets), we prove that the new integral formulation is equivalent to the differential-measure formulation. This yields a unified bounded-variation notion of solution for prox-regular sweeping processes. We further establish a Br\'ezis-Ekeland-Nayroles-type variational characterization via a prox-regular variational residual: the residual is nonpositive along every admissible trajectory, and solutions are exactly those trajectories for which this residual attains its maximal value, namely zero. As a consequence, we prove a stability result: a uniform limit of admissible trajectories with vanishing residual is a solution of the limit sweeping process. The resulting variational framework provides a robust tool for stability and approximation analyses in the prox-regular, nonconvex setting.

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arXiv2026

Convergence Rates for Stochastic Proximal and Projection Estimators

In this paper, we establish explicit convergence rates for the stochastic smooth approximations of infimal convolutions introduced and developed in \cite{MR4581306,MR4923371}. In particular, we quantify the convergence of the associated barycentric estimators toward proximal mappings and metric projections. We prove a dimension-explicit $\sqrt{\delta}$ bound, with explicit constants for the proximal mapping, in the $\rho$-weakly convex (possibly nonsmooth) setting, and we also obtain a dimension-explicit $\sqrt{\delta}$ rate for the metric projection onto an arbitrary convex set with nonempty interior. Under additional regularity, namely $C^{2}$ smoothness with globally Lipschitz Hessian, we derive an improved linear $O(\delta)$ rate with explicit constants, and we obtain refined projection estimates for convex sets with local $C^{2,1}$ boundary. Examples demonstrate that these rates are optimal.

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arXiv2026

Penalty-Based Smoothing of Convex Nonsmooth Supremum Functions with Accelerated Inertial Dynamics

We propose a penalty-based smoothing framework for convex nonsmooth functions with a supremum structure. The regularization yields a differentiable surrogate with controlled approximation error, a single-valued dual maximizer, and explicit gradient formulas. We then study an accelerated inertial dynamic with vanishing damping driven by a time-dependent regularized function whose parameter decreases to zero. Under mild integrability and boundedness conditions on the regularization schedule, we establish an accelerated $\mathcal{O}(t^{-2})$ decay estimate for the regularized residual and, in the regime $\alpha>3$, a sharper $o(t^{-2})$ decay together with weak convergence of trajectories to a minimizer of the original nonsmooth problem via an Opial-type argument. Applications to multiobjective optimization (through Chebyshev/max scalarization) and to distributionally robust optimization (via entropic regularization over ambiguity sets) illustrate the scope of the framework.

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arXiv2026

Stochastic Perturbation of Sweeping Processes Driven by Continuous Uniformly Prox-Regular Moving Sets

In this paper, we study the existence of solutions to sweeping processes in the presence of stochastic perturbations, where the moving set takes uniformly prox-regular values and varies continuously with respect to the Hausdorff distance, without smoothness assumptions. We propose a minimal geometric framework for such moving sets, make precise the logical implications between several standard hypotheses in the literature, and provide practical sufficient conditions that apply in particular to constraints defined as finite intersections of sublevel sets. Within this setting, we establish existence of weak and strong solutions and prove pathwise uniqueness for the associated stochastic differential equations reflected in time-dependent domains.

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arXiv2025

A Projected Variable Smoothing for Weakly Convex Optimization and Supremum Functions

In this paper, we address two main topics. First, we study the problem of minimizing the sum of a smooth function and the composition of a weakly convex function with a linear operator on a closed vector subspace. For this problem, we propose a projected variable smoothing algorithm and establish a complexity bound of $\mathcal{O}(ε^{-3})$ to achieve an $ε$-approximate solution. Second, we investigate the Moreau envelope and the proximity operator of functions defined as the supremum of weakly convex functions, and we compute the proximity operator in two important cases. In addition, we apply the proposed algorithm for solving a distributionally robust optimization problem, the LASSO with linear constraints, and the max dispersion problem. We illustrate numerical results for the max dispersion problem.

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arXiv2025

Inexact Catching-Up Algorithm for Moreau's Sweeping Processes

In this paper, we develop an inexact version of the catching-up algorithm for sweeping processes. We define a new notion of approximate projection, which is compatible with any numerical method for approximating exact projections, as this new notion is not restricted to remain strictly within the set. We provide several properties of the new approximate projections, which enable us to prove the convergence of the inexact catching-up algorithm in three general frameworks: prox-regular moving sets, subsmooth moving sets, and merely closed sets. Additionally, we apply our numerical results to address complementarity dynamical systems, particularly electrical circuits with ideal diodes. In this context, we implement the inexact catching-up algorithm using a primal-dual optimization method, which typically does not necessarily guarantee a feasible point. Our results are illustrated through an electrical circuit with ideal diodes. Our results recover classical existence results in the literature and provide new insights into the numerical simulation of sweeping processes.

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arXiv2024

A fixed-point approach to history-dependent sweeping processes

In this paper, we study the well-posedness of state-dependent and state-independent sweeping processes driven by prox-regular sets and perturbed by a history-dependent operator. Our approach, based on an enhanced version of Gronwall's lemma and fixed-point arguments, provides an efficient framework for analyzing sweeping processes. In particular, our findings recover all existing results for the class of Volterra sweeping processes and provide new insights into history-dependent sweeping processes. Finally, we apply our theoretical results to establish the well-posedness of a viscoelastic model with long memory.

