A Polyhedral Approach for Nonconvex Quadratic Programming Problems with Box Constraints

Journal of Global Optimization

Volumn 13, Issue 2, 1998, Pages 151-170

  Yajima, Yasutoshi ^a   Fujie, Tetsuya ^a  

^a TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

Author keywords

Cutting plane method; Linearization technique; Nonconvex quadratic programs; Valid inequalities

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; BOOLEAN FUNCTIONS; CONSTRAINT THEORY; INTEGRAL EQUATIONS; MATRIX ALGEBRA; OPTIMIZATION; PROBLEM SOLVING; SET THEORY; THEOREM PROVING; VECTORS;

CUTTING PLANE METHOD; LINEARIZATION TECHNIQUE; NONCONVEX QUADRATIC PROGRAMS; VALID INEQUALITIES;

QUADRATIC PROGRAMMING;

EID: 1842740743     PISSN: 09255001     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008293029350     Document Type: Article

References (27)

1. Alizadeh, W.F. (1995), Interior point methods in semidefinite programming with application to combinatorial optimization, SIAM Journal on Optimization 5: 13-51.
2. Barahona, F., Jünger, M. and Reinelt, G. (1989), Experiments in quadratic 0-1 programming, Mathematical Programming 44: 127-137.
3. Barahona, F. and Mahjoub, A.R. (1986), On the cut polytope, Mathematical Programming 36: 157-173.
4. Bazaraa, M., Sherali, H.D. and Shetty, C.M. (1993), Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York.
5. Boros, E. and Hammer, P.L. (1991), The max-cut problem and quadratic 0-1 optimization; polyhedral aspect, relaxations and bounds, Annals of Operations Research 33: 151-180.
6. Boros, E. and Hammer, P.L. (1993), Cut polytopes, Boolean quadric polytopes and nonnegative pseudo-Boolean functions, Mathematics of Operations Research 18: 245-253.
7. Coleman, T.F. and Hulbert, L.A. (1989), A direct active set algorithm for large sparse quadratic programs with simple bounds, Mathematical Programming, 45: 373-406.
8. De Angelis, P.L., Pardalos, P.M. and Toraldo, G. (1997), Quadratic Programming with box constraints, in I.M. Bomze et al. (eds.), Developments in Global Optimization (pp. 73-93), Kluwer Academic Publishers, Dordrecht, Boston, London.
9. Delorme, C. and Poljak, S. (1993), Laplacian eigenvalues and the maximum cut problem, Mathematical Programming 62: 557-574.
10. Fujie, T. and Kojima, M. (1997), Semidefinite programming relaxation for nonconvex quadratic programs, Journal of Global Optimization 10: 367-380.
11. Hansen, P., Jaumard, B., Ruiz, M. and Xiong, J. (1993), Global minimization of indefinite quadratic functions subject to box constraints, Naval Research Logistics Quarterly 40: 373-392.
12. 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.
13. Helmberg, C. and Rendl, F. (1995), Solving quadratic (0,1)-problems by semidefinite programs and cutting planes, ZIB Preprint SC-95-35.
14. Kalantari, B. and Bagchi, A. (1990), An algorithm for quadratic zero-one programs, Naval Research Logistics Quarterly 37: 527-538.
15. Kojima, M., Shindoh, S. and Hara, S. (1997), Interior-point methods for the monotone linear complementarity problem in symmetric matrices, SIAM Journal on Optimization 7: 86-125.
16. Padberg, M. (1989), The Boolean quadric polytope: Some characteristics, facets and relatives, Mathematical Programming 45: 139-172.
17. Pardalos, P.M. and Rodgers, G.P. (1990), Computational aspects of a branch and bound algorithm for quadratic zero-one programming, Computing 45: 131-144.
18. Pardalos, P.M. and Vavasis, S.A. (1991), Quadratic programming with one negative eigenvalue is NP-hard, Journal of Global Optimization 1: 15-22.
19. Poljak, S. and Rendl, F. (1995), Solving the max-cut problem using eigenvalues, Discrete Applied Mathematics 62: 249-278.
20. Ramana, M. (1993), An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems, PhD thesis, Johns Hopkins University, Baltimore, MD.
21. Sherali, H. and Alameddine, A. (1990), An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes, Annals of Operations Research 25: 197-214.
22. Sherali, H. and Alameddine, A. (1992), A new reformulation-linearization for solving bilinear programming problems, Journal of Global Optimization 2: 397-410.
23. Sherali, H., Lee, Y. and Adams, W.P. (1995), A simultaneous lifting strategy for identifying new class of facets for Boolean quadric polytope, Operations Research Letters 17: 19-26.
24. Sherali, H. and Tuncbilek, C.H. (1995), A reformulation-convexification approach for solving nonconvex quadratic programming problems, Journal of Global Optimization 7: 1-31.
25. Simone, C.D. (1989), The cut polytope and the Boolean quadric polytope, Discrete Mathematics 79: 71-75.
26. Vavasis, S.A. (1992), Approximate algorithms for indefinite quadratic programming, Mathematical Programming 57: 279-311.
27. Ye, Y (1992), On the affine scaling algorithm for nonconvex quadratic programming, Mathematical Programming 56: 285-300.