Abstract
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.
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References
Aubin, J. P. and Ekeland, I. (1976), Estimates of the duality gap in nonconvex optimization,Math. Oper. Res. 1, 225–245.
Burkard, R. E., Hamacher, H. W. and Tind, J. (1982), On abstract duality in mathematical programming,Zeitschrift fur Oper. Res. 26, 197–209.
Falk, J. E. and Huffman, K. L. (1976), A successive underestimating method for concave minimization problems,Math. Oper. Res. 1, 251–259.
Hillestad, R. J. and Jacobsen, S. E. (1980), Reverse convex programming,Appl. Math. Optim. 6, 63–78.
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.
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.
Horst, R. and Tuy, H. (1990),Global Optimization, Springer-Verlag.
Konno, H. and Kuno, T. (1992), Linear multiplicative programming,Math. Prog. 56, 51–64.
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.
Muu, L. D., (1985), A convergent algorithm for solving linear programs with an additional reverse convex constraint,Kybernetika (Praha) 21, 428–435.
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.
Oettli, W. (1982), Optimality condition for programming problems involving multivalued mapping,Modern Applied Mathematics, ed. by B. Korte, North-Holland Publishing Company, 196–226.
Pardalos, P. M. and Rosen, J. B. (1987), Constrained global optimization: algorithms and applications,Lecture Notes in Computer Science, Springer-Verlag, 268.
Pshenichnyyi, B. N. (1971), Lecons sur jeux differentials, controle optimal et jeux differentiels,Cahiers de IIRIA, no. 4.
Rockafellar, R. T. (1970),Convex Analysis, Princeton University Press, Princeton, NJ.
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.
Singer, I. (1980), Minimization of continuous convex functionals on complements of convex sets of locally convex spaces,Optimization 11, 221–234.
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.
Thach, P. T., Burkard, R., and Oettli, W. (1991), Mathematical programs with a two-dimensional reverse convex constraint,J. Global Optimization 1, 145–154.
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.
Thach, P. T. (1991b),Global optimality criterions and a duality with a zero gap in nonconvex optimization problems, Preprint, Department of Mathematics, Trier University.
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.
Thoai, N. V. and Tuy, H. (1980), Convergent algorithms for minimizing a concave function,Math. Oper. Res. 5, 556–566.
Tind, J. and Wolsey, L. A. (1981), An elementary survey of general duality theory in mathematical programming,Math. Prog. 21, 241–261.
Toland, J. F. (1978), Duality in nonconvex optimization,J. Math. Anal. Appl. 66, 399–415.
Tuy, H. (1964), Concave programming under linear constraints,Doklady Akademia Nauka SSSR 159, 32–35.
Tuy, H. (1987), Convex programs with an additional reverse convex constraint,J. Optim. Theory and Appl. 52, 463–486.
Tuy, H. (1987), A general deterministic approach to global optimization via d.c. programming,Mathematics Studies 129, 273–303.
Tuy, H. (1991), Polyhedral annexation, dualization and dimension reduction technique in global optimization,J. Global Optimization 1, 229–244.
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On leave from the Institute of Mathematics, Hanoi, Vietnam.
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Thach, P.T. A generalized duality and applications. J Glob Optim 3, 311–324 (1993). https://doi.org/10.1007/BF01096773
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DOI: https://doi.org/10.1007/BF01096773