Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs

Optimization Methods and Software

Volumn 24, Issue 4-5, 2009, Pages 485-504

  Bao, Xiaowei ^a   Sahinidis, Nikolaos V ^b   Tawarmalani, Mohit ^c  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

^c PURDUE UNIVERSITY   (United States)

Author keywords

Convex envelope; Cutting plane; Global optimization; Nonconvex quadratic programming; Polyhedral relaxation

Indexed keywords

BILINEAR FUNCTIONS; COMPUTATIONAL RESULTS; CONVEX ENVELOPE; CUTTING PLANE; CUTTING PLANE ALGORITHMS; CUTTING PLANES; MULTILINEAR FUNCTIONS; NONCONVEX; NONCONVEX QUADRATIC PROGRAMMING; POLYHEDRAL RELAXATION; QUADRATIC CONSTRAINT; QUADRATICALLY-CONSTRAINED QUADRATIC PROGRAMS;

GLOBAL OPTIMIZATION; OPTIMIZATION; QUADRATIC PROGRAMMING;

CONVEX OPTIMIZATION;

EID: 68949144575     PISSN: 10556788     EISSN: 10294937     Source Type: Journal    
DOI: 10.1080/10556780902883184     Document Type: Article

References (68)

1. N. Adhya, M. Tawarmalani, and N.V. Sahinidis, A Lagrangian approach to the pooling problem, Ind. Eng. Chem. 38 (1999), pp. 1956-1972.
2. F.A. Al-Khayyal, Generalized bilinear programming: Part I, models, applications and linear programming relaxation, Eur. J. Oper. Res. 60 (1992), pp. 306-314.
3. F.A. Al-Khayyal and J.E. Falk, Jointly constrained biconvex programming, Math. Oper. Res. 8 (1983), pp. 273-286.
4. F.A. Al-Khayyal, C. Larsen, and T. Van Voorhis, A relaxation method for nonconvex quadratically constrained quadratic programs, J. Global Optim. 6 (1995), pp. 215-230.
5. K.M. Anstreicher, Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming, Optimization Online Digest (2007). Available at http://www.optimization-online.org/DB-HTML/05/1655.html.
6. F. Barahona, M. Jünger, and G. Reinelt, Experiments in quadratic 0-1 programming, Math. Program. 44 (1989), pp. 127-137.
7. C.B. Barber, D.P. Dobkin, and H.T. Huhdanpaa, Lectures on modern convex optimization, ACMTrans. Math. Softw. 22 (1996), pp. 469-483. Available at http://www.qhull.org
8. H.P. Benson, On the construction of convex and concave envelope formulas for bilinear and fractional functions on quadrilaterals, Comput. Optim. Appl. 27 (2007), pp. 5-22.
9. D. Bertsimas and J.N. Tsitsiklis, Introduction to Linear Optimization , Athena Scientific, Nashua, NH, 1997.
10. A. Billionnet and S. Elloumi, Using a mixed integer quadratic programming solver for the unconstrained quadratic 0-1 problem, Math. Program. 109 (2007), pp. 55-68.
11. E. Boros, P.L. Hammer, and G. Tavares, Local search heuristics for quadratic unconstrained binary optimization (QUBO), J. Heuristics 13 (2007), pp. 99-132.
12. A. Brooke, D. Kendrick, and A. Meeraus, GAMS-A User's Guide, The Scientific Press, Redwood City, CA, 1988.
13. M.C. Dorneich and N.V. Sahinidis, Global optimization algorithms for chip layout and compaction, Eng. Optim. 25 (1995), pp. 131-154.
14. S.C. Fang, D.Y. Gao, R.L. Sheu, and S.Y. Wu, Canonical dual approach to solving 0-1 quadratic programming problems, J. Ind. Manag. Optim. 4 (2008), pp. 124-142.
15. C.A. Floudas and V. Visweswaran, Quadratic optimization, in Handbook of Global Optimization, Nonconvex Optimization and its Applications, Vol. 2, Chapter 5, Kluwer Academic Publishers, Dordrecht, 1995, 2, pp. 217-269.
16. M.X. Goemans and D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42 (1995), pp. 1115-1145.
17. E.P. Hao, Quadratically constrained quadratic programming: Some applications and a method for solution, Math. Methods Oper. Res. 26 (1982), pp. 105-119.
18. ILOG, CPLEX 9.0 User's Manual, ILOG CPLEX Division, Incline Village, NV, 2003.
