A Proximal Dynamic Approach to Equilibrium Problems With Finite-Time Convergence

This article proposes a finite-time converging proximal dynamic model (FPD) to deal with equilibrium problems. A distinctive feature of the FPD is its fast and finite-time convergence, in contrast to conventional proximal dynamic methods. It is shown that the solution of the proposed FPD converges to the solution of the corresponding equilibrium problems in finite-time under some mild conditions. Then the proposed FPD is applied to solve problems of nonsmooth composite optimization and absolute value equations. It is further shown in the case of solving composite optimization problems that the equilibrium point of the proposed proximal gradient dynamic model is globally finite-time stable under the so-called proximal Polyak-Lojasiewicz condition, which is weaker than strong convexity. Finally, numerical examples are presented to illustrate the effectiveness of the proposed methods.