
Journal of Convex Analysis 17 (2010), No. 1, 211227 Copyright Heldermann Verlag 2010 A Mathematical Programming Approach to Strong Separation in Normed Spaces Marco A. López Dept. of Statistics and Operations Research, Alicante University, Ap. de Correos 99, 03080 Alicante, Spain marco.antonio@ua.es SoonYi Wu National Cheng Kung University, Tainan, Taiwan Chen Ling Zhejiang University of Finance and Economics, Hangzhou, P. R. China Liqun Qi Polytechnic University of Hong Kong, Hong Kong, P. R. China [Abstractpdf] This paper deals with an infinitedimensional optimization approach to the strong separation of two bounded sets in a normed space. We present an approximation procedure, called Algorithm (A), such that a semiinfinite optimization problem must be solved at each step. Its global convergence is established under certain natural assumptions, and a stopping criterion is also provided. The particular case of strong separation in the space $L_{p} (\mathbb{X}, \mathcal{A}, \mu )$ is approached in detail. We also propose Algorithm (B), which is an implementable modification of Algorithm (A) for separating two bounded sets in $L_{p}([a,b])$, with $[a,b]$ being an interval in $\mathbb{R}$. Some illustative computational experience is reported, and a particular stopping criterion is provided for the case of functions of bounded variation in $L_{2}([a,b])$. Keywords: Strong separation, infinite dimensional optimization, semiinfinite programming. MSC: 90C48, 46A22, 90C90 [ Fulltextpdf (178 KB)] for subscribers only. 