19. S. Kim and M. Kojima, Second order cone programming relaxation of nonconvex quadratic optimization problems, Optim. Methods Softw. 15 (2001), pp. 201-224.
20. S. Kim and M. Kojima, Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations, Comput. Optim. Appl. 26 (2003), pp. 143-154.
21. M. Kojima and L. Tunçel, Some fundamental properties of successive convex relaxation methods on LCP and related problems, J. Global Optim. 24 (2002), pp. 333-348.
22. Y. Lin and L. Schrage, The global solver in the LINDO API, Optim. Methods Softw. 24 (2009), pp. 657-668.
23. J. Linderoth, A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Math. Program. 103 (2005), pp. 251-282.
24. M.L. Liu and N.V. Sahinidis, Process planning in a fuzzy environment, Eur. J. Oper. Res. 100 (1997), pp. 142-169.
25. M. Locatelli and U. Raber, Packing equal circles in a square: A deterministic global optimization approach, Discrete Appl. Math. 122 (2002), pp. 139-166.
26. R.D. McBride and J.S. Yormark, An implicit enumeration algorithm for quadratic integer programming, Manag. Sci. 26 (1980), pp. 282-296.
27. C.A. Meyer and C.A. Floudas, Convex hull of trilinear monomials with positive or negative domains: Facets of the convex and concave envelopes, in Frontiers in Global Optimization, C.A. Floudas and P.M. Pardalos, eds., Kluwer, Boston, 2003, pp. 327-352.
28. C.A. Meyer and C.A. Floudas, Trilinear monomials with mixed sign domains: Facets of the convex and concave envelopes, J. Glob. Optim. 29 (2004), pp. 125-155.
29. C.A. Meyer and C.A. Floudas, Convex envelopes for edge-concave functions, Math. Program. 103 (2005), pp. 207-224.
30. G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, Series in Discrete Mathematics and Optimization, Wiley Interscience, NewYork, NY, 1988.
31. I. Nowak, A new semidefinite programming bound for indefinite quadratic forms over a simplex, J. Global Optim. 14 (1999), pp. 357-364.
32. I. Nowak, Dual bounds and optimality cuts for all-quadratic programs with convex constraints, J. Global Optim. 18 (2000), pp. 337-356.
33. G. Palubeckis, Multistart tabu search strategies for the unconstrained binary quadratic optimization problem, Ann. Oper. Res. 131 (2004), pp. 259-282.
34. P.M. Pardalos and S.A.Vavasis, Quadratic programming with one negative eigenvalue is NP-hard, J. Global Optim. 1 (1991), pp. 15-22.
35. G. Parija, R. Gadidov, and W. Wilhelm, A facet generation procedure for solving 0/1 integer programs, Oper. Res. 47 (1999), pp. 789-791.
36. J.C. Picard and H.D. Ratliff, Minimum cuts and related problems, Networks 5 (1974), pp. 357-370.
37. S. Qu, H. Yin, and K. Zhang, A global optimization algorithm using linear relaxation, Appl. Math. Comput. 178 (2006), pp. 510-518.
38. U. Raber, Nonconvex all-quadratic global optimization problems: Solution method, application and related topics, Ph.D. thesis, Universitär Trier, Germany, 1999.
39. A.D. Rikun, A convex envelope formula for multilinear functions, J. Global Optim. 10 (1997), pp. 425-437.
40. G. Rote, The convergence rate of the sandwich algorithm for approximating convex functions, Computing 48 (1992), pp. 337-361.
41. N.V. Sahinidis and M. Tawarmalani, Accelerating branch-and-bound through a modeling language construct for relaxation-specific constraints, J. Global Optim. 32 (2005), pp. 259-280.
42. N.V. Sahinidis and M. Tawarmalani, BARON 8.1.1: Global Optimization of Mixed-Integer Nonlinear Programs, User's Manual, 2008. Available at http://www.gams.com/dd/docs/solvers/baron.pdf.
