Abstract
We consider a polynomial programming problem P on a compact basic semi-algebraic set K ⊂ ℝn, described by m polynomial inequalities g j (X)≥0, and with criterion f ∈ ℝ[X]. We propose a hierarchy of semidefinite relaxations that take sparsity of the original data into account, in the spirit of those of Waki et al. [7]. The novelty with respect to [7] is that we prove convergence to the global optimum of P when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the running intersection property in graph theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Curto, R.E., Fialkow, L.A.: The truncated complex K-moment problem. Trans. Amer. Math. Soc. 352, 2825–2855 (2000)
Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework. SIAM J. Optim. 11, 647–674 (2001)
Henrion, D., Lasserre, J.B.: GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi. ACM Trans. Math. Soft. 29, 165–194 (2003)
Henrion, D., Lasserre, J.B.: Detecting global optimality and extracting solutions in GloptiPoly. In: Henrion, D., Garulli, A. (eds.) Positive Polynomials in Control. Lecture Notes on Control and Information Sciences, vol. 312, pp. 293–310. Springer, Berlin (2005)
Kim, S., Kojima, M., Waki, H.: Generalized Lagrangian Duals and Sums of Squares Relaxations of Sparse Polynomial Optimization Problems. SIAM J. Optim. 15, 697–719 (2005)
Kojima, M., Kim, S., Waki, H.: Sparsity in sums of squares of polynomials. Math. Prog. (to appear)
Waki, H., Kim, S., Kojima, M., Maramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems witth structured sparsity. Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo (2004)
Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)
Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonliner 0-1 programs. SIAM J. Optim. 12, 756–769 (2002)
Lasserre, J.B.: Convergent SDP-relaxations for polynomial optimization with sparsity. SIAM J. Optim (to appear)
Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting Sparsity in Semidefinite Programming via Matrix Completion II: Implementation and Numerical Results. Math. Progr. 95, 303–327 (2003)
Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 969–984 (1993)
Schweighofer, M.: Optimization of polynomials on compact semialgebraic sets. SIAM J. Optim. 15, 805–825 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lasserre, J.B. (2006). Convergent SDP-Relaxations for Polynomial Optimization with Sparsity. In: Iglesias, A., Takayama, N. (eds) Mathematical Software - ICMS 2006. ICMS 2006. Lecture Notes in Computer Science, vol 4151. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11832225_27
Download citation
DOI: https://doi.org/10.1007/11832225_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-38084-9
Online ISBN: 978-3-540-38086-3
eBook Packages: Computer ScienceComputer Science (R0)