Canonical dual solutions for fixed cost quadratic programs

Springer Optimization and Its Applications

Volumn 39, Issue , 2010, Pages 139-156

  Gao, David Yang ^a   Ruan, Ning ^b   Sherali, Hanif D ^b  

^a UNIVERSITY OF BALLARAT   (Australia)

^b Virginia Polytechnic Institute and State University   (United States)

Author keywords

Canonical duality; Fixed charge objective function; Global optimization; Lagrangian duality; Mixed integer programming

Indexed keywords

EID: 84976351028     PISSN: 19316828     EISSN: 19316836     Source Type: Book Series    
DOI: 10.1007/978-0-387-89496-6_7     Document Type: Chapter

References (45)

1. Aardal, K.: Capacitated facility location: separation algorithms and computational experience. Math. Program. 81(2, Ser. B), 149-175 (1998)
2. Akrotirianakis, I.G., Floudas, C.A.: Computational experience with a new class of convex underestimators: Box-constrained NLP problems, J. Global Optim. 29, 249-264 (2004)
3. Akrotirianakis, I.G., Floudas, C.A.: A new class of improved convex underestimators for twice continuously differentiable constrained NLPs. J. Global Optim. 30, 367-390 (2004)
4. Atamtürk, A.: Flow pack facets of the single node fixed-charge flow polytope. Oper. Res. Lett. 29(3), 107-114 (2001)
5. Barany, I., Van Roy, T.J., Wolsey, L.A.: Strong formulations for multi-item capacitated lot sizing. Manage. Sci. 30, 1255-1261 (1984)
6. Contesse, L.: Une caractérisation compléte des minima locaux en programmation quadratique. Numer. Math. 34, 315-332 (1980)
7. Fang, S.-C., Gao, D.Y., Shue, R.L., Wu, S.Y.: Canonical dual approach to solving 0-1 quadratic programming problems. J. Ind. Manage. Optim. 4(1), 125-142 (2008)
8. Fang, S.-C., Gao, D.Y., Sheu, R.-L., Xing, W.X.: Global optimization for a class of fractional programming problems. J. Global Optim. 45, 337-353 (2009)
9. Floudas, C.A.: Deterministic Optimization. Theory, Methods, and Applications, Kluwer, Dordrecht (2000)
10. Floudas, C.A., Akrotirianakis, I.G., Caratzoulas, S., Meyer, C.A., Kallrath, J.:Global optimization in the 21st century: advances and challenges. Comput. Chem. Eng. 29, 1185-1202 (2005)
11. Floudas, C.A., Visweswaran, V.: A primal-relaxed dual global optimization approach. J. Optim. Theory Appl., 78(2), 187-225 (1993)
12. Floudas, C.A., Visweswaran, V.: Quadratic optimization. In: R. Horst, P.M. Pardalos (Eds.) Handbook of Global Optimization Kluwer, Dordrecht 217-270 (1995)
13. Gao, D.Y.: Duality, triality and complementary extremum principles in nonconvex parametric variational problems with applications. IMA J. Appl. Math. 61, 199-235 (1998)
14. Gao, D.Y.: Analytic solution and triality theory for nonconvex and nonsmooth variational problems with applications. Nonlinear Anal. 42(7), 1161-1193 (2000a)
15. Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications, Kluwer, Dordrecht (2000b)
16. Gao, D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Global Optim. 17(1/4), 127-160 (2000c)
17. Gao, D.Y.: Perfect duality theory and complete solutions to a class of global optimization problems. Optimization 52(4-5), 467-493 (2003a)
18. Gao, D.Y.: Nonconvex semi-linear problems and canonical dual solutions. In: D.Y. Gao, R.W. Ogden (Ed.) Advances in Mechanics and Mathematics (Vol. II, pp. 261-312), Kluwer, Dordrecht (2003b)
19. Gao, D.Y.: Canonical duality theory and solutions to constrained nonconvex quadratic programming. J. Global Optim. 29, 377-399 (2004)
20. Gao, D.Y.: Complete solutions and extremality criteria to polynomial optimization problems. J. Global Optim. 35, 131-143 (2006)
21. Gao, D.Y.: Solutions and optimality to box constrained nonconvex minimization problems. J. Ind. Manage Optim. 3(2), 293-304 (2007a)
22. Gao, D.Y.: Duality-Mathematics. Wiley Encyclopedia of Electrical and Electronics Engineering Vol. 6, pp. 68-77 (1st ed., 1999), Electronic edition, Wiley, New York (2007b)
