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.

Appendix

The capacity of a subset $A \subset \Omega$ can be defined as follows, see e.g. [19] Proposition 3.3.12:

Definition 5.1

$$\begin{aligned} \text {cap}(A) = \inf \left\{ \int _{\Omega } |\nabla v|^2 dx,\, \text {over } v \in H^1_0(\Omega ) \,|\, v \ge 1,\,\,\text { a.e. on an open neighborhood of } A\right\} . \end{aligned}$$

Unlike the Lebesgue measure, $(n-1)$-dimensional manifolds may have positive capacity. The subadditive function cap$(\cdot )$ is used to define the notion of quasi-continuity: A function $f:\Omega \rightarrow \mathbb R$ is said to be quasi-continuous provided there exists a nonincreasing sequence of open sets $\Omega _n \subset \Omega$ with cap$(\Omega _n) \rightarrow 0$ such that f is continuous on $\Omega \setminus \Omega _n$ (cf. Chapter 3.3 in [19], Chapter 6.4.3 in [7], or [9]). It is well-known that every $H^1_0(\Omega )$-function f has a unique quasi-continuous representative $\tilde{f}$. Finally, we say that a property holds “quasi-everywhere”, denoted here by q.e., provided it holds up to a set of capacity zero. This leads to the following result.

Lemma 5.2

Let $\Omega \subset \mathbb R^n$ be open and bounded with $n \ge 1$. Let $\mathcal {A} \subset \Omega$ be compact and assume (18a) holds for $\mathcal {A}$. If $d \in H^1_0(\Omega )$ with $d \ge 0,\,\,a.e.\,{\mathcal {A}}$, then $\tilde{d} \ge 0,\,\,q.e.\,\text {int}({\mathcal {A}})$.

Remark 5.3

In other words, in such a situation, there is no difference between almost everywhere and quasi-everywhere on open sets.

Proof

Define $\mathcal {B} \subset \Omega$ by $\mathcal {B} := \left\{ x \in \Omega | \tilde{d}(x) < 0\right\} ,$ and note that $\mathcal {B} \cap \mathcal {A}$ has Lebesgue measure zero. By definition of quasi-continuity, there exists a non-increasing sequence of open sets $\Omega _n \subset \Omega$ such that $\tilde{d}|_{\Omega \setminus \Omega _n}$ is continuous and $\text {cap}(\Omega _n) \rightarrow 0$ as $n \rightarrow +\infty$. It follows then that $\mathcal {B} \setminus \Omega _n$ is open in $\Omega \setminus \Omega _n$ and, consequently, $\mathcal {B} \cup \Omega _n$ is open in $\Omega$. By (18a), $\mathcal {B} \cap \text {int}(\mathcal {A}) \cup \Omega _n$ is open as well. Following the proof of Lemme 3.3.30 in [19] or Lemma 6.49 in [7] we deduce $\tilde{d} \ge 0, \,\,q.e.\,\text {int}({\mathcal {A}}).$ $\square$