A bundle-free implicit programming approach for a class of elliptic MPECs in function space

Abstract

Using a standard first-order optimality condition for nonsmooth optimization problems, a general framework for a descent method is developed. This setting is applied to a class of mathematical programs with equilibrium constraints in function space from which a new algorithm is derived. Global convergence of the algorithm is demonstrated in function space and the results are then illustrated by numerical experiments.Keywords Elliptic variational inequality · Elliptic MPEC · Implicit programming · Optimal control of variational inequalities · Nonsmooth optimization Mathematical Subject Classification 49J52 · 49K20 · 49K21 · 65Kxx · 90C33 · 49M05

Optimal control of a non-smooth semilinear elliptic equation

“…where χ is as defined in (17). Note that according to assumption (16), the case y(x) = ϕ(x) = 0 is negligible here. Using Lemma 3.7, we now obtain that δ εn = S (u εn )h converges strongly in Y to G χ h. Since h ∈ L 2 (Ω) was arbitrary, this proves the claim.…”

“…The latter two results show that a comparison of an optimality system involving dual variables with conditions of type (ii) is in general far from evident, and we are not aware of any contributions in this direction for the case of optimal control of non-smooth PDEs. This is of particular interest, however, since optimality conditions of type (ii) may be satisfied by accumulation points of sequences generated by optimization algorithms; see [16]. The aim of our paper is to investigate this connection for the particular optimal control problem (P).…”

Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization

“…[17,18,22] Finally, we mention that although the methods in [17,18], when used in conjunction with a non-linear PATH strategy or a heuristic line search argument exhibit globally convergent behavior experimentally, Algorithm 1 is the only proven globally convergent function-space-based algorithm for this problem class.…”

“…Of course, when S is smooth, (3.2) has a unique solution and a descent direction can be obtained by solving the first-order optimality conditions associated with (3.2). Otherwise, we proposed in [22] a new method for obtaining a descent direction in nonsmooth settings when S 0 .uI h/ has an explicit form. With these ideas, we develop a first order method for solving (3.1) and discuss its convergence properties.…”

“…Y is Lipschitz continuous and directionally differentiable and J is Fréchet differentiable in both arguments. We demonstrate in [22] that it is theoretically possible to obtain a descent direction h for J at u by solving the following regularized auxiliary problem:…”