Advertisement

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
Article

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.

Keywords

Generalized convexity Duality Optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Asplund, Differentiability of the metric projection in finite dimensional Euclidean space.Proc. Amer. Math. Soc. 38 (1973) 218–219.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J.-P. Aubin and I. Ekeland, Estimates of the duality gap in nonconvex optimization.Math. Oper. Res. 1 (1976) 225–245.zbMATHMathSciNetGoogle Scholar
  3. [3]
    E. Balas, J.M. Tama and J. Tind, Sequential convexification in reverse convex and disjunctive programming.Math. Prog. 44 (1989) 337–350.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    B.C. Eaves and W.I. Zangwill, Generalized cutting plane algorithm.SIAM J. Control 9 (1971) 529–542.CrossRefMathSciNetGoogle Scholar
  5. [5]
    I. Ekeland and R. Temam,Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976).zbMATHGoogle Scholar
  6. [6]
    P. Hartman, On functions representable as a difference of convex functions.Pacific J. Math. 9 (1959) 707–713.zbMATHMathSciNetGoogle Scholar
  7. [7]
    J.B. Hiriart-Urruty. Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. in: J. Ponstain, ed.,Lectures Notes in Economics and Mathematical Systems. Vol. 256 (Springer, New York, 1984) 37–70.Google Scholar
  8. [8]
    J.B. Hiriart-Urruty, Conditions necessaires et suffisantes d’optimalite globale en optimization de differences de deux fonctions convexes.C.R. Acad. Sci. Paris 309 (1989) 459–462.zbMATHMathSciNetGoogle Scholar
  9. [9]
    R. Horst and H. Tuy,Global Optimization: Deterministic Approaches (Springer, Berlin, 1990).zbMATHGoogle Scholar
  10. [10]
    T.C. Hu, V. Klee and D. Larman, Optimization of globally convex functions.SIAM Control Optim. 27 (1989) 1026–1047.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    A.D. Ioffe and V.M. Tikhomirov,Theory of External Problems (North-Holland, Amsterdam, 1979). (A translation from the Russian edition NAUKA, Moscow, 1974.)Google Scholar
  12. [12]
    G.L. Nemhauser and LA. Wolsey,Integer and Combinatorial Optimization (John Wiley and Sons, New York, 1988).zbMATHGoogle Scholar
  13. [13]
    J.-P. Penot and M.L. Bougeard, Approximation and decomposition properties of some classes of locally d.c. functions.Math. Prog. 41 (1988) 195–227.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).zbMATHGoogle Scholar
  15. [15]
    I. Singer, Maximization of lower semi-continuous convex functionals on bounded subsets of locally convex spaces I: Hyperplane theorems.Appl. Math. Optim. 5 (1979), 349–362.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    P.T. Thach, D.c. sets. d.c. functions and nonlinear equations.Math. Prog. 58 (1987) 415–428.CrossRefMathSciNetGoogle Scholar
  17. [17]
    P.T. Thach, Quasiconjugates of functions and duality relationship between quasi-convex mmimization under a reverse convex constraint and quasiconvex maximization under a convex constraint, and applications.J. Math. Anal. Appl. 159 (1991) 299–322.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    J.F. Toland, Duality in nonconvex optimization,J. Math. Anal. Appl. 66 (1978) 399–415.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    H. Tuy, Global minimization of a difference of two convex functions.Math. Prog. Study 30 (1987) 150–182.zbMATHGoogle Scholar
  20. [20]
    S.A. Vavasis,Nonlinear Optimization. Complexity Issues (Oxford University Press, Oxford, UK, 1991).zbMATHGoogle Scholar
  21. [21]
    J.-P. Vial. Strong and weak convexity of sets and functions.Math. Oper. Res. 8 (1983) 231–259.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    Y. Yajima and H. Konno, Efficient algorithms for solving rank two and rank three bilinear programming problems. J. Global Optimization 1 (1991) 65–82.zbMATHCrossRefMathSciNetGoogle Scholar

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

Personalised recommendations