Abstract
This paper applies the SDP (semidefinite programming)relaxation originally developed for a 0-1 integer program to ageneral nonconvex QP (quadratic program) having a linear objective functionand quadratic inequality constraints, and presents some fundamental characterizations of the SDP relaxation including its equivalence to arelaxation using convex-quadratic valid inequalities for the feasible regionof the QP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alizadeh, W. F. (1995), Interior point methods in semidefinite programming with application to combinatorial optimization, SIAM Journal on Optimization 5, 13–51.
Borwein, J. M. (1981), Direct theorems in semi-infinite convex programming, Mathematical Programming 21, 301–318.
Boyd, S., Ghaoui, L. E., Feron, E. and Balakrishnan, V. (1994), Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia.
Freund, R. M. (1994), Complexity of an algorithm for finding an approximate solution of a semidefinite program with no regularity assumption, Technical report OR 302-94, Operations Research Center, MIT, 1994.
Goemans, M. X. and Williamson, D. P. (1995), Improved approximation algorithms formaximum cut and satisfiability problems using semidefinite programming, Journal of ACM 42, 1115–1145.
Grötschel, M., Lovász, L. and Schrijver, A. (1981), The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1, 169–197.
Grötschel, M., Lovász, L. and Schrijver, A. (1986), Relaxations of vertex packing, Journal of combinatorial Theory, Series B 40, 330–343.
Grötschel, M., Lovász, L. and Schrijver, A. (1988), Geometric algorithms and combinatorial optimization, Springer, New York.
Helmberg, C., Rendl, F., Vanderbei, R. J. and Wolkowicz, H. (1996), An interior-point method for semidefinite programming, SIAM Journal on Optimization 6, 342–361.
Helmberg, C., Poljak, S., Rendl, F. and Wolkowicz, H. (1995), Combining semidefinite and polyhedral relaxation for integer programs, Lecture Notes in Computer Science 920, 124–134.
Karmarkar, N. (1984), A new polynomial-time algorithm for linear programming, Combinatorica 4, 373–395.
Knuth, D. E. (1995), The sandwich theorem, Electronic Journal of Combinatorics 1, 48pp.
Kojima, M., Mizuno, S. and Yoshise, A. (1989), A primal-dual interior point algorithm for linear programming, in Megiddo, N. (ed.), Progress in Mathematical Programming, Interior-Point and Related Methods, Springer-Verlag, New York, 29–47.
Kojima, M., Shindoh, S. and Hara, S. (1995), Interior-point methods for the monotone semidefinite linear complementarity problems, Research Report #282, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 152, Japan, April 1994, Revised October 1995. To appear in SIAM Journal on Optimization.
Lovász, L. (1979), On the Shannon capacity of a graph, IEEE Transactions on Information Theory 25, 1–7.
Lovász, L. and Schrijver, A. (1991), Cones of matrices and set functions and 0-1 optimization, SIAM Journal on Optimization 1, 166–190.
Mangasarian, O. (1969), Nonlinear Programming, McGraw-Hill Book Company, New York.
Megiddo, N. (1989), Pathways to the optimal set in linear programming, in Megiddo, N. (ed.), Progress in Mathematical Programming, Interior-Point and Related Methods, Springer-Verlag, New York, 131–158.
Nesterov, Ju. E. and Nemirovskii, A. S. (1993), Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia.
Poljak, S., Rendl, F. and Wolkowicz, H. (1995), A recipe for semidefinite relaxation for (0,1)-quadratic programming, Journal of Global Optimization 7, 51–73.
Ramana, M. (1993), An algorithmic analysis of multiquadratic and semidefinite programming problems, PhD thesis, Johns Hopkins University, Baltimore, MD.
Ramana, M. and Pardalos, P.M. (1996), Semidefinite programming, in Terlaky, T. (ed.), Interior Point Algorithms, Kluwer Academic Publishers, Dordrecht, The Netherlands, 369–398.
Pardalos, P. M. and Resende, M. G. C. (1996), Interior point methods for global optimization problems, in Terlaky, T. (ed.), Interior Point Algorithms, Kluwer Academic Publishers, Dordrecht, The Netherlands, 467–500.
Rockafellar, T. (1970), Convex Analysis, Princeton University Press, Princeton, New Jersey.
Sherali, H. D. and Alameddine, A. R. (1992), A new reformulation-linearization technique for bilinear programming problems, Journal of Global Optimization 2, 379–410.
Sherali, H. D. and Tuncbilek, C. H. (1995), A reformulation-convexification approach for solving nonconvex programming problems, Journal of Global Optimization 7, 1–31.
Shor, N. Z. (1987), Quadratic optimization problems, Soviet Journal of Computer and Systems Sciences 25, 1–11.
Shor, N. Z. (1990), Dual quadratic estimates in polynomial and boolean programming, Annals of Operations Research 25, 163–168.
Vandenberghe, L. and Boyd, S. (1995), A primal-dual potential reduction method for problems involving matrix inequalities, Mathematical Programming, Series B 69, 205–236.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fujie, T., Kojima, M. Semidefinite Programming Relaxation for Nonconvex Quadratic Programs. Journal of Global Optimization 10, 367–380 (1997). https://doi.org/10.1023/A:1008282830093
Issue Date:
DOI: https://doi.org/10.1023/A:1008282830093