In this presentation, I will discuss two key topics. Firstly, I will delve into the resolution of the Pareto eigenvalue complementarity problem and its inverse counterpart. Our approach employs interior point methods, supplemented by a non-parametric smoothing technique. The efficacy of these proposed methodologies is underscored through an array of numerical experiments.
Shifting our focus to continuous optimization, we adopt a dynamical systems perspective. Specifically, we study various proximal gradient inertial algorithms, discretized from a non-regular inertial dynamical system featuring elements of dry friction and Hessian-driven damping. Additionally, we examine a doubly nonlinear evolution equation governed by two potentials, accelerating its convergence through the application of time scaling and averaging techniques. This results in inertial dynamics featuring dry friction and implicit Hessian-driven damping. The numerical tests corroborate the superior performance of inertial systems over their first-order counterparts, aligning with our theoretical results.
