A new Relaxation Method for Optimal Control of Semilinear Elliptic Variational Inequalities Obstacle Problems

TL;DR

This paper introduces a novel relaxation method for solving optimal control problems constrained by semilinear elliptic variational inequalities, specifically obstacle problems, combining mathematical programming and penalization techniques with numerical validation.

Contribution

The paper presents a new relaxation approach that simplifies the feasible domain and derives smooth optimality conditions for obstacle problems, supported by numerical experiments.

Findings

Efficient solution of obstacle problems using the proposed relaxation method.

Derivation of smooth Lagrange multipliers for optimality conditions.

Numerical experiments demonstrate the approach's effectiveness with IPOPT.

Abstract

In this paper, we investigate optimal control problems governed by semilinear elliptic variational inequalities involving constraints on the state, and more precisely the obstacle problem. Since we adopt a numerical point of view, we first relax the feasible domain of the problem, then using both mathematical programming methods and penalization methods we get optimality conditions with smooth Lagrange multipliers. Some numerical experiments using IPOPT algorithm are presented to verify the efficiency of our approach.