First-order necessary optimality conditions for general bilevel programming problems
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We formulate in this paper several versions of the necessary conditions for general bilevel programming problems. The technique used is related to standard methods of nonsmooth analysis. We treat separately the following cases: Lipschitz case, differentiable case, and convex case. Many typical examples are given to show the efficiency of theoretical results. In the last section, we formulate the general multilevel programming problem and give necessary conditions of optimality in the general case. We illustrate then the application of these conditions by an example.
Key WordsBilevel programming value functions existence theorems necessary conditions nonsmooth analysis Lagrange multipliers
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- 1.Bard, J. F.,Optimality Conditions for the Bilevel Programming Problem, Naval Research Logistics Quarterly, Vol. 31, pp. 13–24, 1984.Google Scholar
- 2.Bard, J. F., andFalk, J. E.,An Explicit Solution to the Multilevel Programming Problem, Computers and Operations Research, Vol. 9, pp. 77–100, 1982.Google Scholar
- 3.Benson, H. P.,On the Structure and Properties of a Linear Multilevel Programming Problem, Journal of Optimization Theory and Applications, Vol. 60, pp. 353–373, 1989.Google Scholar
- 4.Clark, P. A., andWesterberg, A. W.,A Note on the Optimality Conditions for the Bilevel Programming Problem, Naval Research Logistics, Vol. 35, pp. 414–418, 1988.Google Scholar
- 5.Florian, M., andChen, Y.,The Nonlinear Bilevel Programming Problem: Formulation, Regularity, and Optimality Conditions, CRT Publication 794, University of Montreal, Montreal, Canada, 1993.Google Scholar
- 6.Yezza, A.,Some Properties of the Bilevel Programming Problems and Existence, Preprint, University of Montreal, Montreal, Canada, 1993.Google Scholar
- 7.Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, New York, 1983.Google Scholar
- 8.Mäkelä, M. M., andNeittaanmäki, P.,Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control, World Scientific Publishing, Singapore, Republic of Singapore, 1992.Google Scholar
- 9.Bazaraa, M. S., andShetty, C. M.,Nonlinear Programming, John Wiley and Sons, New York, New York, 1979.Google Scholar