Mathematical Programming

, Volume 77, Issue 3, pp 23–47 | Cite as

On the degree and separability of nonconvexity and applications to optimization problems

  • Phan Thien Thach
  • Hiroshi Konno


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.


Generalized convexity Duality Optimization 


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Copyright information

© The Mathematical Programming Society, Inc 1997

Authors and Affiliations

  • Phan Thien Thach
    • 1
    • 2
  • Hiroshi Konno
    • 1
  1. 1.Department of Industrial Engineering and ManagementTokyo Institute of TechnologyTokyoJapan
  2. 2.Institute of MathematicsHanoiViet Nam

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