Decomposition-based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems

Article

Abstract

We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in Lasserre (SIAM J. Optim. 17(3):822–843, 2006) and Waki et al. (SIAM J. Optim. 17(1):218–248, 2006) that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decomposition-based method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs (Benders, Comput. Manag. Sci. 2(1):3–19, 2005).

Keywords

Polynomial optimization Semidefinite programming Sparse SDP relaxations Benders decomposition 

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References

  1. 1.
    Lasserre, J.B.: Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–242 (2006) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Comput. Manag. Sci. 2(1), 3–19 (2005). Reprinted from Numer. Math. 4, 238–252 (1962) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ramana, M., Goldman, A.J.: Some geometric results in semidefinite programming. J. Glob. Optim. 7(1), 33–50 (1995) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kim, S., Kojima, M., Toint, P.: Recognizing underlying sparsity in optimization. Technical Report, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology (2006) Google Scholar
  6. 6.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: SparsePOP: a sparse semidefinite programming relaxation of polynomial optimization problems. ACM Trans. Math. Softw. 35(2) (2008) Google Scholar
  7. 7.
    Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5(1), 13–51 (1995) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Pardalos, P.M., Wolkowicz, H. (eds.): Topics in Semidefinite and Interior-point Methods. Fields Institute Communications, vol. 18. American Mathematical Society, Providence (1998) MATHGoogle Scholar
  9. 9.
    Ramana, M.V., Pardalos, P.M.: Semidefinite programming. In: Interior Point Methods of Mathematical Programming. Appl. Optim., vol. 5, pp. 369–398. Kluwer Academic, Dordrecht (1996) Google Scholar
  10. 10.
    Pataki, G.: The geometry of semidefinite programming. In: Handbook of Semidefinite Programming. Internat. Ser. Oper. Res. Management Sci., vol. 27, pp. 29–65. Kluwer Academic, Dordrecht (2000) Google Scholar
  11. 11.
    Yajima, Y., Ramana, M.V., Pardalos, P.M.: Cuts and semidefinite relaxations for nonconvex quadratic problems. In: Generalized Convexity and Generalized Monotonicity, Karlovassi, 1999. Lecture Notes in Econom. and Math. Systems, vol. 502, pp. 48–70. Springer, Berlin (2001) Google Scholar
  12. 12.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2000/01) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program., Ser. B 96(2), 293–320 (2003). Algebraic and geometric methods in discrete optimization MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging Applications of Algebraic Geometry. IMA Vol. Math. Appl., vol. 149, pp. 157–270. Springer, New York (2009) Google Scholar
  15. 15.
    Floudas, C.A., Pardalos, P.M.: A collection of test problems for constrained global optimization algorithms. Lecture Notes in Computer Science, vol. 455. Springer, Berlin (1990) MATHGoogle Scholar
  16. 16.
    LP_SOLVE Description (2008). tech.groups.yahoo.com/group/lp_solve
  17. 17.
    Geoffrion, A.M.: Generalized Benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Meyer, R.: The validity of a family of optimization methods. SIAM J. Control Optim. 8, 41–54 (1970) MATHCrossRefGoogle Scholar
  19. 19.
    Grothey, A., Leyffer, S., Mckinnon, K.I.M.: A note on feasibility in benders decomposition (1999) Google Scholar
  20. 20.
    Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior point method for semidefinite programming. SIAM J. Optim. 6(2), 342–361 (1996) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
  22. 22.
    Davis, T.A.: User Guide for CHOLMOD: a sparse Cholesky factorization and modification package. Technical Report, Dept. of Computer and Information Science and Engineering, University of Florida (2006) Google Scholar
  23. 23.
    Borchers, B.: CSDP, A C library for semidefinite programming. Optim. Methods Softw. 11, 613–623 (1999) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Benson, S.J., Ye, Y.: DSDP5: Software for semidefinite programming. Technical Report, Mathematics and Computer Science Division, Argonne National Laboratory (2005) Google Scholar
  25. 25.
    Fujisawa, K., Kojima, M., Nakata, K., Yamashita, M.: SDPA (SemiDefinite Programming Algorithm) User’s Manual—Version 5.01. Technical Report, Tokyo Institute of Technology (1995) Google Scholar
  26. 26.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11/12(1–4), 625–653 (1999) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Kleniati, P.M.: Decomposition Schemes for Polynomial Optimization, Semidefinite Programming and Applications to Nonconvex Portfolio Decisions. Ph.D. Thesis, Imperial College London (2009) Google Scholar
  28. 28.
    Zhao, G.: A log-barrier method with Benders decomposition for solving two-stage stochastic linear programs. Math. Program., Ser. A 90(3), 507–536 (2001) MATHCrossRefGoogle Scholar
  29. 29.
    Sivaramakrishnan, K.K., Plaza, G., Terlaky, T.: A conic interior point decomposition approach for large scale semidefinite programming (2005) Google Scholar
  30. 30.
    Sivaramakrishnan, K.K.: A parallel interior point decomposition algorithm for block angular semidefinite programs. Comput. Optim. Appl. (2008) Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of ComputingImperial College LondonLondonUK

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