Skip to main content
SpringerLink
Log in

The hierarchy of local minimums in polynomial optimization

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

This paper studies the hierarchy of local minimums of a polynomial in the vector space \(\mathbb {R}^n\). For this purpose, we first compute \(H\)-minimums, for which the first and second order necessary optimality conditions are satisfied. To compute each \(H\)-minimum, we construct a sequence of semidefinite relaxations, based on optimality conditions. We prove that each constructed sequence has finite convergence, under some generic conditions. A procedure for computing all local minimums is given. When there are equality constraints, we have similar results for computing the hierarchy of critical values and the hierarchy of local minimums.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Series: Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2003)

  2. Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1995)

  3. Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)

  4. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Undergraduate Texts in Mathematics. Springer, New York (1997)

  5. Cui, C., Dai, Y., Nie, J.: All Real Eigenvalues of Symmetric Tensors. SIAM J. Matrix Anal. Appl. (to appear)

  6. Curto, R., Fialkow, L.: Truncated K-moment problems in several variables. J. Oper. Theory 54, 189–226 (2005)

    MATH  MathSciNet  Google Scholar 

  7. De Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084–1127 (2008)

    Article  MathSciNet  Google Scholar 

  8. Greuet, A., Safey El Din, M.: Deciding reachability of the infimum of a multivariate polynomial. In: Proceedings of ISSAC, pp. 131–138 (2011)

  9. Guo, F., Safey El Din, M., Zhi, L.: Global optimization of polynomials using generalized critical values and sums of squares. In: Proceedings of ISSAC, pp. 107–114 (2010)

  10. Henrion, D., Lasserre, J., Loefberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Henrion, D., Lasserre, J.B.: Detecting global optimality and extracting solutions in GloptiPoly. In: Positive polynomials in Control, 293–310, Lecture Notes in Control and Information Sciences, vol. 312. Springer, Berlin (2005)

  12. Henrion, D., Lasserre, J.: Convergent relaxations of polynomial matrix inequalities and static output feedback. IEEE Trans. Autom. Control 51, 192–202 (2006)

    Article  MathSciNet  Google Scholar 

  13. Hu, S., Qi, L.: Algebraic connectivity of an even uniform hypergraphs. J. Comb. Optim. 24, 564–579 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Klep, I., schweighofer, M.: Pure states, positive matrix polynomials and sums of hermitian squares. Indiana Univ. Math. J. 59(3), 857–874 (2010)

  15. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lasserre, J., Laurent, M., Rostalski, P.: Semidefinite characterization and computation of zero-dimensional real radical ideals. Found. Comput. Math. 8(5), 607–647 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)

    Book  Google Scholar 

  18. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, Vol. 149 of IMA Volumes in Mathematics and its Applications, pp. 157–270. Springer, Berlin (2009)

    Chapter  Google Scholar 

  19. Li, G., Qi, L., Yu, G.: The Z-eigenvalues of a symmetric tensor and its applications to spectral hypergraph theory. Numer. Linear Algebra Appl. 20, 1001–1029 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Löfberg, J.: YALMIP: a toolbox for modeling and optimization in Matlab. In: Proceedings of the IEEE CACSD Symposium, Taiwan (2004). www.control.isy.liu.se/johanl

  21. Luo, Z., Pang, J., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

  22. Marshall, M.: Positive Polynomials and Sums of Squares. Mathematical Surveys and Monographs, vol. 146. American Mathematical Society, Providence, RI (2008)

  23. Mohar, B.: Laplance eigenvalues of graphs: a survey. Discret. Math. 109, 171–183 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39, 117–129 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  25. Nesterov, Y.: Squared functional systems and optimization problems. In: Frenk et al. H. (ed.) High Performance Optimization, pp. 405–440. Kluwer, Dordrecht (2000)

  26. Nie, J., Demmel, J., Sturmfels, B.: Minimizing polynomials via sum of squares over the gradient ideal. Math. Program. Ser. A 106(3), 587–606 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Nie, J., Ranestad, K.: Algebraic degree of polynomial optimization. SIAM J. Optim. 20(1), 485–502 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Nie, J.: An exact Jacobian SDP relaxation for polynomial optimization. Math. Program. Ser. A 137, 225–255 (2013)

    Article  MATH  Google Scholar 

  29. Nie, J.: Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program. Ser. A 142(1–2), 485–510 (2013)

  30. Nie, J.: Optimality conditions and finite convergence of Lasserre’s hierarchy. Math. Program. Ser. A 146(1–2), 97–121 (2014)

  31. Parrilo, P.: Semidefinite programming relaxations for semialgebraic problems. Math. Prog. Ser. B 96(2), 293–320 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Parrilo, P.A., Sturmfels, B.: Minimizing polynomial functions. In: Basu, S., Gonzalez-Vega, L. (eds.) Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, volume 60 of DIMACS Series in Discrete Mathematics and Computer Science, pp. 83–99. AMS (2003)

  33. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 969–984 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  34. Qi, L., Teo, K.L.: Multivariate polynomial minimization and its application in signal processing. J. Glob. Optim. 26, 419–433 (2003)

  35. Qi, L., Yu, G., Wu, E.X.: Higher order positive semidefinite diffusion tensor imaging. SIAM J. Imaging Sci. 3, 416–433 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  36. Reznick, B.: Some concrete aspects of Hilbert’s 17th problem. In: Contemporary Mathematics, vol. 253, pp. 251–272. American Mathematical Society (2000)

  37. Safey El Din, M.: Computing the global optimum of a multivariate polynomial over the reals. In: Proceedings of ISSAC, pp. 71–78 (2008)

  38. Schweighofer, M.: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17(3), 920–942 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Scheiderer, C.: Positivity and sums of squares: a guide to recent results. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry. IMA Volumes in Mathematics and Its Applications, vol. 149, pp. 271–324. Springer, Berlin (2009)

  40. Sturm, J.F.: SeDuMi 1.02: a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11&12, 625–653 (1999). http://sedumi.ie.lehigh.edu

  41. Sturmfels, B.: Solving Systems of Polynomial Equations. CBMS Regional Conference Series in Mathematics, vol. 97. American Mathematical Society, Providence, RI (2002)

  42. Vui, H.H., Són, P.T.: Global optimization of polynomials using the truncated tangency variety and sums of squares. SIAM J. Optim. 19(2), 941–951 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author would like to thank Didier Henrion for his comments on using polynomial matrix inequalities in GlotpiPoly and YALMIP. He also likes to thank the editors and anonymous referees for the useful suggestions. The research was partially supported by the NSF Grants DMS-0844775 and DMS-1417985.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Jiawang Nie

Authors
  1. Jiawang Nie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiawang Nie.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nie, J. The hierarchy of local minimums in polynomial optimization. Math. Program. 151, 555–583 (2015). https://doi.org/10.1007/s10107-014-0845-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-014-0845-2

Keywords

Mathematics Subject Classification

Search

Navigation