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Some properties of the bilevel programming problem. (English) Zbl 0696.90086
The purpose of this paper is to elaborate on the difficulties accompanying the development of efficient algorithms for solving the bilevel programming problem (BLPP). We begin with a pair of examples showing that, even under the best of circumstances, solutions may not exist. This is followed by a proof that the BLPP is NP-hard.
Reviewer: J.F.Bard

90C99Mathematical programming
91A052-person games
90C60Abstract computational complexity for mathematical programming problems
93A13Hierarchical systems
Full Text: DOI
[1] Bard, J. F.,Optimality Conditions for the Bilevel Programming Problem, Naval Research Logistics Quarterly, Vol. 31, pp. 13-26, 1984. · Zbl 0537.90087 · doi:10.1002/nav.3800310104
[2] Hasen, P., Jaumard, B., andSavard, G.,A Variable Elimination Algorithm for Bilevel Programming, RUTCOR Research Report RRR-17-89, Rutgers University, New Brunswick, New Jersey, 1989.
[3] Jeroslow, R. G.,The Polynomial Hierarchy and a Simple Model for Competitive Analysis, Mathematical Programming, Vol. 32, pp. 146-164, 1985. · Zbl 0588.90053 · doi:10.1007/BF01586088
[4] Hogan, W. W.,Point-to-Set Maps in Mathematical Programming, SIAM Review, Vol. 15, pp. 591-603, 1973. · Zbl 0256.90042 · doi:10.1137/1015073
[5] Garey, M. R. andJohnson, D. S.,Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, New York, New York, 1979.
[6] Ben-Ayed, O., andBlair, C. E.,Computational Difficulties of Bilevel Programming, Working Paper WP-1432, College of Commerce and Business Administration, University of Illinois, Urbana, Illinois, 1988.