Journal of Global Optimization

, Volume 48, Issue 4, pp 549–567 | Cite as

Partitioning procedure for polynomial optimization

  • Polyxeni-Margarita Kleniati
  • Panos Parpas
  • Berç Rustem
Article

Abstract

We consider the problem of finding the minimum of a real-valued multivariate polynomial function constrained in a compact set defined by polynomial inequalities and equalities. This problem, called polynomial optimization problem (POP), is generally nonconvex and has been of growing interest to many researchers in recent years. Our goal is to tackle POPs using decomposition, based on a partitioning procedure. The problem manipulations are in line with the pattern used in the generalized Benders decomposition, namely projection followed by relaxation. Stengle’s and Putinar’s Positivstellensätze are employed to derive the feasibility and optimality constraints, respectively. We test the performance of the proposed partitioning procedure on a collection of benchmark problems and present the numerical results.

Keywords

Polynomial optimization Positivstellensatz Sum of squares Benders decomposition 

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References

  1. 1.
  2. 2.
    Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization. MPS/SIAM Series on Optimization, SIAM, Philadelphia, PA (2001)Google 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(1962), pp. 238–252CrossRefGoogle Scholar
  4. 4.
    Boyd S., Vandenberghe L.: Convex Optimization. Cambridge University Press, Cambridge (2004)Google Scholar
  5. 5.
    Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, New York (1992)Google Scholar
  6. 6.
    Floudas, C.A.: Deterministic global optimization, Nonconvex Optimization and its Applications, vol 37. Kluwer, Dordrecht, Theory, methods and applications (2000)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Floudas C.A., Visweswaran V.: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs: I. Theory Comput. Chem. Eng. 14(12), 1397–1417 (1990)CrossRefGoogle Scholar
  9. 9.
    Geoffrion A.M.: Elements of large-scale mathematical programming. I. Concepts. Manag. Sci. 16, 652–675 (1970)CrossRefGoogle Scholar
  10. 10.
    Geoffrion A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972)CrossRefGoogle Scholar
  11. 11.
    Henrion D., Lasserre J.B.: GloptiPoly: global optimization over polynomials with Matlab and SeDuMi. ACM Trans. Math. Softw. 29(2), 165–194 (2003)CrossRefGoogle Scholar
  12. 12.
    Hogan W.W.: Point-to-set maps in mathematical programming. SIAM Rev. 15, 591–603 (1973)CrossRefGoogle Scholar
  13. 13.
    Kleniati, P.M., Parpas, P., Rustem, B.: Decomposition-based method for sparse semidefinite relaxations of polynomial optimization problems. J. Optim. Theory Appl. doi:10.1007/s10957-009-9624-2 (2009)
  14. 14.
    Lasserre J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2000/2001)CrossRefGoogle Scholar
  15. 15.
    Lasserre J.B.: Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006)CrossRefGoogle Scholar
  16. 16.
    Laurent M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds) Emerging applications of algebraic geometry, IMA Vol. in Math. and its Appl., vol 149., pp. 157–270. Springer, New York (2009)Google Scholar
  17. 17.
    Meyer R.: The validity of a family of optimization methods. SIAM J. Control 8, 41–54 (1970)CrossRefGoogle Scholar
  18. 18.
    Nie J., Schweighofer M.: On the complexity of Putinar’s Positivstellensatz. J. Complex. 23(1), 135–150 (2007)CrossRefGoogle Scholar
  19. 19.
    Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, California Institute of Technology (2000)Google Scholar
  20. 20.
    Parrilo P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96(2, Ser. B), 293–320 (2003)CrossRefGoogle Scholar
  21. 21.
    Prestel A., Delzell C.N.: Positive Polynomials. Springer Monographs in Mathematics. Springer, Berlin (2001)Google Scholar
  22. 22.
    Putinar M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)CrossRefGoogle Scholar
  23. 23.
    Schmüdgen K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289(2), 203–206 (1991)CrossRefGoogle Scholar
  24. 24.
    Schweighofer M.: Optimization of polynomials on compact semialgebraic sets. SIAM J. Optim. 15(3), 805–825 (2005)CrossRefGoogle Scholar
  25. 25.
    Stengle G.: A nullstellensatz and a positivstellensatz in semialgebraic geometry. Math. Ann. 207, 87–97 (1974)CrossRefGoogle 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)CrossRefGoogle Scholar
  27. 27.
    Tind J., Wolsey L.A.: An elementary survey of general duality theory in mathematical programming. Math. Program. 21(3), 241–261 (1981)CrossRefGoogle Scholar
  28. 28.
    Tuy H.: Convex Analysis and Global Optimization, Nonconvex Optimization and its Applications, vol 22. Kluwer, Dordrecht (1998)Google Scholar
  29. 29.
    Visweswaran V., Floudas C.A.: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs: II. Application of theory and test problems. Comput. Chem. Eng. 14(12), 1419–1434 (1990)CrossRefGoogle Scholar
  30. 30.
    Visweswaran V., Floudas C.A.: Unconstrained and constrained global optimization of polynomial functions in one variable. J. Global Optim. 2(1), 73–99 (1992)CrossRefGoogle Scholar
  31. 31.
    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)CrossRefGoogle Scholar
  32. 32.
    Wolsey L.A.: A resource decomposition algorithm for general mathematical programs. Math. Program. Stud. 14, 244–257 (1981)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • Polyxeni-Margarita Kleniati
    • 1
  • Panos Parpas
    • 2
  • Berç Rustem
    • 1
  1. 1.Department of ComputingImperial College LondonLondonUK
  2. 2.Energy Initiative Engineering Systems DivisionMassachusetts Institute of TechnologyCambridgeUK

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