×

zbMATH — the first resource for mathematics

Duality principles for optimization problems dealing with the difference of vector-valued convex mappings. (English) Zbl 1041.90068
Consider the following equivalent problems: \[ \min f(x)+g\bigl( G(x)-H(x) \bigr), \text{ on a vectorial space},\tag{P} \] and \[ \min f(x), \text{ for } G(x)-H(x)\leq 0,\tag{R} \] where \(f,g\) are convex functions and \(G,H\) are vector-valued mappings that are convex with respect to a partial vectorial order for which \(g\) is nondecreasing.
In this paper the author obtains duality formulas for problems (P) and (R) in terms of the Legendre-Fenchel conjugates of the data functions. The author also provides relations between the optimal solutions of primal and dual problems and a general necessary optimality condition. In particular the author applies the results to the problem of minimization of the composite of a convex mapping with a nonincreasing convex function and the minimization of the upper envelope of a family of concave functions.

MSC:
90C46 Optimality conditions and duality in mathematical programming
49J52 Nonsmooth analysis
49N15 Duality theory (optimization)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] HIRIART-URRUTY, J. B., Generalized Differentiability, Duality, and Optimization for Problems Dealing with Differences of Convex Functions, Convexity and Duality in Optimization (Groningen, 1984), Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, Vol. 256, pp. 37–70, 1985.
[2] TUY, H., DC Optimization: Theory, Methods, and Algorithms, Handbook of Global Optimization; Nonconvex Optimization and Its Aplications, Kluwer Academic Publishers, Dordrecht, Holland, Vol. 2, pp. 149–216, 1995. · Zbl 0832.90111
[3] HORST, R., and THOAI, N. V., DC Programming Overview, Journal of Optimization Theory and Applications, Vol. 103, pp. 1–43, 1999. · Zbl 1073.90537 · doi:10.1023/A:1021765131316
[4] TOLAND, J. F., Duality in Nonconvex Optimization, Journal of Mathematical Analysis and Applications, Vol. 66, pp. 399–415, 1978. · Zbl 0403.90066 · doi:10.1016/0022-247X(78)90243-3
[5] SINGER, I., A Fenchel-Rockafellar Type Duality Theorem for Maximization, Bulletin of the Australian Mathematical Society, Vol. 20, pp. 193–198, 1979. · Zbl 0404.90101 · doi:10.1017/S0004972700010844
[6] VOLLE, M., Concave Duality: Application to Problems Dealing with Difference of Functions, Mathematical Programming, Vol. 41A, pp. 261–278, 1988. · Zbl 0665.49016 · doi:10.1007/BF01580767
[7] LEMAIRE, B., and VOLLE, M., Duality in DC Programming, Nonconvex Optimization and Its Applications: Generalized Convexity, Generalized Monotonicity, Edited by J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, Kluwer, Dordrecht, Holland, Vol. 27, pp. 331–345, 1998. · Zbl 0961.90127
[8] LEMAIRE, B., and VOLLE, M., A General Duality Scheme for Nonconvex Minimization Problems with a Strict Inequality Constraint, Journal of Global Optimization, Vol. 13, pp. 317–327, 1998. · Zbl 0934.90058 · doi:10.1023/A:1008285930026
[9] MARTINEZ-LEGAZ, J. E., and VOLLE, M., Duality in DC Programming: The Case of Several DC Constraints, Journal of Mathematical Analysis and Applications, Vol. 237, pp. 657–671, 1999. · Zbl 0946.90064 · doi:10.1006/jmaa.1999.6496
[10] MARTINEZ-LEGAZ, J. E., and VOLLE, M., Duality for DC Optimization over Compact Sets, Optimization Theory, Edited by F. Giannessi, P. Pardalos, and T. Rapcsák, Kluwer Academic Publishers, pp. 139–146, 2001. · Zbl 1041.90067
[11] LEVIN, V. L., Sur le Sous-Différentiel de Fonctions Composées, Doklady Akademia Nauk, Belorussian Academy of Sciences, Vol. 194, pp. 28–29, 1970.
[12] LEMAIRE, B., Subdifferential of a Convex Composite Functional: Application to Optimal Control in Variational Inequalities, Nondifferentiable Optimization, Sopron, Hungary, 1984.
[13] COMBARI, C., LAGHDIR, M., and THIBAULT, L., Sous-Différentiels de Fonctions Convexes Composeés, Annales des Sciences Mathématiques du Québec, Vol. 18, pp. 119–148, 1994. · Zbl 0840.49009
[14] ROCKAFELLAR, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.18401
[15] LAURENT, P. J., Approximation et Optimisation, Hermann, Paris, France, 1972. · Zbl 0238.90058
[16] LAGHDIR, M., and VOLLE, M., A General Formula for the Horizon Fuction of a Convex Composite Function, Archiv der Mathematik, Vol. 73, pp. 291–302, 1999. · Zbl 1028.90068 · doi:10.1007/s000130050401
[17] PENNANEN, T., Graph Convex Mappings and K-Convex Functions, Journal of Convex Analysis, Vol. 6, pp. 235–266, 1999. · Zbl 0953.90046
[18] RAFFIN, C., Sur les Programmes Convexes Définis dans les Espaces Vectoriels Topologiques, Annales de l ’Institut Fourier, Vol. 20, pp. 457–491, 1970. · Zbl 0195.49601 · doi:10.5802/aif.347
[19] VALADIER, M., Sous-Différentiabilité de Fonctions Convexes à Valeurs dans un Espace Vectoriel Ordonné, Mathematica Scandinavica, Vol. 30, pp. 65–74, 1972. · Zbl 0239.46038
[20] THERA, M., Subdifferential Calculus for Convex Operators, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 78–91, 1981. · Zbl 0467.90078 · doi:10.1016/0022-247X(81)90093-7
[21] BORWEIN, M., Continuity and Differentiability Properties of Convex Operators, Proceedings of the London Mathematical Society, Vol. 44, pp. 420–444, 1982. · Zbl 0487.46026 · doi:10.1112/plms/s3-44.3.420
[22] LEMAIRE, B., Duality in Reverse Convex Optimization, SIAM Journal on Optimization, Vol. 8, pp. 1029–1037, 1998. · Zbl 0914.90263 · doi:10.1137/S1052623495295857
[23] SION, M., On General Minimax Theorems, Pacific Journal of Mathematics, Vol. 8, pp. 171–176, 1958. · Zbl 0081.11502 · doi:10.2140/pjm.1958.8.171
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.