Journal of Convex Analysis 17 (2010), No. 1, 211--227
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
Soon-Yi 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
[Abstract-pdf]
This paper deals with an infinite-dimensional 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 semi-infinite 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, semi-infinite programming.
MSC: 90C48, 46A22, 90C90
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