An adaptive exact penalty method for nonsmooth optimal control problems with nonsmooth nonconvex state and control constraints

TL;DR

This paper introduces an adaptive exact penalty method for nonsmooth, nonconvex optimal control problems, combining classical DCA with adaptive penalty and line search techniques, and proves its convergence with practical numerical demonstrations.

Contribution

It develops a novel adaptive penalty approach for nonsmooth nonconvex optimal control, integrating DCA with adaptive parameter updates and convergence guarantees.

Findings

Method converges under approximate subproblem solutions.

Adaptive parameters improve optimization efficiency.

Numerical example demonstrates practical effectiveness.

Abstract

A class of exact penalty-type local search methods for optimal control problems with nonsmooth cost functional, nonsmooth (but continuous) dynamics, and nonsmooth state and control constraints is presented, in which the the penalty parameter and several line search parameters are adaptively adjusted during the optimisation process. This class of methods is applicable to problems having a known DC (Difference-of-Convex functions) structure and in its core is based on the classical DCA method, combined with the steering exact penalty rules for updating the penalty parameter and an adaptive nonmonotone line search procedure. Under the assumption that all auxiliary subproblems are solved only approximately (that is, with finite precision), we prove the correctness of the proposed family of methods and present its detailed convergence analysis. The performance of several different versions of the method is illustrated by means of a numerical example, in which the methods are applied to a semi-academic optimal control problem with a nonsmooth nonconvex state constraint.