A Relaxation Approach to Vector-Valued Allen–Cahn MPEC Problems

Abstract

In this paper we consider a vector-valued Allen-Cahn MPEC problem. To derive optimality conditions we exploit a regularization-relaxation technique. The optimality system of the regularized-relaxed subproblems are investigated by applying the classical result of Zowe and Kurcyusz. Finally we show that the stationary points of the regularizedrelaxed subproblems converge to weak stationary points of the limit problem.

Optimal Control of Allen-Cahn Systems

“…The optimization problem (P 0 ) belongs to the problem class of so-called MPECs (Mathematical Programs with Equilibrium Constraints) which violate classical NLP constraint qualifications. In the next two subsections we present results concerning first-order necessary optimality systems obtained by the penalization approach, see [11], or the relaxation approach, see [10]. These techniques have been discussed also in [2,16,17].…”

“…f ≡ 0 in general, and without elasticity we use a relaxation approach. Details for our presented results can be found in [10]. After reformulating as in (2.5) − (2.6) the Allen-Cahn system with the help of a slack variable ξ into an MPEC, we add to the problem (P 0 ) an additional constraint 1 2 ξ 2 L 2 (Ω T ) ≤ R and denote this modified optimization problem by (P R ).…”

Optimal Boundary Control of a Simplified Ericksen–Leslie System for Nematic Liquid Crystal Flows in 2D

“…In this direction, we refer to [48,49] for local models in one and two spatial dimensions and to the recent paper [38], in which the first-order necessary optimality conditions were derived for the nonlocal convective Cahn-Hilliard system in three dimensions, in the case of degenerate mobilities and singular potentials. At last, regarding the Allen-Cahn type equations (i.e., the scalar version of the director equation (1.3) with zero velocity), distributed and boundary optimal control problems with various types of dynamic boundary conditions have been studied in a number of recent papers, in particular for the case of double obstacle potentials (see [8,12,13,28]).…”

A Deep Quench Approach to the Optimal Control of an Allen–Cahn Equation with Dynamic Boundary Conditions and Double Obstacles

“…It is a well-known fact that the differential inclusion conditions (1.3)-(1.5) occurring as constraints in (P 0 ) violate all of the known classical NLP (nonlinear programming) constraint qualifications. Hence, the existence of Lagrange multipliers cannot be inferred from standard theory, and the derivation of first-order necessary condition becomes very difficult, as the treatments in [6,7,8,9] for the case of standard Neumann boundary conditions show (note that [9] deals with the more difficult case of the Cahn-Hilliard equation).…”