Dynamical System Approach for Optimal Control Problems with Equilibrium Constraints Using Gap-Constraint-Based Reformulation

This study focuses on using direct methods (first-discretize-then-optimize) to solve optimal control problems for a class of nonsmooth dynamical systems governed by differential variational inequalities (DVI), called optimal control problems with equilibrium constraints (OCPEC). In the discretization step, we propose a class of novel approaches to smooth the DVI. The generated smoothing approximations of DVI, referred to as gap-constraint-based reformulations, have computational advantages owing to their concise and semismoothly differentiable constraint system. In the optimization step, we propose an efficient dynamical system approach to solve the discretized OCPEC, where a sequence of its smoothing approximations is solved approximately. This system approach involves a semismooth Newton flow, thereby achieving fast local exponential convergence. We confirm the effectiveness of our method using a numerical example.