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Stochastic Differential Inclusions driven by Maximal Monotone Operators with empty interiors

Juan Guillermo Garrido, Pedro Pérez-Aros, Mathias Staudigl

Abstract

This paper studies the long-time behavior of stochastic differential inclusions driven by maximal monotone operators, motivated by continuous-time models of first-order optimization methods under noisy or approximate operator information. We first address well-posedness and show that existence and uniqueness can be established without the customary requirement that the operator's domain has nonempty interior, by adopting an appropriate notion of solution. We then analyze asymptotic properties of the resulting stochastic dynamics, extending convergence guarantees beyond previously studied settings that rely on smooth potentials, full-domain subdifferentials, or Lipschitz monotone operators. In addition, we consider a Tikhonov-type regularization of the stochastic inclusion and prove corresponding well-posedness and long-time convergence results.

Stochastic Differential Inclusions driven by Maximal Monotone Operators with empty interiors

Abstract

This paper studies the long-time behavior of stochastic differential inclusions driven by maximal monotone operators, motivated by continuous-time models of first-order optimization methods under noisy or approximate operator information. We first address well-posedness and show that existence and uniqueness can be established without the customary requirement that the operator's domain has nonempty interior, by adopting an appropriate notion of solution. We then analyze asymptotic properties of the resulting stochastic dynamics, extending convergence guarantees beyond previously studied settings that rely on smooth potentials, full-domain subdifferentials, or Lipschitz monotone operators. In addition, we consider a Tikhonov-type regularization of the stochastic inclusion and prove corresponding well-posedness and long-time convergence results.
Paper Structure (12 sections, 16 theorems, 147 equations)

This paper contains 12 sections, 16 theorems, 147 equations.

Table of Contents

  1. Keywords.
  2. AMS subject classifications.
  3. Introduction
  4. Notation and Preliminaries in Convex Analysis
  5. Notions in Convex Analysis
  6. Measure theory and stochastic processes
  7. Integral average and like-Opial’s Lemma
  8. The SDI Model for Monotone Inclusions and Definition of Solutions
  9. SDI related to monotone differential inclusions
  10. Stochastic Subdifferential Inclusion
  11. Tikhonov Regularization
  12. Conclusions

Key Result

Theorem 2.1

Let $X,Y$ be two finite dimensional Hilbert spaces, and $A \colon Y \rightrightarrows Y$ be maximal monotone mapping, $P \colon X \to Y$ a be a linear mapping. Define $T \colon X \rightrightarrows X$ by If $\operatorname{rge} P \cap \operatorname{rint} \left( \operatorname{dom} A \right) \neq \emptyset$, then $T$ is maximal monotone.

Theorems & Definitions (32)

  • Theorem 2.1: Maximal monotonicity under composition
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 3.1
  • Example 3.1
  • Theorem 3.2
  • proof
  • Example 3.2
  • ...and 22 more