Abstract
Indefinite quadratic programs with quadratic constraints can be reduced to bilinear programs with bilinear constraints by duplication of variables. Such reductions are studied in which: (i) the number of additional variables is minimum or (ii) the number of complicating variables, i.e., variables to be fixed in order to obtain a linear program, in the resulting bilinear program is minimum. These two problems are shown to be equivalent to a maximum bipartite subgraph and a maximum stable set problem respectively in a graph associated with the quadratic program. Non-polynomial but practically efficient algorithms for both reductions are thus obtaine.d Reduction of more general global optimization problems than quadratic programs to bilinear programs is also briefly discussed.
Similar content being viewed by others
References
Al-Khayyal, F. A. (1990), Generalized Bilinear Programming: Part I. Models, Applications, and Linear Programming Relaxation, Research Report, Georgia Institute of Technology.
Al-Khayyal, F. A. and J. E., Falk (1983), Jointly Constrained Biconvex Programming, Mathematics of Operations Research 8 (2), 273–286.
Al-Khayyal, F., R. Horst, and P. Pardalos (1991), Global Optimization of Concave Functions Subject to Separable Quadratic Constraints and of All-Quadratic Separable Problems, Annals of Operations Research.
Avriel, M., W. E., Diernert, S., Schaible, and I., Zang (1988), Generalized Concavity, New York: Plenum Press.
Avriel, M. and A. C., Williams (1971), An Extension of Geometric Programming with Applications in Engineering Optimization, Journal of Engineering Mathematics 5 (3), 187–194.
Balas, E. and C. S., Yu (1986), Finding a Maximum Clique in an Arbitrary Graph, SIAM Journal on Computing 15, 1054–1068.
Baron, D. P. (1972), Quadratic Programming with Quadratic Constraints, Naval Research Logistics Quarterly 19, 253–260.
Bartholomew-Biggs, M. C. (1976), A Numerical Comparison between Two Approaches to Nonlinear Programming Problems, Technical Report #77, Numerical Optimization Center, Hatfield, England.
Benacer, R. and T., Pham Dinh (1986), Global Maximization of a Nondefinite Quardatic Function over a Convex Polyhedron, pp. 65–76 in J.-B., Hirriart-Urruty (ed.), Fermat Days 1985: Mathematics for Optimization, Amsterdam: North-Holland.
Berge, C. (1983), Graphes, 3rd ed., Paris: Gauthier-Villars.
Berge, C. (1987), Hypergraphes, Paris, Gauthier-Villars.
Bernard, J. C. and J. A., Ferland (1989), Convergence of Interval-Type Algorithms for Generalized Fractional Programming, Mathematical Programming 43, 349–363.
Carraghan, R. and P. M., Pardalos (1990), An Exact Algorithm for the Maximum Clique Problem, Operations Research Letters 9, 375–382.
Colville, A. R. (1986), A Comparative Study on Nonlinear Programming Codes, IBM Scientific Center Report 320-2949, New York.
Dembo, R. S. (1972), Solution of Complementary Geometric Programming Problems, M.Sc. Thesis, Technion, Haifa.
Dembo, R. S. (1976), A Set of Geometric Programming Test Problems and Their Solutions, Mathematical Programming 10, 192–213.
Duffin, R. J., E. L., Peterson, and C., Zener (1967), Geometric Programming: Theory and Applications, New York: Wiley.
Ecker, J. G. and R. D., Niemi (1975), A Dual Method for Quadratic Programs with Quadratic Constraints, SIAM Journal on Applied Mathematics 28, 568–576.
Evans, D. H. (1963), Modular Design—A Special Case in Nonlinear Programming, Operations Research 11, 637–647.
Flippo, O. E. (1989), Stability, Duality and Decomposition in General Mathematical Programming, Rotterdam: Erasmus University Press.
Floudas, C. A., A., Aggarwal, and A. R., Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems, Computers and Chemical Engineering 13 (10), 1117–1132.
Floudas, C. A. and P., Pardalos (1990), A Collection of Test Problems for Constrained Global Optimization, Lecture Notes in Computer Science, 455, Berlin: Springer-Verlag.
Floudas, C. A. and V., Visweswaran (1990), A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs—I. Theory, Computers and Chemical Engineering 14 (12), 1397–1417.
