Solving some optimal control problems using the Barrier penalty function method

Abstract

In this paper we present a new approach to solve a two-level optimization problem arising from an approximation by means of the finite element method of optimal control problems governed by unilateral boundary-value problems. The problem considered is to find a minimum of a functional with respect to the control variablesu. The minimized functional depends on control variables and state variablesx. The latter are the optimal solution of an auxiliary quadratic programming problem, whose parameters depend onu.

Our main idea is to replace this QP problem by its dual and then apply the barrier penalty method to this dual QP problem or to the primal one if it is in an appropriate form. As a result we obtain a problem approximating the original one. Its good property is the differentiable dependence of state variables with respect to the control variables. Furthermore, we propose a method for finding an approximate solution of a penalized lower-level problem if the optimal solution of the original QP problem is known. We apply the result obtained to some optimal shape design problems governed by the Dirichlet-Signorini boundary-value problem.