OS Numerical Optimization: Fast optimistic methods for monotone equations and convex optimization problems
Dienstag, 28. November 2023
15:15 bis 16:45 Uhr
B. Azmi & S. Volkwein
Vortragende Person/Vortragende Personen:
Prof. Dr. Radu Ioan Boţ
In this talk we discuss continuous in time dynamics for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V . Such problems are motivated by minimax convex- concave and, in particular, by convex optimization problems with linear constraints. The central role is played by a second order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. We show that these methods exhibit fast convergence rates for ||V (z(t))|| as t →+∞, where z(·) denotes the generated trajectory, and for the restricted gap function, and that z(·) converges to a zero of the operator V . For the corresponding implicit and explicit discrete time models with Nesterov’s momentum we prove that they share the asymptotic features of the continuous dynamics.
Extensions to variational inequalities and fixed point problems are also addressed. The theoretical re- sults are illustrated by numerical experiments on bilinear games and the training of generative adversarial networks.