Density of convex intersections and applications.

M Hintermüller,S Rösel,C N Rautenberg

doi:10.1098/rspa.2016.0919

Abstract

In this paper, we address density properties of intersections of convex sets in several function spaces. Using the concept of Γ-convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite-element discretizations of sets associated with convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems.

Highlights

Convex constraint sets K as subsets of an infinitedimensional Banach space X are common to many fields in mathematics such as calculus of variations, variational inequalities and control theory
We investigate the stability of a large number of perturbation and dualization approaches to variational inequality and constrained optimization problems in the context of density properties of a convex constraint set
If the intersection with certain dense subspaces is dense in the feasible set, one may prove the unconditional consistency of various perturbation schemes including Galerkin approximations

Summary

Introduction

Convex constraint sets K as subsets of an infinitedimensional Banach space X are common to many fields in mathematics such as calculus of variations, variational inequalities and control theory. [1,2,3] for fundamental concepts in variational analysis In this vein, given a set of functions satisfying an arbitrary convex constraint, density properties of more regular functions satisfying the same restriction are of utmost importance. Making use of the density results provided by the preceding sections, we prove several new Mosco convergence results in the Hilbert spaces L2, H1 and H(div) for different types of finite-element discretizations of K, even for discontinuous obstacles α. The density property arises as an essential condition for the equivalent reformulation of the problem in the Hilbert space H(div) by means of Fenchel duality

Motivation

Density results for continuous obstacles

Conclusion