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Generalized semi-infinite programming: Theory and methods. (English) Zbl 0933.90063

Summary: Generalized semi-infinite optimization problems (GSIP) are considered. The difference between GSIP and standard semi-infinite problems (SIP) is illustrated by examples. By applying the ‘Reduction Ansatz’, optimality conditions for GSIP are derived. Numerical methods for solving GSIP are considered in comparison with methods for SIP. From a theoretical and a practical point of view it is investigated, under which assumptions a GSIP can be transformed into an SIP.

MSC:

90C34 Semi-infinite programming
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[1] Coope, I. D.; Price, C. J., Computational numerical experiments in semi-infinite programming, Comput. Optimiz. Appl., 6, 169-189 (1996) · Zbl 0860.90118
[2] S. Görner, R. Reemtsen, Numerical methods for semi-infinite programming: A survey, in: Reemtsen, Rückmann (Eds.), Semi-infinite Programming, Kluwer, Boston, 1998; S. Görner, R. Reemtsen, Numerical methods for semi-infinite programming: A survey, in: Reemtsen, Rückmann (Eds.), Semi-infinite Programming, Kluwer, Boston, 1998
[3] Guddat, J.; Jongen, H. T.; Rückmann, J., On stability and stationary points in nonlinear optimization, J. Australian Math. Soc. Ser. B, 28, 36-56 (1986) · Zbl 0621.49016
[4] R. Hettich, P. Zencke, Numerische Methoden der Approximation und der semi-infiniten Optimierung, Teubner, Stuttgart, 1982; R. Hettich, P. Zencke, Numerische Methoden der Approximation und der semi-infiniten Optimierung, Teubner, Stuttgart, 1982 · Zbl 0481.65033
[5] R. Hettich, G. Still, Semi-infinite programming models in Robotics, in: Guddat et al. (Eds.), Parametric Optimization and Related Topics II, Akademie Verlag, Berlin, 1991; R. Hettich, G. Still, Semi-infinite programming models in Robotics, in: Guddat et al. (Eds.), Parametric Optimization and Related Topics II, Akademie Verlag, Berlin, 1991 · Zbl 0737.90068
[6] Hettich, R.; Kortanek, K., Semi-infinite programming: Theory, methods and applications, SIAM Rev., 35, 3, 380-429 (1993) · Zbl 0784.90090
[7] Hettich, R.; Still, G., Second order optimality conditions for generalized semi-infinite programming problems, Optimization, 34, 195-211 (1995) · Zbl 0855.90129
[8] A. Hoffmann, R. Reinhardt, On reverse Chebyshev approximation problems, Preprint No. M08/94, Technical University of Illmenau, 1994; A. Hoffmann, R. Reinhardt, On reverse Chebyshev approximation problems, Preprint No. M08/94, Technical University of Illmenau, 1994
[9] H. Th Jongen, J.-J. Rückmann, O. Stein, Generalized semi-infinite optimization: A first order optimality condition and examples, Mathematical Programming, 83 (1998) 145-158; H. Th Jongen, J.-J. Rückmann, O. Stein, Generalized semi-infinite optimization: A first order optimality condition and examples, Mathematical Programming, 83 (1998) 145-158 · Zbl 0949.90090
[10] Th Jongen, H.; Rückmann, J.-J.; Stein, O., Disjunctive optimization: Critical point theory, J. Optim. Theory Appl., 93, 2, 321-336 (1997) · Zbl 0901.90164
[11] A. Kaplan, R. Tichatschke, On a class of terminal variational problems, in: Guddat et al. (Eds.), Parametric Optimization and Related Topics IV, Peter Lang Verlag, Frankfurt a.M., 1997; A. Kaplan, R. Tichatschke, On a class of terminal variational problems, in: Guddat et al. (Eds.), Parametric Optimization and Related Topics IV, Peter Lang Verlag, Frankfurt a.M., 1997 · Zbl 0885.49018
[12] E. Levitin, R. Tichatschke, A branch and bound approach for solving a class of generalized semi-infinite programming problems, Journal of Global Optimization 13 (1998) 299-315; E. Levitin, R. Tichatschke, A branch and bound approach for solving a class of generalized semi-infinite programming problems, Journal of Global Optimization 13 (1998) 299-315 · Zbl 0912.90274
[13] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970; R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970 · Zbl 0193.18401
[14] W. Weber, Generalized semi-infinite optimization: On some foundations, Preprint, University of Darmstadt, 1996; W. Weber, Generalized semi-infinite optimization: On some foundations, Preprint, University of Darmstadt, 1996
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