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

TL;DR

This paper introduces a novel smoothing reformulation for differential variational inequalities in optimal control problems with equilibrium constraints, enabling efficient solution via a dynamical system approach with fast convergence.

Contribution

It proposes a gap-constraint-based reformulation for DVI smoothing and an efficient semismooth Newton flow method for solving discretized OCPECs.

Findings

The method achieves fast local exponential convergence.

Numerical example confirms the effectiveness of the approach.

Reformulation simplifies the constraint system for better computational performance.

Abstract

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.