Stable Proximal Dynamical System for Mixed Variational Inequalities in Hilbert Spaces

Abstract

In this paper we focus on solving mixed variational inequalities by proximal fixed-time dynamical systems, in which the solution of the proximal dynamical system is uniquely described and converges to the solution of the associated mixed variational inequality under the assumptions of strong monotonicity and Lipschitz continuity on the operator in the variational inequality, where the time of convergence is finite and is uniformly bounded for all initial conditions. The proposed technique can be reduced to the fixed-time stable projected dynamical system and results can still be applied even with relaxing the assumption of strong monotonicity. The modified continuous-time proximal dynamical system is designed to solve convex minimizing problems on a (possibly infinite-dimensional) Hilbert spaces. Finally, some qualitative properties with fixed-time stability of equilibrium points to the proposed scheme for solving continuous-time nonsmooth convex optimization problems or, in a general setting mixed variational inequalities are presented.