Skip to main content
SpringerLink
Log in

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The optimal value of a polynomial optimization over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semi-algebraic feasible sets, for which an associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. For such problems, we show that the corresponding hierarchy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions, the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.

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

Notes

  1. A polynomial \(f\) is SOS-convex if its Hessian \(\nabla ^2 f(x)\) factors as \(L(x)L(x)^T\) for some matrix polynomial \(L(x)\).

  2. The hierarchy of semidefinite programs (3) was studied in [15] for convex POPs for which the monotone convergence, \(f_k\uparrow f^*\) as \(k\rightarrow \infty \), has been proved to be true always (i.e. without any regularity condition).

  3. As shown in [15], for a convex polynomial \(f\), the positive definiteness of the Hessian of \(f\) at a single point guarantees coercivity of \(f.\)

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

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

  4. Schmüdgen, K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289, 203–206 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Putinar, N.: Positive polynomials on compact semi-algebraic sets. Ind. Uni. Math. J. 41, 49–95 (1993)

    MathSciNet  Google Scholar 

  6. Klerk, E., Laurent, M.: On the Lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems. SIAM J. Optim. 21(3), 824–832 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nie, J.: Optimality conditions and finite convergence of Lasserre’s hierarchy. Math. prog. Ser. A. doi:10.1007/s10107-013-0680-x (2013)

  8. Marshall, M.: Representation of non-negative polynomials, degree bounds and applications to optimization. Canadian J. Math. 61, 205–221 (2009)

    Article  MATH  Google Scholar 

  9. Scheiderer, C.: Sums of squares on real algebraic curves. Math. Zeit. 245, 725–760 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Hanzon, B., Jibetean, D.: Global minimization of a multivariate polynomial using matrix methods. J. Global Optim. 27, 1–23 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jibetean, D., Laurent, M.: Semidefinite approximations for global unconstrained polynomial optimization. SIAM J. Optim. 16, 490–514 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Jeyakumar, V., Són, P.T., Li, G.: Convergence of the Lasserre hierarchy of SDP relaxations for convex polynomial programs without compactness. Oper. Res. Lett. 42(1), 34–40 (2014)

  16. Löfberg, J.: Pre- and post-processing sum-of-squares programs in practice. IEEE Trans. Autom. Cont. 54, 1007–1011 (2009)

    Article  Google Scholar 

  17. Löfberg, J.: YALMIP : A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan, (2004)

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of New South Wales, Sydney , 2052, Australia

    V. Jeyakumar & G. Li

  2. LAAS-CNRS and Institute of Mathematics, Toulouse, France

    J. B. Lasserre

Authors
  1. V. Jeyakumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. B. Lasserre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Jeyakumar.

Additional information

Communicated by Lionel Thibault.

The work of J. B. Lasserre was partially done while he was a Faculty of Science Visiting Fellow at UNSW.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jeyakumar, V., Lasserre, J.B. & Li, G. On Polynomial Optimization Over Non-compact Semi-algebraic Sets. J Optim Theory Appl 163, 707–718 (2014). https://doi.org/10.1007/s10957-014-0545-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0545-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation