Abstract
We consider the local and global topological structure of the feasible set M of a generalized semi-infinite optimization problem. Under the assumption that the defining functions for M are affine-linear with respect to the index variable and separable with respect to the index and the state variable, M can globally be written as the finite union of certain open and closed sets. Here, it is not necessary to impose any kind of constraint qualification on the lower level problem.
In fact, these sets are level sets of the lower level Lagrangian, and the open sets are generated exactly by Lagrange multiplier vectors with vanishing entry corresponding to the lower level objective function. This result gives rise to a first order necessary optimality condition for the considered generalized semi-infinite problem.
Finally it is shown that the description of M by open and closed level sets of the lower level Lagrangian locally carries over to points of the so-called mai-type, where neither the linearity nor the separability assumption is satisfied.
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References
C. Berge, Topological Spaces (Oliver and Boyd, Edinburgh, London, 1963).
T.J. Graettinger and B.H. Krogh, The acceleration radius: a global performance measure for robotic manipulators, IEEE J. of Robotics and Automation 4 (1988) 60–69.
R. Hettich and K.O. Kortanek, Semi–infinite programming: theory, methods, and applications, SIAM Review 35 (1993) 380–429.
R. Hettich and G. Still, Second order optimality conditions for generalized semi–infinite programming problems, Optimization 34 (1995) 195–211.
R. Hettich and P. Zencke, Numerische Methoden der Approximation und semi–infiniten Optimierung (Teubner Studienbücher, Stuttgart, 1982).
H.Th. Jongen, J.–J. Rückmann and O. Stein, Disjunctive optimization: critical point theory, JOTA 93 (1997) 321–336.
H.Th. Jongen, J.–J. Rückmann and O. Stein, Generalized semi–infinite optimization: a first order optimality condition and examples, Mathematical Programming 83 (1998) 145–158.
A. Kaplan and R. Tichatschke, On a class of terminal variational problems, in: Parametric Optimization and Related Topics IV, eds. J. Guddat, H.Th. Jongen, F. Nožička, G. Still and F. Twilt (Peter Lang, Frankfurt am Main, 1997) pp. 185–199.
W. Krabs, On time–minimal heating or cooling of a ball, in: International Series of Numerical Mathematics, Vol. 81 (Birkhäuser, Basel, 1987) pp. 121–131.
S. Pickl and G.–W. Weber, Generalized semi–infinite optimization: An iterative approach based on approximative linear programming problems, Preprint, Technical University Darmstadt (1997).
R. Reemtsen and J.–J. Rückmann, eds., Semi–Infinite Programming (Kluwer Academic, Boston, 1998).
J.–J. Rückmann and A. Shapiro, First–order optimality conditions in generalized semi–infinite programming, JOTA 101 (1999) 677–691.
J.–J. Rückmann and A. Shapiro, Second–order optimality conditions in generalized semi–infinite programming, submitted.
O. Stein, The Reduction Ansatz in absence of lower semi–continuity, in: Parametric Optimization and Related Topics V (Peter Lang, Frankfurt am Main, 2000) pp. 165–178, to appear.
O. Stein, On level sets of marginal functions, Optimization, to appear.
O. Stein and G. Still, On optimality conditions for generalized semi–infinite programming problems, JOTA 104 (2000) 443–458.
J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimensions I (Springer, Berlin, 1970).
G.–W. Weber, Generalized semi–infinite optimization: On some foundations, Journal of Computational Technologies 4 (1999) 41–61.
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Rückmann, JJ., Stein, O. On Linear and Linearized Generalized Semi-Infinite Optimization Problems. Annals of Operations Research 101, 191–208 (2001). https://doi.org/10.1023/A:1010972524021
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DOI: https://doi.org/10.1023/A:1010972524021