Fixed-Time Stable Proximal Dynamical System for Solving Mixed Variational Inequality Problems
Kunal Garg, Mayank Baranwal, Rohit Gupta, Ram Vasudevan, Dimitra Panagou · arXiv (Cornell University) · 2019
In this paper, a novel modified proximal dynamical system is proposed to compute the solution of a {mixed variational inequality problem} within a fixed time, where the time of convergence is finite, and is uniformly bounded for all initial conditions. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that a solution of the modified proximal dynamical system exists, is uniquely determined and converges to the unique solution of the associated mixed variational inequality problem (MVIP) within a fixed time. As a special case for solving variational inequality problems, the modified proximal dynamical system reduces to a fixed-time stable projected dynamical system. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumptions of strong monotonicity are relaxed to that of strong pseudomonotonicity. Connections to convex optimization problems are discussed, and commonly studied dynamical systems in the continuous-time optimization literature are shown as special cases of the modified proximal dynamical system proposed in this paper. Finally, it is shown that the discretization of the proposed scheme converges to an arbitrarily small neighborhood of the solution of the associated MVIP within a fixed number of steps, independent of the initial conditions. Two numerical examples are presented, which give numerical evidence in support of the fixed-time convergent behavior of the proposed modified proximal dynamical system.