Abstract
Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. In this paper, we consider the bilevel programming problem in which the objective function of both levels is indefinite quadratic and the feasible region is a convex polyhedron. At first, a method similar to simplex method to solve indefinite quadratic programming problem is developed. It is shown that the given problem is equivalent to maximizing a quasi-convex function over a feasible region comprised of faces of the polyhedron. Hence, there is an extreme point of the polyhedron that solves the problem. Finally, same method is employed to solve the problem when the variables are required to be integers. The solution method is illustrated with the help of an example.
Similar content being viewed by others
References
Al-Khayyal, F.A. (1986) “Linear, quadratic and bilinear programming approaches to linear complimentarity problem”, European Journal of Operational Research 24, 216–227.
Cabot, A.V. and Francis, R.L. (1970) “Solving certain non-convex, quadratic minimization problems by ranking the extreme points”, Operations Research 18, 82–86.
Calvete, H.I. and Gale, C. (1998) “On the quasi-concave bilevel programming problem”, Journal of Optimization Theory and Applications, Vol. 98, No.3, 613–622.
Falk, J.E. and Soland, R.M. (1969) “An algorithm for solving separable non-convex programming problems”, Management Science 15, 550–569.
Konno, H. (1976) “A cutting plane algorithm for solving bilinear programs”, Mathematical Programming 11, 14–27.
Konno, H. and Kuno, T. (1992) “Linear multiplicative programming”, Mathematical Programming 56, 51–64.
Liu, Y.H. and Spencer, T.H. (1995) “Solving a bilevel linear program when the inner decision maker controls few variables”, European Journal of Operational Research, 81, 644–651.
Bialas, W.F. and Karwan, M.H. (1982) “On two level optimization”, IEEE Transactions on Automatic Control, Vol. 27, No. 1, 211–214.
Bialas, W.F. and Karwan, M.H. (1984) “Two level linear programming”, Management Science, Vol. 30, No. 8, 1004–1020.
Murty, K.G. (1969) “Solving the fixed charge problem by ranking the extreme points”, Operations Research 16, 268–279.
Polijak, S. and Wolkowicz, H. (1995) “Convex relaxations of (0, 1) quadratic programming”, Mathematics of Operations Research Vol. 20, No. 3, 550–561.
Thoi, N.V. and Tuy, H. (1980) “Convergent algorithms for minimizing a concave function”, Mathematics of Operations Research, 5, 556–566.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alemayehu, G., Narang, R. & Arora, S.R. On the indefinite quadratic bilevel programming problem. OPSEARCH 41, 264–277 (2004). https://doi.org/10.1007/BF03398849
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03398849