23. Gao, D.Y.: Canonical duality theory: Unified understanding and generalized solution for global optimization problems. Comput. Chem. Eng. 33, 1964-1972 (2009) doi: 10.1016/j.compchemeng.2009.06.009
24. Gao, D.Y., Ogden, R.W.: Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem. Zeits. Angewandte Math. Phy. 59(3), 498-517 (2008a)
25. Gao, D.Y., Ogden, R.W.: Multi-solutions to nonconvex variational problems with implications for phase transitions and numerical computation. Quart. J. Mech. Appl. Math. 61(4), 497-522 (2008b)
26. Gao, D.Y., Ruan, N.: Complete solutions and optimality criteria for nonconvex quadratic-exponential minimization problem. Math. Methods Oper. Res. 67(3), 479-496 (2008c)
27. Gao, D.Y., Ruan, N.: On the solutions to quadratic minimization problems with box and integer constraints. J. Global Optim. to appear (2009a)
28. Gao, D.Y., Ruan, N., Sherali, H.D.: Solutions and optimality criteria for nonconvex constrained global optimization problems. J. Global Optim. to appear (2009b)
29. Gao, D.Y., Sherali, H.D.: Canonical duality theory: connections between nonconvex mechanics and global optimization. In: D.Y. Gao, H.D. Sherali (Eds.), Advances in Applied Mathematics and Global Optimization, 257-326, Springer (2009)
30. Gao, D.Y., Strang, G.: Geometric nonlinearity: potential energy, complementary energy, and the gap function. Quart. Appl. Math. 47(3), 487-504 (1989)
31. Gao, D.Y., Yu, H.: Multi-scale modelling and canonical dual finite element method in phase transitions of solids. Int. J. Solids Struct. 45, 3660-3673 (2008)
32. Glover, F., Sherali H.D.: Some classes of valid inequalities and convex hull characterizations for dynamic fixed-charge problems under nested constraints. Ann. Oper. Res. 40(1), 215-234 (2005)
33. Grippo, L., Lucidi, S.: A differentiable exact penalty function for bound constrained quadratic programming problems. Optimization 22(4), 557-578 (1991)
34. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted flow cover inequalities for mixed 0-1 integer programs. Math. Program. 85(3, Ser. A), 439-467 (1999)
35. Han, C.G., Pardalos, P.M., Ye, Y.: An interior point algorithm for large-scale quadratic problems with box constraints. In: A. Bensoussan, J.L. Lions (Eds.), Springer-Verlag Lecture Notes in Control and Information (Vol. 144, pp. 413-422) (1990)
36. Hansen, P., Jaumard, B., Ruiz, M., Xiong, J.: Global minimization of indefinite quadratic functions subject to box constraints. Nav. Res. Logist. 40, 373-392 (1993)
37. Li, S.F., Gupta, A.: On dual configuration forces. J. Elasticity 84, 13-31 (2006)
38. Murty, K.G., Kabadi, S.N.: Some NP-hard problems in quadratic and nonlinear programming. Math. Program. 39, 117-129 (1987)
39. Padberg, M.W., Van Roy, T.J., Wolsey, L.A.: Valid linear inequalities for fixed charge problems. Oper. Res. 33, 842-861 (1985)
40. Pardalos, P.M., Schnitger, G.: Checking local optimality in constrained quadratic and nonlinear programming. Oper. Res. Lett. 7, 33-35 (1988)
41. Sherali, H.D., Smith, J.C.: An improved linearization strategy for zero-one quadratic programming problems. Optim. Lett. 1(1), 33-47 (2007)
42. Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problem using a reformulation-linearization technique. J. Global Optim. 2, 101-112 (1992)
43. Sherali, H.D., Tuncbilek, C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Global Optim. 7, 1-31 (1995)
44. Sherali, H.D., Tuncbilek, C.H.: New reformulation-linearization technique based relaxation for univariate and multivariate polynominal programming problems. Oper. Res. Lett. 21(1), 1-10 (1997)
45. Wang, Z.B., Fang, S.-C., Gao, D.Y., Xing, W.X.: Global extremal conditions for multi-integer quadratic programming. J. Ind. Manage. Optim. 4(2), 213-225 (2008)