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Zero duality gaps in infinite-dimensional programming. (English) Zbl 0687.90077
We study the following infinite-dimensional programming problem $$ (P)\quad \mu:=\inf f\sb 0(x),\quad subject\quad to\quad x\in C,\quad f\sb i(x)\le,\quad i\in I, $$ where I is an index set with possibly infinite cardinality and C is an infinite-dimensional set. Zero duality gap results are presented under suitable regularity hypotheses for convex- like (nonconvex) and convex infinitely constrained programs (P). Various properties of the value function of the convex-like program and its connections to the regularity hypotheses are studied. Relationships between the zero duality gap property, semicontinuity, and $\epsilon$- subdifferentiability of the value function are examined. In particular, a characterization for the value function without convexity is given, using the $\epsilon$-subdifferential of the value function.
Reviewer: V.Jeyakumar

90C30Nonlinear programming
90C34Semi-infinite programming
49N15Duality theory (optimization)
90C48Programming in abstract spaces
Full Text: DOI
[1] Charnes, A., Cooper, W. W., andKortanek, K. O.,Duality, Harr Programs, and Finite Sequence Spaces, Proceedings of the National Academy of Sciences, Vol. 48, pp. 782-786, 1962. · Zbl 0105.12804 · doi:10.1073/pnas.48.5.783
[2] Glashoff, K., andGustafson, G. A.,Linear Optimization and Approximation, Springer-Verlag, Berlin, Germany, 1983.
[3] Duffin, R. J., andKarlovitz, L. A.,An Infinite Linear Program with a Duality Gap, Management Science, Vol. 12, pp. 122-134, 1965. · Zbl 0133.42604 · doi:10.1287/mnsc.12.1.122
[4] Ekeland, I., andTemam, R.,Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976. · Zbl 0322.90046
[5] Rockafellar, R. T.,Conjugate Duality and Optimization, CBMS Lecture Notes Series, Vol. 162, SIAM Publications, Philadelphia, Pennsylvania, 1974. · Zbl 0296.90036
[6] Charnes, A., Cooper, W. W., andKortanek, K. O.,On Representations of Semi-Infinite Programs Which Have No Duality Gaps, Management Science, Vol. 12, pp. 113-121, 1965. · Zbl 0143.42304 · doi:10.1287/mnsc.12.1.113
[7] Karney, D. F.,A Duality Theorem for Semi-Infinite Convex Programs and Their Finite Subprograms, Mathematical Programming, Vol. 27, pp. 75-82, 1983. · Zbl 0527.49027 · doi:10.1007/BF02591965
[8] Borwein, J. M.,How Special is Semi-Infinite Programming?, Semi-Infinite Programming and Applications, Edited by A. V. Fiacco and K. O. Kortanek, Springer-Verlag, Berlin, Germany, pp. 10-36, 1983. · Zbl 0514.49019
[9] Jeroslow, R. J.,Uniform Duality in Semi-Infinite Convex Optimization, Mathematical Programming, Vol. 27, pp. 144-154, 1983. · Zbl 0556.49008 · doi:10.1007/BF02591942
[10] Ben-Israel, A., Charnes, A., andKortanek, K. O.,Duality and Asymptotic Solvability over Cones, Bulletin of the American Mathematical Society, Vol. 75, pp. 318-324, 1969. · Zbl 0187.17504 · doi:10.1090/S0002-9904-1969-12153-1
[11] Ben-Israel, A., Charnes, A., andKortanek, K. O.,Asymptotic Duality in Semi-Infinite Programming and Convex Core Topology, Rendiconti di Mathematica, Vol. 4, pp. 751-767, 1971. · Zbl 0237.90041
[12] Jeroslow, R. G.,A Limiting Lagrangian for Infinitely Constrained Convex Optimization, Journal of Optimization Theory and Applications, Vol. 33, pp. 479-495, 1981. · Zbl 0427.49030 · doi:10.1007/BF00935754
[13] Kortanek, K. O.,Constructing a Perfect Duality in Infinite Programming, Applied Mathematics and Optimization, Vol. 3, pp. 358-372, 1977. · Zbl 0375.90078
[14] Borwein, J. M.,A Note on Perfect Duality and Limiting Lagrangians, Mathematical Programming, Vol. 18, pp. 330-337, 1980. · Zbl 0438.90072 · doi:10.1007/BF01588327
[15] Fan, K., Minimax Theorems,Proceedings of the National Academy of Sciences, Vol. 39, pp. 42-47, 1953. · Zbl 0050.06501 · doi:10.1073/pnas.39.1.42
[16] Jeyakumar, V.,Convex-Like Alternative Theorems and Mathematical Programming, Optimization, Vol. 16, pp. 643-652, 1985. · Zbl 0581.90079 · doi:10.1080/02331938508843061
[17] Craven, B. D., Gwinner, J., andJeyakumar, V.,Nonconvex Theorems of the Alternative and Minimization, Optimization, Vol. 18, pp. 151-163, 1987. · Zbl 0613.49026 · doi:10.1080/02331938708843228
[18] Borwein, J. M., andJeyakumar, V.,On Convex-Like Lagrangian and Minimax Theorems, University of Waterloo, Research Report No. 24, 1988.
[19] Holmes, R. B.,Geometric Functional Analysis, Springer-Verlag, New York, New York, 1975. · Zbl 0336.46001
[20] Karney, D. F.,Clark’s Theorem for Semi-infinite Convex Programs, Advances in Applied Mathematics, Vol. 2, pp. 7-12, 1981. · Zbl 0456.90069 · doi:10.1016/0196-8858(81)90036-1
[21] Clarke, F.,Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1983. · Zbl 0582.49001
[22] Hiriart-Urruty, J. B.,?-Subdifferential Calculus, Convex Analysis and Applications, Edited by J. P. Aubin and R. B. Vinter, Vol. 57, pp. 43-92, 1982.
[23] Ponstein, J.,Applying Some Modern Developments to Choosing Your Own Lagrange Multipliers, SIAM Review, Vol. 25, pp. 183-199, 1983. · Zbl 0518.90073 · doi:10.1137/1025044
[24] Wolkowicz, H.,Optimality Conditions and Shadow Prices, Mathematical Programming with Data Perturbations, II, Edited by A. V. Fiacco, Marcel Dekker, New York, New York, 1983. · Zbl 0506.90076
[25] Rockafellar, R. T.,Generalized Directional Derivatives and Subgradients of Nonconvex Functions, Canadian Journal of Mathematics, Vol. 32, pp. 257-280, 1980. · Zbl 0447.49009 · doi:10.4153/CJM-1980-020-7
[26] Borwein, J. M., andStrojwas, H. M.,Directional Lipschitzian Mappings and Baire Spaces, Canadian Journal of Mathematics, Vol. 36, pp. 95-130, 1984. · Zbl 0534.46031 · doi:10.4153/CJM-1984-008-7
[27] Geoffrion, A. M.,Duality in Nonlinear Programming: A Simplified Applications-Oriented Development, SIAM Review, Vol. 13, pp. 1-37, 1971. · Zbl 0232.90049 · doi:10.1137/1013001
[28] Kortanek, K. O., andRom, W. O.,Classification Schemes for the Strong Duality of Linear Programming over Cones, Operations Research, Vol. 7, pp. 1571-1585. · Zbl 0236.90045
[29] Kallina, C., andWilliams, A. C.,Linear Programming in Reflexive Spaces, SIAM Review, Vol. 13, pp. 350-376, 1971. · Zbl 0224.90042 · doi:10.1137/1013065
[30] Kortanek, K. O.,Classifying Convex Extremum Problems over Linear Topologies Having Separation Properties, Journal of Mathematical Analysis and Applications, Vol. 46, pp. 725-755, 1974. · Zbl 0283.90043 · doi:10.1016/0022-247X(74)90270-4
[31] Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1969. · Zbl 0186.23901
[32] Dedieu, J. P., Critères de Fermeture pour l’Image d’un Fermé Nonconvex par une Multiplication, Comptes Rendus de l’Académie des Sciences, Vol. 287, pp. 941-943, 1978. · Zbl 0401.46005
[33] Anderson, E., andNash, P.,Linear Programming in Infinite-Dimensional Spaces, John Wiley and Sons, Chichester, England, 1987.