On the degree and separability of nonconvexity and applications to optimization problems
Thach, Phan Thien
Konno, Hiroshi

Date: 1997
Abstract: We study qualitative indications for d. c. representations of closed sets in and functions on Hilbert spaces. The first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity. We show that a closed set is weakly closed if this indication is finite. Using this result we can prove the solvability of nonconvex minimization problems. By duality a minimization problem on a feasible set in which this indication is low, can be reduced to a quasi-concave minimization over a convex set in a low-dimensional space. The second indication is the separability which can be incorporated in solving dual problems. Both the index of nonconvexity and the separability can be characteristics to "good" d. c. representations. For practical computation we present a notion of clouds which enables us to obtain a good d. c. representation for a class of nonconvex sets. Using a generalized Caratheodory's theorem we present various applications of clouds. .
Rights: Tots els drets reservats.
Language: Anglès
Document: Article ; recerca ; Versió publicada
Subject: Generalized convexity ; Duality ; Optimization
Published in: Mathematical Programming, vol. 77 n. 1 (1997) p. 23-47, ISSN 0025-5610

25 p, 1.0 MB
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 Record created 2006-03-13, last modified 2023-06-03

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