Multiscale dynamical systems for constrained monotone inclusions
Multiscale Dynamical Systems for Constrained Monotone Inclusions
Variational inequalities and constrained optimization problems form an important class of mathematical models used in a range of applications, including equilibrium theory, control systems, game theory, and machine learning. These problems are frequently characterized by nonsmoothness, constraints, nonuniqueness of equilibria, and uncertainty in the available information, making the analysis of asymptotic feasibility and solution selection particularly challenging.
This thesis develops a unified multiscale dynamical systems framework for constrained monotone inclusions and variational inequalities. Exterior penalization is used to enforce constraints asymptotically, while Tikhonov regularization promotes convergence toward the least-norm solution. We first study how increasing penalization and vanishing regularization can be balanced to obtain strong convergence toward the least-norm solution. Based on a family of auxiliary monotone inclusions and the associated multiscale central path, the analysis covers both forward-backward and forward-backward-forward dynamical systems. We then extend the framework to stochastic differential inclusions perturbed by Brownian motion, where we establish asymptotic convergence and finite-time convergence rates, together with concentration around the solution set. We also relate the continuous-time stochastic dynamics to discrete stochastic forward-backward schemes. The final part develops an inertial forward-backward algorithm combining exterior penalization, Tikhonov regularization, and inertial extrapolation. The framework accommodates smooth, nonsmooth, and mixed smooth-nonsmooth penalty operators. Within this framework, we establish weak ergodic convergence. By introducing a discrete Tikhonov central path, we further prove strong convergence to the least-norm solution under an additional assumption.