Juan Guillermo Garrido

In this paper, we study the existence of solutions to sweeping processes in the presence of stochastic perturbations, where the moving set takes uniformly proxregular 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.

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.

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.

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.

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.