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Necessary optimality conditions for Stackelberg problems. (English) Zbl 0802.49007
Summary: First-order necessary optimality conditions are derived for a class of two-level Stackelberg problems in which the followers’ lower-level problems are convex programs with unique solutions. To this purpose, generalized Jacobians of the marginal maps corresponding to followers’ problems are estimated. As illustrative examples, two discretized optimum design problems with elliptic variational inequalities are investigated. The theoretical results may be used also for the numerical solution of the Stackelberg problems considered by nondifferentiable optimization methods.

MSC:
49J40 Variational inequalities
Software:
NLPQL
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[1] Von Stackelberg, H.,The Theory of the Market Economy, Oxford University Press, Oxford, England, 1952.
[2] Simaan, M., andCruz, J. B.,On the Stackelberg Strategy in Nonzero-Sum Games, Journal of Optimization Theory and Applications, Vol. 11, pp. 535-555, 1973. · Zbl 0243.90056
[3] Loridan, P., andMorgan, J.,A Theoretical Approximation Scheme for Stackelberg Problems, Journal of Optimization Theory and Applications, Vol. 61, pp. 95-110, 1989. · Zbl 0642.90107 · doi:10.1007/BF00940846
[4] Basar, T., andOlsder, G. J.,Dynamic Noncooperative Game Theory, Academic Press, New York, New York, 1982. · Zbl 0479.90085
[5] Shimizu, K., andAiyoshi, E.,A New Computational Method for Stackelberg and Minimax Problems by Use of a Penalty Method, IEEE Transactions on Automatic Control, Vol. 26, pp. 460-466, 1981. · Zbl 0472.93004 · doi:10.1109/TAC.1981.1102607
[6] Outrata, J. V.,On the Numerical Solution of a Class of Stackelberg Problems, Zeitschrift für Operations Research, Vol. 4, pp. 255-278, 1990. · Zbl 0714.90077
[7] Mignot, F.,Contrôle dans les Inéquations Variationelles Elliptiques, Journal of Functional Analysis, Vol. 22, pp. 130-185, 1976. · Zbl 0364.49003 · doi:10.1016/0022-1236(76)90017-3
[8] Mignot, F., andPuel, J. P.,Optimal Control in Some Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 22, pp. 466-476, 1984. · Zbl 0561.49007 · doi:10.1137/0322028
[9] Shi, S. H.,Optimal Control of Strongly Monotone Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 26, pp. 274-290, 1988. · Zbl 0644.49007 · doi:10.1137/0326016
[10] Shi, S. H.,Erratum: Optimal Control of Strongly Monotone Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 28, pp. 243-249, 1990. · Zbl 0692.49007 · doi:10.1137/0328012
[11] Hiriart-Urruty, J. B., andThibault, L.,Existence et Caractérization de Differentielles Généralisées, Comptes Rendus de l’Académie des Sciences de Paris, Vol. 290, pp. 1091-1094, 1980.
[12] Clarke, F. H.,Optimization and Nonsmooth Analysis, J. Wiley and Sons, New York, New York, 1983. · Zbl 0582.49001
[13] Cornet, B., andLaroque, G.,Lipschitz Properties of Solutions in Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 53, pp. 407-427, 1987. · Zbl 0595.90081 · doi:10.1007/BF00938947
[14] Jitorntrum, K.,Solution Point Differentiability without Strict Complementarity in Nonlinear Programming, Mathematical Programming Study, Vol. 21, pp. 127-138, 1984. · Zbl 0571.90080
[15] Kyparisis, J.,Sensitivity Analysis for Nonlinear Programs and Variational Inequalities with Nonunique Multipliers, Mathematics of Operations Research, Vol. 15, pp. 286-298, 1990. · Zbl 0708.90086 · doi:10.1287/moor.15.2.286
[16] Fiacco, A. V.,Sensitivity Analysis for Nonlinear Programming Using Penalty Methods, Mathematical Programming, Vol. 10, pp. 287-311, 1976. · Zbl 0357.90064 · doi:10.1007/BF01580677
[17] Malanowski, K.,Differentiability with Respect to Parameters of Solutions to Convex Programming Problems, Mathematical Programming, Vol. 33, pp. 352-361, 1985. · Zbl 0582.49015 · doi:10.1007/BF01584382
[18] Hiriart-Urruty, J. B.,Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces, Mathematics of Operations Research, Vol. 4, pp. 79-97, 1979. · Zbl 0409.90086 · doi:10.1287/moor.4.1.79
[19] Ko?vara, M., andOutrata, J. V.,A Nondifferentiable Approach to the Solution of Optimum Design Problems with Variational Inequalities, System Modelling and Optimization, Edited by P. Kall, Springer-Verlag, Berlin, Germany, pp. 364-373, 1992. · Zbl 0789.49027
[20] Haslinger, J., andNeittaanmäki, P.,Finite-Element Approximations for Optimal Shape Design, J. Wiley and Sons, Chichester, England, 1988. · Zbl 0713.73062
[21] Hlavá?ek, I.,Shape Optimization of an Elastic Perfectly Plastic Body, Aplikace Matematiky, Vol. 32, pp. 381-400, 1987. · Zbl 0632.73082
[22] Schramm, H., andZowe, J.,A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results, SIAM Journal on Optimization, Vol. 2, pp. 121-152, 1992. · Zbl 0761.90090 · doi:10.1137/0802008
[23] Schittkowski, K.,nlpql: A fortran Subroutine Solving Constrained Nonlinear Programming Problems, Annals of Operations Research, Vol. 5, pp. 485-500, 1985.
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