Friden, C., A., Hertz, and D.de, Werra (1990), Tabaris: An Exact Algorithm Based on Tabu Search for Finding a Maximum Independent Set in a Graph, Computers and Operations Research 17 (5), 437–445.
Garey, M. R., D. S., Johnson, and L., Stockmeyer (1976), Some Simplified NP-Complete Graph Problems, Theoretical Computer Science 1, 237–267.
Geoffrion, A. M. (1972), Generalized Benders Decomposition, Journal of Optimization Theory and Its Applications 10, 237–260.
Gochet, W. and Y., Smeers (1979), A Branch and Bound Method for Reversed Geometric Programming, Operations Research 27, 982–996.
Hock, W. and K., Schittkowski (1981), Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems #187, Berlin: Springer-Verlag.
Horst, R. and H., Tuy (1990), Global Optimization, Deterministic Approaches, Berlin: Springer-Verlag.
Konno, H. (1976), Maximizing a Convex Quadratic Function Subject to Linear Constraints Mathematical Programming 11, 117–127.
Konno, H. and T. Kuno (1989), Linear Multiplicative Programming, Preprint IHSS 89-13, Tokyo Institute of Technology.
Kough, P. F. (1979), The Indefinite Quadratic Programming Problem, Operations Research 27, 516–533.
Mladineo, R. H. (1986), An Algorithm for Finding the Global Maximum of a Multimodal, Multivariate Function, Mathematical Programming 34, 188–200.
Pardalos, P. M., J. H., Glick, and J. B., Rosen (1987), Global Minimization of Indefinite Quadratic Problems, Computing 39, 281–291.
Pardalos, P. M. and J. B., Rosen (1986), Methods for Global Concave Minimization: A Bibliographic Survey, SIAM Review 28 (3), 367–379.
Pardalos, P. M. and J. B., Rosen (1987), Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science #268, Berlin: Springer Verlag.
Pham Dinh, T. and S.El, Bernoussi (1989), Numerical Methods for Solving a Class of Global Nonconvex Optimization Problems, International Series of Numerical Mathematics 87, 97–132.
Phan, H. (1982), Quadratically Constrained Quadratic Programming: Some Applications and a Method of Solution, Zeitschrift für Operations Research 26, 105–119.
Phillips, A. T. and J. B., Rosen, (1990), Guaranteed ε-Approximate Solution to Indefinite Global Optimization, Naval Research Logistics 37, 499–514.
Ragavachari, M. (1989), On Connections between Zero-One Integer Programming and Concave Programming under Linear Constraints, Operations Research 17, 680–684.
Reeves, G. R. (1975), Global Minimization in Nonconvex All-Quadratic Programming, Management Science 22, 76–86.
Sherali, H. and A. Alameddine (1990), A New Reformulation-Linearization Technique for Bilinear Programming Problems, Research Report, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute.
Stephanopoulos, G. and A. W., Westerberg (1975), The Use of Hestenes' Method of Multipliers to Resolve Dual Gaps in Engineering System Optimization, Journal of Optimization Theory and Applications 15, 285–309.
Simões, L. M. C. (1987), Search for the Global Optimum of Least Volume Trusses, Engineering Optimization 11, 49–67.
Thoai, N. V. (1990), Application of Decomposition Techniques in Global Optimization to the Convex Multiplicative Programming Problem, Paper presented at the Second Workshop on Global Optimization, Sopron, Hungary.
Tuy, H. (1986), A General Deterministic Approach to Global Optimization via d.-c. Programming, pp. 137–162, in J. B., Hirriart-Urruty (ed.), Fermat Days 1985: Mathematics for Optimization, Amsterdam: North-Holland.
Visweswaran, V. and C. A., Floudas (1990), A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs—II. Application of Theory and Test Problems, Computers and Chemical Engineering 14 (12), 1419–1434.
Wolsey, L. A. (1981), A Resource Decomposition Algorithm for General Mathematical Programs, Mathematical Programming Study 14, 244–257.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hansen, P., Jaumard, B. Reduction of indefinite quadratic programs to bilinear programs. J Glob Optim 2, 41–60 (1992). https://doi.org/10.1007/BF00121301
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00121301