Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme

Here we present well-posedness results for first order stochastic differential inclusions, more precisely for sweeping process with a stochastic perturbation. These results are provided in combining both deterministic sweeping process theory and methods concerning the reflection of a Brownian motion. In addition, we prove convergence results for a Euler scheme, discretizing theses stochastic differential inclusions.Comment: 30 page

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.Comment: 23 page

About regularity properties in variational analysis and applications in optimization

Regularity properties lie at the core of variational analysis because of their importance for stability analysis of optimization and variational problems, constraint qualications, qualication conditions in coderivative and subdierential calculus and convergence analysis of numerical algorithms. The thesis is devoted to investigation of several research questions related to regularity properties in variational analysis and their applications in convergence analysis and optimization. Following the works by Kruger, we examine several useful regularity properties of collections of sets in both linear and Holder-type settings and establish their characterizations and relationships to regularity properties of set-valued mappings. Following the recent publications by Lewis, Luke, Malick (2009), Drusvyatskiy, Ioe, Lewis (2014) and some others, we study application of the uniform regularity and related properties of collections of sets to alternating projections for solving nonconvex feasibility problems and compare existing results on this topic. Motivated by Ioe (2000) and his subsequent publications, we use the classical iteration scheme going back to Banach, Schauder, Lyusternik and Graves to establish criteria for regularity properties of set-valued mappings and compare this approach with the one based on the Ekeland variational principle. Finally, following the recent works by Khanh et al. on stability analysis for optimization related problems, we investigate calmness of set-valued solution mappings of variational problems.Doctor of Philosoph

Stochastic Variational Inequalities on Non-Convex Domains

The objective of this work is to prove, in a first step, the existence and the uniqueness of a solution of the following multivalued deterministic differential equation: $dx(t)+\partial ^-\varphi (x(t))(dt)\ni dm(t),\ t>0$, $x(0)=x_0$, where $m:\mathbb{R}_+\rightarrow\mathbb{R}^d$ is a continuous function and $\partial^-\varphi$ is the Fr\'{e}chet subdifferential of a semiconvex function $\varphi$; the domain of $\varphi$ can be non-convex, but some regularities of the boundary are required. The continuity of the map $m\mapsto x:C([0,T];\mathbb{R}^{d})\rightarrow C([0,T] ;\mathbb{R}^{d})$, which associate the input function $m$ with the solution $x$ of the above equation, as well as tightness criteria allow to pass from the above deterministic case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion: $X_t+K_t = \xi+\int_0^t F(s,X_{s})ds + \int_0^t G(s,X_s) dB_s,\; t\geq0$, $\;$ with $dK_{t}(\omega)\in\partial^-\varphi( X_t (\omega))(dt)$.Comment: 39 page