Advertisement

Canonical Dual Solutions for Fixed Cost Quadratic Programs

  • David Yang Gao
  • Ning Ruan
  • Hanif D. Sherali
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 39)

Summary

This chapter presents a canonical dual approach for solving a mixed-integer quadratic minimization problem with fixed cost terms. We show that this well-known NP-hard problem in \(\mathbb{R}^{2n}\) can be transformed into a continuous concave maximization dual problem over a convex feasible subset of \(\mathbb{R}^{2n}\) with zero duality gap. The resulting canonical dual problem can be solved easily, under certain conditions, by traditional convex programming methods. Both existence and uniqueness of global optimal solutions are discussed. Application to a decoupled mixed-integer problem is illustrated and analytic solutions for both a global minimizer and a global maximizer are obtained. Examples for both decoupled and general nonconvex problems are presented. Furthermore, we discuss connections between the proposed canonical duality theory approach and the classical Lagrangian duality approach. An open problem is proposed for future study.

Keywords

canonical duality Lagrangian duality global optimization mixed-integer programming fixed-charge objective function 

References

  1. 1.
    Aardal, K.: Capacitated facility location: separation algorithms and computational experience. Math. Program. 81(2, Ser. B), 149–175 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Atamtürk, A.: Flow pack facets of the single node fixed-charge flow polytope. Oper. Res. Lett. 29(3), 107–114 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Barany, I., Van Roy, T.J., Wolsey, L.A.: Strong formulations for multi-item capacitated lot sizing. Manage. Sci. 30, 1255–1261 (1984)zbMATHCrossRefGoogle Scholar
  6. 6.
    Contesse, L.: Une caractérisation compléte des minima locaux en programmation quadratique. Numer. Math. 34, 315–332 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Floudas, C.A.: Deterministic Optimization. Theory, Methods, and Applications, Kluwer, Dordrecht (2000)Google Scholar
  10. 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)CrossRefGoogle Scholar
  11. 11.
    Floudas, C.A., Visweswaran, V.: A primal-relaxed dual global optimization approach. J. Optim. Theory Appl., 78(2), 187–225 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Floudas, C.A., Visweswaran, V.: Quadratic optimization. In: R. Horst, P.M. Pardalos (Eds.) Handbook of Global Optimization Kluwer, Dordrecht 217–270 (1995)Google Scholar
  13. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gao, D.Y.: Analytic solution and triality theory for nonconvex and nonsmooth variational problems with applications. Nonlinear Anal. 42(7), 1161–1193 (2000a)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications, Kluwer, Dordrecht (2000b)zbMATHGoogle Scholar
  16. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gao, D.Y.: Perfect duality theory and complete solutions to a class of global optimization problems. Optimization 52(4–5), 467–493 (2003a)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 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)CrossRefGoogle Scholar
  19. 19.
    Gao, D.Y.: Canonical duality theory and solutions to constrained nonconvex quadratic programming. J. Global Optim. 29, 377–399 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gao, D.Y.: Complete solutions and extremality criteria to polynomial optimization problems. J. Global Optim. 35, 131–143 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gao, D.Y.: Solutions and optimality to box constrained nonconvex minimization problems. J. Ind. Manage Optim. 3(2), 293–304 (2007a)zbMATHCrossRefGoogle Scholar
  22. 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)Google Scholar
  23. 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.2009.06.009Google Scholar
  24. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Gao, D.Y., Ruan, N.: On the solutions to quadratic minimization problems with box and integer constraints. J. Global Optim. to appear (2009a)Google Scholar
  28. 28.
  29. 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)CrossRefGoogle Scholar
  30. 30.
    Gao, D.Y., Strang, G.: Geometric nonlinearity: potential energy, complementary energy, and the gap function. Quart. Appl. Math. 47(3), 487–504 (1989)MathSciNetzbMATHGoogle Scholar
  31. 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)zbMATHCrossRefGoogle Scholar
  32. 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)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Grippo, L., Lucidi, S.: A differentiable exact penalty function for bound constrained quadratic programming problems. Optimization 22(4), 557–578 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 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)Google Scholar
  36. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Li, S.F., Gupta, A.: On dual configuration forces. J. Elasticity 84, 13–31 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Murty, K.G., Kabadi, S.N.: Some NP-hard problems in quadratic and nonlinear programming. Math. Program. 39, 117–129 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Padberg, M.W., Van Roy, T.J., Wolsey, L.A.: Valid linear inequalities for fixed charge problems. Oper. Res. 33, 842-861 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Pardalos, P.M., Schnitger, G.: Checking local optimality in constrained quadratic and nonlinear programming. Oper. Res. Lett. 7, 33–35 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Sherali, H.D., Smith, J.C.: An improved linearization strategy for zero-one quadratic programming problems. Optim. Lett. 1(1), 33–47 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Sherali, H.D., Tuncbilek, C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Global Optim. 7, 1–31 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 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)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Graduate School of Information Technology and Mathematical SciencesUniversity of BallaratMt HelenAustralia
  2. 2.Grado Department of Industrial and Systems EngineeringVirginia TechBlacksburgUSA

Personalised recommendations