D.C. Representability of closed sets in reflexive Banach spaces and applications to optimization problems

  • P. T. Thach
  • H. Konno
Contributed Papers


A closed subsetM of a Hausdorff locally convex space is called d.c. representable if there are an extended-real valued lsc convex functionf and a continuous convex functionh such that
$$M = \{ x \in X:f(x) - h(x) \leqslant 0\} .$$

Using the existence of a locally uniformly convex norm, we prove that any closed subset in a reflexive Banach space is d.c. representable. For d.c. representable subsets, we define an index of nonconvexity, which can be regarded as an indicator for the degree of nonconvexity. In fact, we show that a convex closed subset is weakly closed when it has a finite index of nonconvexity, and optimization problems on closed subsets with a low index of nonconvexity are less difficult from the viewpoint of computation.

Key Words

Generalized convexity d.c. functions optimization 


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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • P. T. Thach
    • 1
  • H. Konno
    • 1
  1. 1.Department of Industrial Engineering and ManagementTokyo Institute of TechnologyTokyoJapan

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