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Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems. (English) Zbl 1188.90261
Summary: This paper is devoted to the study of nonsmooth generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be locally Lipschitz. We introduce a constraint qualification which is based on the Mordukhovich subdifferential. Then, we derive a Fritz-John type necessary optimality condition. Finally, interrelations between the new and the existing constraint qualifications such as the Mangasarian-Fromovitz, linear independent, and the Slater are investigated.

90C34Semi-infinite programming
90C40Markov and semi-Markov decision processes
49J52Nonsmooth analysis (other weak concepts of optimality)
Full Text: DOI
[1] Aubin, J. P.: Optima and equilibria, an introduction to nonlinear analysis, (1998) · Zbl 0930.91001
[2] Borwein, J. M.; Lewis, A. S.: Convex analysis and nonlinear optimization: theory and examples, (2000) · Zbl 0953.90001
[3] Bank, B.; Guddat, J.; Klatte, D.; Kummer, B.; Tammer, K.: Non-linear parametric optimization, (1982) · Zbl 0502.49002
[4] Clarke, F. H.: Optimization and nonsmooth analysis, (1983) · Zbl 0582.49001
[5] Ergenc, T.; Pickl, S. W.; Radde, N.; Weber, G. W.: Generalized semi-infinite optimization and anticipatory systems, International journal of computing anticipatory systems 15, 3-30 (2004)
[6] N. Kanzi, S. Nobakhtian, Optimality conditions for non-smooth semi-infinite programming, in Optimization, (2009) 1 -- 11. · Zbl 1172.90019
[7] Kanzi, N.; Nobakhtian, S.: Nonsmooth semi-infinite programming problems with mixed constraints, Journal of mathematical analysis and applications 131, 170-181 (2009) · Zbl 1172.90019 · doi:10.1016/j.jmaa.2008.10.009
[8] Graettinger, T. J.; Krogh, B. H.: The acceleration radius: A global performance measure for robotic manipulators, IEEE journal of robotics and automation 4, 60-69 (1998)
[9] A. Hoffman, R. Reinhardt, On reverse Chebyshev approximation problems, Preprint M 08/94, Facualty of Mathematics and Natural Sciences, Technical University of Ilmenau, 1994.
[10] Jongen, H. T.; Rückmann, J. J.; Stein, O.: Generalized semi-infinite optimization: A first-order optimality condition and examples, Mathematical programming 83, 145-158 (1998) · Zbl 0949.90090 · doi:10.1007/BF02680555
[11] Mangasarian, O. L.; Fromovitz, S.: The fritz -- John necessary optimality conditions in the presence of equality and inequality constraints, Journal of mathematical analysis and applications 17, 37-47 (1967) · Zbl 0149.16701 · doi:10.1016/0022-247X(67)90163-1
[12] Mordukhovich, B. S.: Variational analysis and generalized differentiation. I. basic theory, (2006)
[13] Mordukhovich, B. S.: Variational analysis and generalized differentiation. II. applications, (2006)
[14] Mordukhovich, B. S.; Nam, N. M.: Variational stability and marginal functions via generalized differentiation, Mathematics of operations research 30, 800-816 (2005) · Zbl 1284.90083
[15] Mordukhovich, B. S.; Nam, N. M.; Yen, N. D.: Subgradients of marginal functions in parametric mathematical programming, Mathematical programming 116, No. 1 -- 2, 369-396 (2009) · Zbl 1177.90377 · doi:10.1007/s10107-007-0120-x
[16] Robinson, S. M.: Generalized equations and their solutions, part II: Applications to nonlinear programming, Mathematical programming study 19, 200-221 (1982) · Zbl 0495.90077
[17] Rockafellar, R. T.: Convex analysis, Princeton mathematical series 28 (1970) · Zbl 0193.18401
[18] Rockafellar, R. T.; Wets, J. B.: Variational analysis, (1998) · Zbl 0888.49001
[19] Rückmann, J. J.; Stein, O.: On convex lower level problems in generalized semi-infinite optimization, Semi-infinite programming-recent advances, 121-134 (2001) · Zbl 1073.90561
[20] Rückman, J. J.; Shapiro, A.: First-order optimality conditions in generalized semi-infinite programming, Journal of optimization theory and applications 101, No. 3, 677-691 (1999) · Zbl 0956.90055 · doi:10.1023/A:1021746305759
[21] Rückmann, J. J.; Shapiro, A.: Second-order optimality conditions in generalized semi-infinite programming, Set-valued analysis 9, 169-186 (2001) · Zbl 0984.90056 · doi:10.1023/A:1011239607220
[22] Tanino, T.; Ogawa, T.: An algorithm for solving two-level convex optimization problem, International journal of system science 15, 163-174 (1984) · Zbl 0542.90075 · doi:10.1080/00207728408926552
[23] M. Slater, Lagrange multipliers revisited: A contribution to non-linear programming, Cowles Commission Discussion Paper Math, 1950, p. 430.
[24] Stein, O.: First order optimality conditions for degenerate index sets in generalized semi-infinite programming, Mathematics of operations research 26, 565-582 (2001) · Zbl 1073.90562 · doi:10.1287/moor.26.3.565.10583
[25] Stein, O.: Bi-level strategies in semi-infinite programming, (2003) · Zbl 1103.90094
[26] Stein, O.; Still, G.: On optimality conditions for generalized semi-infinite programming problems, Journal of optimization theory and applications 104, No. 2, 443-458 (2000) · Zbl 0964.90047 · doi:10.1023/A:1004622015901
[27] Still, G.: Generalized semi-infinite programming: theory and methods, European journal of operational research 119, 301-313 (1999) · Zbl 0933.90063 · doi:10.1016/S0377-2217(99)00132-0
[28] Still, G.: Solving generalized semi-infinite programs by reduction to simpler problems, Optimization 53, No. 1, 19-38 (2004) · Zbl 1079.90144 · doi:10.1080/02331930410001661190
[29] Vazquez, F. G.; Rückmann, J. J.: Extensions of the Kuhn -- Tucker constraint qualification to generalized semi-infinite programming, SIAM journal on optimization 15, No. 3, 926-937 (2005) · Zbl 1114.90141 · doi:10.1137/S1052623403431500
[30] G.W. Weber, Generalized semi-infinite optimization and relative topics, in: Lemgo, K.H. Hofmann, R. Wille (Eds.), Research and Exposition in Mathematics, vol. 29, Heldermann Publishing House, 2003. · Zbl 1056.90134
[31] Weber, G. W.: Generalized semi-infinite optimization: theory and applications in optimal control and discrete optimization, Journal of statistics and management systems 5, 359-388 (2002) · Zbl 1079.90609
[32] Weber, G. W.; Tezel, A.: On generalized semi-infinite optimization of genetic networks, Top 15, No. 1, 65-77 (2007) · Zbl 1123.93018 · doi:10.1007/s11750-007-0003-6
[33] Weber, G. W.; Tezel, A.: New views: generalized semi-infinite optimization of genetic networks, TOP, the operational research journal of SEIO 14, No. 1, 48-55 (2006)
[34] Ye, J. J.; Wu, S. Y.: First order optimality conditions for generalized semi-infinite programming problems, Journal of optimization theory and applications 137, No. 2, 419-434 (2008) · Zbl 1152.90011 · doi:10.1007/s10957-008-9352-z