Abstract
We present an algorithm for finding approximate global solutions to quadratically constrained quadratic programming problems. The method is based on outer approximation (linearization) and branch and bound with linear programming subproblems. When the feasible set is non-convex, the infinite process can be terminated with an approximate (possibly infeasible) optimal solution. We provide error bounds that can be used to ensure stopping within a prespecified feasibility tolerance. A numerical example illustrates the procedure. Computational experiments with an implementation of the procedure are reported on bilinearly constrained test problems with up to sixteen decision variables and eight constraints.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Al-Khayyal, F. A. (1990), Jointly Constrained Bilinear Programs and Related Problems: An Overview,Computers and Mathematics with Applications 19, 53–62.
Al-Khayyal, F. A. (1992), Generalized Bilinear Programming, Part I: Models, Applications and Linear Programming Relaxation,European Journal of Operational Research 60, 306–314.
Al-Khayyal, F. A. and Falk, J. E. (1983), Jointly Constrained Biconvex Programming,Mathematics of Operations Research 8, 273–286.
Al-Khayyal, F. A. and Larsen, C. (1990), Global Optimization of a Quadratic Function Subject to a Bounded Mixed Integer Constraint Set,Annals of Operations Research 25, 169–180.
Al-Khayyal, F. A. and Pardalos, P. M. (1991), Global Optimization Algorithms for VLSI Compaction Problems,Arabian Journal for Science and Engineering 16, 335–339.
Baron, D. P. (1972), Quadratic Programming with Quadratic Constraints,Naval Research Logistics Quarterly 17, 253–260.
Eben Chaime, M. (1990), The Physical Design of Printed Circuit Boards: A Mathematical Programming Approach, Ph.D. Dissertation, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia.
Erenguc, S. S. and Benson, H. P. (1991), An Algorithm for Indefinite Integer Quadratic Programming,Computers & Mathematics with Applications 21, 99–106.
Evans, D. H. (1963), Modular Design — A Special Case in Nonlinear Programming,Operations Research 11, 637–647.
Filar, J. A. and Schultz, T. A. (1995), Bilinear Programming and Structured Stochastic Games,Journal of Optimization Theory and Applications (forthcoming).
Floudas, C. A. and Aggarwal, A. (1990), A Decomposition Strategy for Global Optimum Search in the Pooling Problem,ORSA Journal on Computing 2, 225–235.
Floudas, C. A. and Visweswaran, V. (1993), A Primal-Relaxed Dual Global Optimization Approach,Journal of Optimization Theory and Its Applications 78, 187–225.
Foulds, L. R., Haughland, D., and Johnston, K. (1992), A Bilinear Approach to the Pooling Problem,Optimization 24, 165–180.
Goldberg, J. B. (1984), The Modular Design Problem with Linear Separable Side Constraints: Heuristics and Applications, Ph.D. Dissertation, Industrial and Operations Engineering Department, University of Michigan, Ann Arbor, Michigan.
Hansen, P. (1979), Methods of Nonlinear 0–1 Programming,Annals of Discrete Mathematics 5, 53–70.
Hansen, P. and Jaumard, B. (1992), Reduction of Indefinite Quadratic Programs to Bilinear Programs,Journal of Global Optimization 2, 41–60.
Hao, E. P. (1982), Quadratically Constrained Quadratic Programming: Some Applications and a Method for Solution,ZOR 26, 105–119.
Horst, R. (1986), A General Class of Branch and Bound Methods in Global Optimization with Some New Approaches to Concave Minimization,Journal of Optimization Theory and Applications 51, 271–291.
Horst, R. and Tuy, H. (1993),Global Optimization: Deterministic Approaches, Second Edition, Springer-Verlag, Berlin.
Pardalos, P. M. and Rosen, J. B. (1987),Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science 268. Springer-Verlag, Berlin.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1988),Numerical Recipes in C, Cambridge University Press, Cambridge.
Rutenberg, D. P. and Shaftel, T. L. (1971), Product Design: Sub-Assemblies for Multiple Markets,Management Science 18, B220-B231.
Sherali, H. D. and Tuncbilek, C. H. (1992), A Global Optimization Algorithm for Polynomial Programming Problems Using a Reformulation-Linearization Technique,Journal of Global Optimization 2, 101–112.
Shor, N. Z. (1990), Dual Quadratic Estimates in Polynomial and Boolean Programming,Annals of Operations Research 25, 163–168.
Van de Panne, C. (1966), Programming with a Quadratic Constraint,Management Science 12, 748–815.
Visweswaran, V. and Floudas, C. A. (1993), New Properties and Computational Improvement of the GOP Algorithm for Problems with Quadratic Objective Functions and Constraints,Journal of Global Optimization 3, 439–462.
Visweswaran, V. and Floudas, C. A. (1990), A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLP's: II. Applications of Theory and Test Problems,Computers and Chemical Engineering 14, 1417–1434.
Author information
Authors and Affiliations
Additional information
This research was supported in part by National Science Foundation Grant DDM-91-14489.
Rights and permissions
About this article
Cite this article
Al-Khayyal, F.A., Larsen, C. & Van Voorhis, T. A relaxation method for nonconvex quadratically constrained quadratic programs. J Glob Optim 6, 215–230 (1995). https://doi.org/10.1007/BF01099462
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01099462