Journal of Global Optimization

, Volume 3, Issue 3, pp 311–324 | Cite as

A generalized duality and applications

  • Phan Thien Thach


The aim of this paper is to present a nonconvex duality with a zero gap and its connection with convex duality. Since a convex program can be regarded as a particular case of convex maximization over a convex set, a nonconvex duality can be regarded as a generalization of convex duality. The generalized duality can be obtained on the basis of convex duality and minimax theorems. The duality with a zero gap can be extended to a more general nonconvex problems such as a quasiconvex maximization over a general nonconvex set or a general minimization over the complement of a convex set. Several applications are given.

Key words

Nonconvex duality zero gap global optimization 


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  1. 1.
    Aubin, J. P. and Ekeland, I. (1976), Estimates of the duality gap in nonconvex optimization,Math. Oper. Res. 1, 225–245.Google Scholar
  2. 2.
    Burkard, R. E., Hamacher, H. W. and Tind, J. (1982), On abstract duality in mathematical programming,Zeitschrift fur Oper. Res. 26, 197–209.Google Scholar
  3. 3.
    Falk, J. E. and Huffman, K. L. (1976), A successive underestimating method for concave minimization problems,Math. Oper. Res. 1, 251–259.Google Scholar
  4. 4.
    Hillestad, R. J. and Jacobsen, S. E. (1980), Reverse convex programming,Appl. Math. Optim. 6, 63–78.Google Scholar
  5. 5.
    Hiriart-Urruty, J. B. (1984), Generalized differentiability, duality and optimization for problems dealing with differences of convex functions,Lecture Notes in Economics and Mathematical Systems, ed. by J. Ponstain, 256, 37–70.Google Scholar
  6. 6.
    Horst, R. (1980), A note on the dual gap in nonconvex optimization and a very simple procedure for bild evaluation type problems,European J. Oper. Res. 5, 205–210.Google Scholar
  7. 7.
    Horst, R. and Tuy, H. (1990),Global Optimization, Springer-Verlag.Google Scholar
  8. 8.
    Konno, H. and Kuno, T. (1992), Linear multiplicative programming,Math. Prog. 56, 51–64.Google Scholar
  9. 9.
    Konno, H. and Yajima, Y. (1982), Minimizing and maximizing the product of linear fractional functions,Recent Advances in Global Optimization, Princeton University Press, 259–273.Google Scholar
  10. 10.
    Muu, L. D., (1985), A convergent algorithm for solving linear programs with an additional reverse convex constraint,Kybernetika (Praha) 21, 428–435.Google Scholar
  11. 11.
    Oettli, W. (1981), Optimality condition involving generalized convex mappings,Generalized Concavity in Optimization and Economics, ed. by S. Schaible and W. T. Ziemba, Academic Press, 227–238.Google Scholar
  12. 12.
    Oettli, W. (1982), Optimality condition for programming problems involving multivalued mapping,Modern Applied Mathematics, ed. by B. Korte, North-Holland Publishing Company, 196–226.Google Scholar
  13. 13.
    Pardalos, P. M. and Rosen, J. B. (1987), Constrained global optimization: algorithms and applications,Lecture Notes in Computer Science, Springer-Verlag, 268.Google Scholar
  14. 14.
    Pshenichnyyi, B. N. (1971), Lecons sur jeux differentials, controle optimal et jeux differentiels,Cahiers de IIRIA, no. 4.Google Scholar
  15. 15.
    Rockafellar, R. T. (1970),Convex Analysis, Princeton University Press, Princeton, NJ.Google Scholar
  16. 16.
    Rosen, J. B. and Pardalos, P. M. (1986), Global minimization of large-scale constrained concave quadratic problems by separable programming,Math. Prog. 34, 163–174.Google Scholar
  17. 17.
    Singer, I. (1980), Minimization of continuous convex functionals on complements of convex sets of locally convex spaces,Optimization 11, 221–234.Google Scholar
  18. 18.
    Thach, P. T. (1991a), Quasiconjugates of functions, duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint, and applications,J. Math. Anal. Appl. 159, 299–322.Google Scholar
  19. 19.
    Thach, P. T., Burkard, R., and Oettli, W. (1991), Mathematical programs with a two-dimensional reverse convex constraint,J. Global Optimization 1, 145–154.Google Scholar
  20. 20.
    Thach, P. T. and Tuy, H. (1990),Dual Outer Approximation Methods for Concave Programs and Reverse Convex Programs, IHSS 90-30, Institute of Human and Social Sciences, Tokyo Institute of Technology.Google Scholar
  21. 21.
    Thach, P. T. (1991b),Global optimality criterions and a duality with a zero gap in nonconvex optimization problems, Preprint, Department of Mathematics, Trier University.Google Scholar
  22. 22.
    Thach, P. T. and Konno, H. (1992),A Generalized Dantzig-Wolfe Decomposition Principle for a Class of Nonconvex Programming Problems, IHSS 92-47, Institute of Human and Social Sciences, Tokyo Institute of Technology.Google Scholar
  23. 23.
    Thoai, N. V. and Tuy, H. (1980), Convergent algorithms for minimizing a concave function,Math. Oper. Res. 5, 556–566.Google Scholar
  24. 24.
    Tind, J. and Wolsey, L. A. (1981), An elementary survey of general duality theory in mathematical programming,Math. Prog. 21, 241–261.Google Scholar
  25. 25.
    Toland, J. F. (1978), Duality in nonconvex optimization,J. Math. Anal. Appl. 66, 399–415.Google Scholar
  26. 26.
    Tuy, H. (1964), Concave programming under linear constraints,Doklady Akademia Nauka SSSR 159, 32–35.Google Scholar
  27. 27.
    Tuy, H. (1987), Convex programs with an additional reverse convex constraint,J. Optim. Theory and Appl. 52, 463–486.Google Scholar
  28. 28.
    Tuy, H. (1987), A general deterministic approach to global optimization via d.c. programming,Mathematics Studies 129, 273–303.Google Scholar
  29. 29.
    Tuy, H. (1991), Polyhedral annexation, dualization and dimension reduction technique in global optimization,J. Global Optimization 1, 229–244.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Phan Thien Thach
    • 1
  1. 1.Institute of Human and Social SciencesTokyo Institute of TechnologyJapan

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