The operation of infimal convolution

Strömberg Thomas

GDML_Books, (1996), p.

Résumé

AbstractThis paper is a survey article on the theory and applications of infimal convolution. We consider the convex as well as the nonconvex case. In particular, we provide a detailed investigation of the regularizing effects of infimal convolution, and study continuity properties of the operation with respect to notions of variational convergence. Several examples are included and well-known results are complemented, unified or extended in various ways.

CONTENTS1. Introduction and preliminaries....................................................5 1.1. Introduction............................................................................5 1.2. Organization..........................................................................6 1.3. Prerequisites.........................................................................6 1.4. Introductory examples...........................................................92. Elementary properties.............................................................14 2.1. Basic facts...........................................................................14 2.2. Infimal convolution of subadditive functions.........................17 2.3. Semicontinuity, continuity, and exactness............................19 2.4. Two examples......................................................................223. The convex case.....................................................................23 3.1. Basic results........................................................................23 3.2. Differential calculus, and first order differentiability..............28 3.3. Formulas on f ▫ g.................................................................31 3.4. Loss of differentiability.........................................................324. Continuity of the operation of infimal convolution....................33 4.1. Introduction.........................................................................34 4.2. Epi-convergence.................................................................35 4.3. The Mosco topology and the slice topology.........................36 4.4. The affine topology..............................................................38 4.5. The Attouch-Wets topology.................................................395. Regularization.........................................................................41 5.1. Introduction and first results................................................41 5.2. Approximation in Hilbert spaces...........................................47 5.3. Generalized Moreau-Yosida approximation.........................52References..................................................................................55

1991 Mathematics Subject Classification: 41A65, 46N10, 49J27, 49J45, 49J50, 49L25, 52A40, 52A41, 54B20, 65K10.

Détail

Zbl 0858.49010

EUDML-ID : urn:eudml:doc:271746