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arXiv2024

A Newton-Like Dynamical System for Nonsmooth and Nonconvex Optimization

This work investigates a dynamical system functioning as a nonsmooth adaptation of the continuous Newton method, aimed at minimizing the sum of a primal lower-regular and a locally Lipschitz function, both potentially nonsmooth. The classical Newton method's second-order information is extended by incorporating the graphical derivative of a locally Lipschitz mapping. Specifically, we analyze the existence and uniqueness of solutions, along with the asymptotic behavior of the system's trajectories. Conditions for convergence and respective convergence rates are established under two distinct scenarios: strong metric subregularity and satisfaction of the Kurdyka-Lojasiewicz inequality.

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arXiv2024

Differentiability and Approximation of Probability Functions under Gaussian Mixture Models

In this work, we study probability functions associated with Gaussian mixture models. Our primary focus is on extending the use of spherical radial decomposition for multivariate Gaussian random vectors to the context of Gaussian mixture models, which are not inherently spherical, but conditionally so. Specifically, the conditional probability distribution, given a random parameter of the random vector, follows a Gaussian distribution, which allows us to rewrite the probability function as a tractable integrated Gaussian mixture. This assumption, together with spherical radial decomposition for Gaussian random vectors, enables us to represent the probability function as an integral over the Euclidean sphere. Using this representation, we establish sufficient conditions to ensure the differentiability of the probability function and provide an integral representation of its gradient. Furthermore, we approximate the probability function using random sampling over the parameter space and the Euclidean sphere. Finally, we present a numerical example that illustrates the advantages of this approach over classical approximations based on random vector sampling.

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arXiv2024

Splitting Algorithms for Distributionally Robust Optimization

In this paper, we provide different splitting methods for solving distributionally robust optimization problems in cases where the uncertainties are described by discrete distributions. The first method involves computing the proximity operator of the supremum function that appears in the optimization problem. The second method solves an equivalent monotone inclusion formulation derived from the first-order optimality conditions, where the resolvents of the monotone operators involved in the inclusion are computable. The proposed methods are applied to solve the Couette inverse problem with uncertainty and the denoising problem with uncertainty. We present numerical results to compare the efficiency of the algorithms.

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arXiv2024

Galerkin-like Method for Integro-Differential Inclusions with applications to Volterra Sweeping processes

In this paper, we develop the Galerkin-like method to address first-order integro-differential inclusions. Under compactness or monotonicity conditions, we obtain new results for the existence of solutions for this class of problems, which generalize existing results in the literature and provide new insights for differential inclusions with an unbounded right-hand side. The effectiveness of the proposed approach is illustrated by presenting new existence results for nonconvex state-dependent Volterra sweeping processes, where the right-hand side is unbounded, and the classical theory of differential inclusions is not applicable. This is the first result of its kind. The paper concludes with an application to the existence of an optimal control problem governed by nonconvex state-dependent Volterra sweeping processes in finite dimensions.

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arXiv2024

Well-posedness for integro-differential sweeping processes of Volterra type

In this paper, we study the well-posedness of integro-differential sweeping processes of Volterra type. Using new enhanced versions of Gronwall's inequality, a reparametrization technique, and a fixed point argument for history-dependent operators, we obtain the existence of solutions and provide a fully continuous dependence result for the integro-differential sweeping process. The paper ends with an application to projected dynamical systems.

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arXiv2023

Catching-up Algorithm with Approximate Projections for Moreau's Sweeping Processes

In this paper, we develop an enhanced version of the catching-up algorithm for sweeping processes through an appropriate concept of approximate projections. We establish some properties of this notion of approximate projection. Then, under suitable assumptions, we show the convergence of the enhanced catching-up algorithm for prox-regular, subsmooth, and merely closed sets. Finally, we discuss efficient numerical methods for obtaining approximate projections. Our results recover classical existence results in the literature and provide new insight into the numerical simulation of sweeping processes.

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arXiv2023

Inner Moreau envelope of nonsmooth conic chance constrained optimization problems

Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth. This motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function which covers most of the general classes of probabilistic constraints: $$φ(x)=\mathbb{P}(Φ(x,ξ)\in -\mathcal{K}),$$ where $\mathcal{K}$ is a convex cone of a Banach space. The conic inclusion $Φ(x,ξ) \in - \mathcal{K}$ represents an abstract system of inequalities, and $ξ$ is a random vector. We propose a regularization by applying the Moreau envelope to the scalarization of the function $Φ$. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can thus entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite and probust chance constrained optimization problems.

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arXiv2022

Integral functionals on nonseparable Banach spaces with applications

In this paper, we study integral functionals defined on spaces of functions with values on general (non-separable) Banach spaces. We introduce a new class of integrands and multifunctions for which we obtain measurable selection results. Then, we provide an interchange formula between integration and infimum, which enables us to get explicit formulas for the conjugate and Clarke subdifferential of integral functionals. Applications to expected functionals from stochastic programming, optimality conditions for a calculus of variation problem and sweeping processes are given.

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arXiv2021

Random multifunctions as the set minimizers of infinitely many differentiable random functions

Under mild assumptions, we prove that any random multifunction can be represented as the set of minimizers of an infinitely many differentiable normal integrand, which preserves the convexity of the random multifunction. We provide several applications of this result to the approximation of random multifunctions and integrands. The paper ends with a characterization of the set of integrable selections of a measurable multifunction as the set of minimizers of an infinitely many differentiable integral function.

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