Abstract
In this paper we consider the optimization of a quadratic function subject to a linearly bounded mixed integer constraint set. We develop two types of piecewise affine convex underestimating functions for the objective function. These are used in a branch and bound algorithm for solving the original problem. We show finite convergence to a near optimal solution for this algorithm. We illustrate the algorithm with a small numerical example. Finally we discuss some modifications of the algorithm and address the question of extending the problem to include quadratic constraints.
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W.P. Adams and H.D. Sherali, A tight linearization and an algorithm for zero-one quadratic programming problems, Management Sci. 32 (1986) 1274–1290.
F.A. Al-Khayyal, Jointly constrained bilinear programs and related problems: an overview, Comput. Math. Appl. 19 (1990) 53–62.
F.A. Al-Khayyal and J.E. Falk, Jointly constratined biconvex programming, Math. Oper. Res. 8 (1983) 273–286.
E. Balas, Nonconvex quadratic programming via generalized polars, SIAM J. Appl. Math. 28 (1975) 335–349.
E. Balas, The assignable subgraph polytope of a directed graph, Management Science Research Report No. MSRR-533 (1987), Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA 15213.
E. Balas and W.R. Pulleybland, The perfectly matchable subgraph polytope of a bipartite graph, Networks 13 (1983) 495–516.
E. Balas and W. R. Pulleyblank, The perfectly matchable subgraph polytope of an arbitrary graph, Management Science Research Report No. MSRR-538 (1987), Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA 15213.
S.S. Erenguc and H.P. Benson, An algorithm for indefinite integer quadratic programming, Discussion Paper No. 134 (1987), University of Florida, Gainesville FL 32611.
F. Glover, Improved linear integer programming formulations of nonlinear integer problems, Management Sci. 22 (1975) 455–460.
R. Horst, A general class of branch and bound methods in global optimization with some new approaches for concave minimization, J. Optim. Theory Appl. 51 (1986) 271–291.
P. Kough, The indefinite quadratic programming problem, Oper. Res. 27 (1979) 516–533.
R. Lazimy, Improved algorithm for mixed integer quadratic programs and computational study, Math. Progr. 32 (1985) 100–113.
G.L. Nemhauser and L.A. Wolsey,Integer and Combinatorial Optimization (Wiley, New York, 1988).
P. Pardalos and J. B. Rosen, Constrained global optimization: algorithms and applications,Lecture Notes in Computer Science, vol. 268 (Springer, Berlin 1987).
H.D. Sherali and W.P. Admas, A hierarchy of relaxations between the continuous and convex hull representations for zero one programming problems, Working Paper (1988), Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061–0118.
H.D. Sherali and W.P. Admas, A hierarchy of relaxations and convex hull characterizations for mixed integer programming problems, Working Paper (1989), Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061–0118.
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Supported by grants from the Danish Natural Science Research Council and the Danish Research Academy.
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Al-Khayyal, F.A., Larsen, C. Global optimization of a quadratic function subject to a bounded mixed integer consraint set. Ann Oper Res 25, 169–180 (1990). https://doi.org/10.1007/BF02283693
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DOI: https://doi.org/10.1007/BF02283693