Enhancing RLT relaxations via a new class of semidefinite cuts
 Hanif D. Sherali,
 Barbara M. P. Fraticelli
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In this paper, we propose a mechanism to tighten ReformulationLinearization Technique (RLT) based relaxations for solving nonconvex programming problems by importing concepts from semidefinite programming (SDP), leading to a new class of semidefinite cutting planes. Given an RLT relaxation, the usual nonnegativity restrictions on the matrix of RLT product variables is replaced by a suitable positive semidefinite constraint. Instead of relying on specific SDP solvers, the positive semidefinite stipulation is rewritten to develop a semiinfinite linear programming representation of the problem, and an approach is developed that can be implemented using traditional optimization software. Specifically, the infinite set of constraints is relaxed, and members of this set are generated as needed via a separation routine in polynomial time. In essence, this process yields an RLT relaxation that is augmented with valid inequalities, which are themselves classes of RLT constraints that we call semidefinite cuts. These semidefinite cuts comprise a relaxation of the underlying semidefinite constraint. We illustrate this strategy by applying it to the case of optimizing a nonconvex quadratic objective function over a simplex. The algorithm has been implemented in C++, using CPLEX callable routines, and two types of semidefinite restrictions are explored along with several implementation strategies. Several of the most promising lower bounding strategies have been implemented within a branchandbound framework. Computational results indicate that the cutting plane algorithm provides a significant tightening of the lower bound obtained by using RLT alone. Moreover, when used within a branchandbound framework, the proposed lower bound significantly reduces the effort required to obtain globally optimal solutions.
 Alizadeh, F. (1995) Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization. SIAM Journal of Optimization 5: pp. 1351
 Audet, C., Hansen, P., Jaumard, B., Savard, G. (2000) Branch and Cut Algorithm for Nonconvex Quadratically Constrained Quadratic Programming. Mathematical Programming 87: pp. 131152
 Bazaraa, M. S., Sherali, H. D., Shetty, C. M. (1993) Nonlinear Programming Theory and Applications. Wiley, New York, NY
 Bertsimas, D. and Ye, Y. (1998) Semidefinite Relaxations,Multivariate Normal Distributions and Order Statistics. In: D.Z. Du and P. M. Pardalos (eds.): Handbook of Combinatorial Optimization, Vol. 3. Kluwer Academic Publishers, pp. 1–19.
 Burer, S. and Monteiro, R. (1998) A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Lowrank Factorization. Presented at the ISMP Conference, Atlanta, GA.
 Goemans, M., Williamson, D. (1995) Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. Journal of the Association for Computational Machinery 42: pp. 11151145
 Nowak, I. (1998) A Global Optimality Criterion for NonConvex Quadratic Programming Over a Simplex. Humboldt University, Berlin
 Paige, C.C. (1972) Computational Variants of the Lanczos Method for the Eigenproblems. Journal of the Institute of Mathematics and Its Applications 10: pp. 373381
 Ramana, M., Goldman, A. J. (1995) Some Geometric Results in Semidefinite Programming. Journal of Global Optimization 7: pp. 3350
 Ramana, M., Pardalos, P. M. Semidefinite Programming. In: Terlaky, T. eds. (1996) Interior Point Methods of Mathematical Programming. Kluwer Academic Publishers, Dordrecht, pp. 369398
 Sherali, H. D., Adams, W. P. (1999) A ReformulationLinearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishing, Boston, MA
 Sherali, H. D., Tuncbilek, C. H. (1992) A Global Optimization Algorithm for Polynomial Programming PRoblems Using a ReformulationLinearization Technique. Journal of Global Optimization 2: pp. 101112
 Sherali, H. D., Tuncbilek, C. H. (1997) ReformulationLinearization/Convexification Relaxations for Univariate and Multivariate Polynomial Programming Problems. Operations Research Letters 21: pp. 110
 Sherali, H. D., Wang, H. (2001) Global Optimization of Nonconvex Factorable Programming Problems. Mathematical Programming 89: pp. 459478
 Shor, N. Z. (1998) Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic Publishing, Boston, MA
 Todd, M. J. (1998), Semidefinite Programming Applications, Duality, and InteriorPoint Methods. Presented at the Fall INFORMS meeting, Seattle,WA. Also available on the WorldWide Web at http://www.orie.cornell.edu/miketodd/todd.html.
 Vandenbergh, L., Boyd, S. (1996) Semidefinite Programming. SIAM Review 38: pp. 4995
 Vanderbei, R.J., Benson, H. Y. (2000) On Formulating Semidefinite Programming Problems as Smooth Convex Nonlinear Optimization Problems. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ
 Wolkowicz, H., Saigal, R., Vandenbergh, L. (2000) Handbook of Semidefinite Progamming: Theory and Algorithms, and Applications. Kluwer Academic Publishers, Boston, MA
 Title
 Enhancing RLT relaxations via a new class of semidefinite cuts
 Journal

Journal of Global Optimization
Volume 22, Issue 14 , pp 233261
 Cover Date
 20020101
 DOI
 10.1023/A:1013819515732
 Print ISSN
 09255001
 Online ISSN
 15732916
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 ReformulationLinearization Technique (RLT)
 Semidefinite programming (SDP)
 Semidefinite cuts
 Valid inequalities
 Nonconvex quadratic programming
 Industry Sectors
 Authors

 Hanif D. Sherali ^{(1)}
 Barbara M. P. Fraticelli ^{(1)}
 Author Affiliations

 1. Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, 250 New Engineering Building, Blacksburg, VA, 24061, USA