Multiscale dynamical systems for constrained monotone inclusions

Constrained optimization problems and variational inequalities across machine learning, control systems, and game theory often suffer from nonsmoothness, nonunique equilibria, and uncertainty. To address the resulting difficulties in asymptotic feasibility and solution selection, the thesis establishes a unified multiscale dynamical systems framework tailored to constrained monotone inclusions and variational inequalities. By pairing exterior penalization with Tikhonov regularization, the research identifies how to balance growing penalties and decaying regularization parameters to ensure strong convergence toward the minimal-norm solution. This approach is analyzed across forward-backward and forward-backward-forward dynamical systems via auxiliary monotone inclusions and an associated multiscale central path. The framework is also broadened to stochastic differential inclusions affected by Brownian motion, where the work determines asymptotic behavior, finite-time convergence rates, and solution-set concentration properties.