Minimum Distance to the Complement of a Convex Set: Duality Result

W. Briec
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W. Briec: Institut de Gestion de Rennes

Journal of Optimization Theory and Applications, 1997, vol. 93, issue 2, No 3, 319 pages

Abstract: Abstract The subject of this paper is to study the problem of the minimum distance to the complement of a convex set. Nirenberg has stated a duality theorem treating the minimum norm problem for a convex set. We state a duality result which presents some analogy with the Nirenberg theorem, and we apply this result to polyhedral convex sets. First, we assume that the polyhedral set is expressed as the intersection of some finite collection of m given half-spaces. We show that a global solution is determined by solving m convex programs. If the polyhedral set is expressed as the convex hull of a given finite set of extreme points, we show that a global minimum for a polyhedral norm is obtained by solving a finite number of linear programs.

Keywords: Convex sets; polyhedral convex sets; separation theorem; Banach spaces; duality theory; weak topology; linear programming (search for similar items in EconPapers)
Date: 1997
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Citations: View citations in EconPapers (11)

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DOI: 10.1023/A:1022697822407

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