Journal of Convex Analysis, Vol. 7, No. 1, pp. 73-94 (2000)

Piecewise Affine Selections for Piecewise Polyhedral Multifunctions and Metric Projections

M. Finzel and W. Li

Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054 Erlangen, Germany, and Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529-0077, USA,

Abstract: Piecewise polyhedral multifunctions are the set-valued version of piecewise affine functions. We investigate selections of piecewise polyhedral multifunctions, in particular, the least norm selection and continuous extremal point selections.
A special class of piecewise polyhedral multifunctions is the collection of metric projections $\Pi_{K,P}$ from $\mathbb{R}^n$ (endowed with a polyhedral norm $\|\cdot\|_P$) to a polyhedral subset $K$ of $\mathbb{R}^n$. As a consequence, the two types of selections are piecewise affine selections for $\Pi_{K,P}$. Moreover, if $\Pi_{K,\infty}$ and $\Pi_{K,1}$ are the metric projection onto $K$ in $\mathbb{R}^n$ endowed with the $\ell_\infty$-norm and the $\ell_1$-norm, respectively, we prove that $\Pi_{K,1}$ has a piecewise affine and quasi-linear extremal point selection when $K$ is a subspace, and that the strict best approximation $\operatorname{\mathbf{sba}}_K(x)$ of $x$ in $K$ is a piecewise affine selection for $\Pi_{K,\infty}$.

Full text of the article:

[Previous Article] [Next Article] [Contents of this Number]
© 2000 ELibM for the EMIS Electronic Edition