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Alternative theorems and saddlepoint results for convex programming problems of set functions with values in ordered vector spaces. (English) Zbl 0808.90108
Some alternative theorems and saddle point results for convex set functions with values in an ordered vector space are given. These results are generalizations of some similar convex versions in finite dimensions by Farkas, Ky Fan and two saddle point theorems of the Lagrangian.

90C25Convex programming
90C48Programming in abstract spaces
Full Text: DOI
[1] C. Berge and A. Ghouila-Houri,Programming, Games and Transportation Networks, Wiley and Sons (N. Y., 1965).
[2] J. H. Chou, W. S. Hsia and T. Y. Lee, On multiple objective programming problems with set functions,J. Math. Anal. Appl.,105 (1985), 383--394. · Zbl 0564.90069 · doi:10.1016/0022-247X(85)90055-1
[3] J. H. Chou, W. S. Hsia and T. Y. Lee, Second order optimality conditions for mathematical programming with set functions,J. Austral. Math. Soc. (Ser. B),26 (1985), 284--292. · Zbl 0571.90099 · doi:10.1017/S0334270000004513
[4] J. H. Chou, W. S. Hsia and T. Y. Lee, Epigraphs of convex set functions,J. Math. Anal. Appl.,118 (1986), 247--254. · Zbl 0599.49014 · doi:10.1016/0022-247X(86)90306-9
[5] J. H. Chou, W. S. Hsia and T. Y. Lee, Convex programming with set functions,Rocky Mountain J. Math.,17 (1987), 535--543. · Zbl 0644.90069 · doi:10.1216/RMJ-1987-17-3-535
[6] H. W. Corley, Optimization theory forn-set functions,J. Math. Anal. Appl.,127 (1987), 193--205. · Zbl 0715.90096 · doi:10.1016/0022-247X(87)90151-X
[7] B. D. Craven and J. J. Koliha, Generalizations of Farkas’ theorems,SIAM J. Math. Anal.,8 (1977), 983--997. · Zbl 0408.52006 · doi:10.1137/0508076
[8] Ky Fan, On systems of linear inequalities,Linear Inequalities and Related System (Ann. of Math. Studies 38), Edited by H. W. Kuhn and A. W. Tucker, Princeton Univ. Press (Princeton, N. J., 1956), pp. 99--156. · Zbl 0072.37602
[9] W. S. Hsia and T. Y. Lee, ProperD-solutions of multiobjective programming problems with set functions,J. Optim. Theory Appl.,53 (1987), 247--258. · Zbl 0595.90084 · doi:10.1007/BF00939217
[10] H. C. Lai and S. S. Yang, Saddle point and duality in the optimization theory of convex functions,J. Austral. Math. Soc. (Ser. B),24 (1982), 130--137. · Zbl 0502.49006 · doi:10.1017/S0334270000003635
[11] H. C. Lai, S. S. Yang and Goerge R. Hwang, Duality in mathematical programming of set functions -- On Fenchel duality theorem,J. Math. Anal. Appl.,95 (1983), 223--234. · Zbl 0529.49007 · doi:10.1016/0022-247X(83)90145-2
[12] H. C. Lai and C. P. Ho, Duality theorem of nondifferentiable convex multiobjective programming,J. Optim. Theory Appl.,50 (1986), 407--420. · Zbl 0577.90077 · doi:10.1007/BF00938628
[13] H. C. Lai and L. J. Lin, Moreau-Rockafellar type theorem for convex set functions,J. Math. Anal. Appl.,132 (1988); 558--571. · Zbl 0656.90097 · doi:10.1016/0022-247X(88)90084-4
[14] H. C. Lai and L. J. Lin, The Fenchel-Moreau theorem for set functions,Proc. Amer. Math. Soc.,103 (1988), 85--90. · Zbl 0645.49009 · doi:10.1090/S0002-9939-1988-0938649-4
[15] H. C. Lai and L. J. Lin, Optimality for set functions with values in ordered vector spaces,J. Optim. Theory Appl.,63 (1989), 371--389. · Zbl 0664.90094 · doi:10.1007/BF00939803
[16] H. C. Lai and L. S. Yang, Strong duality for infinite-dimensional vector-valued programming problems,J. Optim. Theory Appl.,62 (1989), 449--466. · Zbl 0654.90098 · doi:10.1007/BF00939816
[17] O. L. Mangasarian,Nonlinear Programming, McGraw-Hill Co. (N. Y., 1969).
[18] R. J. T. Morris, Optimal constrained selection of a measurable subset,J. Math. Anal. Appl.,70 (1979), 546--562. · Zbl 0417.49032 · doi:10.1016/0022-247X(79)90064-7
[19] J. Zowe, A duality theorem for a convex programming problem in order complete vector lattices,J. Math. Anal. Appl.,50 (1975), 273--287. · Zbl 0314.90079 · doi:10.1016/0022-247X(75)90022-0