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Journal of Convex Analysis 07 (2000), No. 2, 375--394
Copyright Heldermann Verlag 2000

Compactly Epi-Lipschitzian Convex Sets and Functions in Normed Spaces

Jonathan Borwein
Dept. of Mathematics and Statistics, Simon Fraser University, Burnaby BC, V5A 1S6, Canada

Yves Lucet
Dept. of Mathematics and Statistics, University of Victoria, Victoria BC, V8W 3P4, Canada

Boris Mordukhovich
Dept. of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.

We provide several characterizations of compact epi-Lipschitzness for closed convex sets in normed vector spaces. In particular, we show that a closed convex set is compactly epi-Lipschitzian if and only if it has nonempty relative interior, finite codimension, and spans a closed subspace. Next, we establish that all boundary points of compactly epi-Lipschitzian sets are proper support points. We provide the corresponding results for functions by using inf-convolutions and the Legendre-Fenchel transform. We also give an application to constrained optimization with compactly epi-Lipschitzian data via a generalized Slater condition involving relative interiors.

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