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Generalized semi-infinite optimization: A first order optimality condition and examples. (English) Zbl 0949.90090

Summary: We consider a Generalized Semi-Infinite Optimization Problem (GSIP) of the form

min{f(x)xM}( GSIP )

where M={x n h i (x)=0, i=1,,m, G(x,y)0, yY(x)} and all appearing functions are continuously differentiable. Furthermore, we assume that the set Y(x) is compact for all x under consideration and the set-valued mapping Y(·) is upper semi-continuous. The difference with a standard semi-infinite problem lies in the x-dependence of the index set Y. We prove a first order necessary optimality condition of Fritz John type without assuming a constraint qualification or any kind of reduction approach. Moreover, we discuss some geometrical properties of the feasible set M.

90C34Semi-infinite programming
[1]C. Berge, Topological Spaces, Oliver & Boyd, Edinburgh, London, 1963.
[2]W.W. Hogan, Point-to-set maps in mathematical programming, SIAM Rev. 15 (1973) 591–603. · Zbl 0256.90042 · doi:10.1137/1015073
[3]R. Hettich, K.O. Kortanek, Semi-infinite programming: Theory, methods, and applications, SIAM Rev. 35 (1993) 380–429. · Zbl 0784.90090 · doi:10.1137/1035089
[4]R. Hettich, P. Zencke, Numerische Methoden der Approximation und semi-infiniten Optimierung, Teubner Studienbücher, Stuttgart, 1982.
[5]R. Hettich, G. Still, Second order optimality conditions for generalized semi-infinite programming problems, Optimization 34 (1995) 195–211. · Zbl 0855.90129 · doi:10.1080/02331939508844106
[6]T.J. Graettinger, B. H. Krogh The acceleration radius: A global performance measure for robotic manipulators, IEEE J. Robotics and Automation 4 (1988) 60–69. · doi:10.1109/56.772
[7]R. Hettich, G. Still, Semi-infinite programming models in robotics, in: J. Guddat, H.Th. Jongen, B. Kummer, F. Nožička (Eds.), Parametric Optimization and Related Topics II, Akademie Verlag, Berlin, 1991, pp. 112–118.
[8]A. Kaplan, R. Tichatschke, On a class of terminal variational problems in: J. Guddat, H.Th. Jongen, F. Nožička, G. Still, F. Twilt (Eds.), Parametric Optimization and Related Topics IV, Peter Lang Verlag, Frankfurt a.M., 1997, pp. 185–199.
[9]R. Hettich, H.Th. Jongen, Semi-infinite programming: conditions of optimality and applications, in: J. Stoer (Ed.), Optimization Techniques, Part 2, Lecture Notes in Control and Information Science 7, Springer, Heidelberg, New York, 1978, pp. 1–11.
[10]H.Th. Jongen, J.-J. Rückmann, O. Stein, Disjunctive optimization: Critical point theory, J. Optim. Theory Appl. 93 (2) (1997) 321–336. · Zbl 0901.90164 · doi:10.1023/A:1022650006477
[11]E.W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966.
[12]H.Th. Jongen, P. Jonker, F. Twilt, Nonlinear Optimization in n. II. Transversality, Flows, Parametric Aspects, Peter Lang Verlag, Frankfurt a.M., 1986.
[13]H.Th. Jongen, F. Twilt, G.-W. Weber, Semi-infinite optimization: Structure and stability of the feasible set, J. Optim. Theory Appl. 72 (1992) 529–552. · Zbl 0807.90113 · doi:10.1007/BF00939841
[14]M. Kojima, Strongly stable stationary solutions in nonlinear programs, in: S.M. Robinson (Ed.), Analysis and Computation of Fixed Points, Academic Press, New York, 1980, pp. 93–138.
[15]J.-J. Rückmann, On existence and uniqueness of stationary points (submitted for publication).