43. A. Saxena, P. Bonami, and J. Lee, Convex relaxations of non-convex mixed integer quadratically constrained problems: Extended formulations, IBM Research Report RC24621, 2008.
44. H.D. Sherali, A constructive proof of the representation theorem for polyhedral sets based on fundamental definitions, Am. J. Math. Manage. Sci. 7 (1987), pp. 253-270.
45. H.D. Sherali, Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets, Acta Math. Vietnam. 22 (1997), pp. 245-270.
46. H.D. Sherali and A. Alameddine, A new reformulation-linearization technique for bilinear programming problems, J. Global Optim. 2 (1992), pp. 379-410.
47. H.D. Sherali and B.M.P. Fraticelli, Enhancing RLT relaxation via a new class of semidefinite cuts, J. Global Optim. 22 (2002), pp. 233-261.
48. H.D. Sherali and C.H. Tuncbilek, A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique, J. Global Optim. 2 (1992), pp. 101-112.
49. H.D. Sherali and C.H. Tuncbilek, A reformulation-convexification approach for solving nonconvex quadratic programming problems, J. Global Optim. 7 (1995), pp. 1-31.
50. N.Z. Shor, Quadratic optimization problems, Sov. J. Comput. and Syst. Sci. 25 (1987), pp. 1-11.
51. N.Z. Shor, Dual estimates in multiextremal problems, J. Global Optim. 2 (1992), pp. 411-418.
52. A. Sutou and Y. Dai, Global optimization approach to unequal sphere packing problems in 3D, J. Optim. Theory Appl. 114 (2002), pp. 671-694.
53. F. Tardella, On the existence of polyhedral convex envelopes, in Frontiers in Global Optimization, C.A. Floudas and P.M. Pardalos, eds., Kluwer, Boston, 2003, pp. 149-188.
54. F. Tardella, Existence and sum decomposition of vertex polyhedral convex envelopes, Optim. Lett. 2 (2008), pp. 363-375.
55. M. Tawarmalani and N.V. Sahinidis, Convex extensions and convex envelopes of l.s.c. functions, Math. Program. 93 (2002), pp. 247-263.
56. M. Tawarmalani and N.V. Sahinidis, Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, Kluwer Academic Publishers, Dordrecht, 2002.
57. M. Tawarmalani and N.V. Sahinidis, Global optimization of mixed-integer nonlinear programs: A theoretical and computational study, Math. Program. 99 (2004), pp. 563-591.
58. N.V. Thoai, Duality bound method for the general quadratic programming problem with quadratic constraints, J. Optim. Theory Appl. 107 (2000), pp. 331-354.
59. H. Tuy, On solving nonconvex optimization problems by reducing the duality gap, J. Global Optim. 32 (2005), pp. 349-365.
60. H. Tuy and N.T. Hoai-Phuong, A robust algorithm for quadratic optimization under quadratic constraints, J. Global Optim. 4 (2007), pp. 557-569.
61. L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38 (1996), pp. 49-95.
62. T. Van Voorhis, A global optimization algorithm using lagrangian underestimates and the interval Newton method, J. Global Optim. 24 (2002), pp. 349-370.
63. T. Van Voorhis and F.A. Al-Khayyal, Difference of convex solution of quadratically constrained optimization problems, Eur. J. Oper. Res. 148 (2003), pp. 349-362.
64. S.A. Vavasis, Quadratic programming is in NP, Inf. Process. Lett. 36 (1990), pp. 73-77.
65. S.A. Vavasis, Nonlinear Optimization: Complexity Issues, Oxford University Press, 1991.
66. A.Weintraub and J. Vera, A cutting plane approach for chance constrained linear programs, Oper. Res. 39 (1991), pp. 776-785.
67. W. Xie and N.V. Sahinidis, A branch-and-bound algorithm for the continuous facility layout problem, Comput. Chem. Eng. 32 (2008), pp. 1016-1028.
68. M.Yamashita, K. Fujisawa, and M.Kojima, Implementation and evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0), Optim. Methods Softw. 18 (2002), pp. 491-505.