Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity

Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity (2006)

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by Hayato Waki , Sunyoung Kim , Masakazu Kojima , Masakazu Muramatsu

Venue:	SIAM Journal on Optimization
Citations:	45 - 16 self

BibTeX

@ARTICLE{Waki06sumsof,
    author = {Hayato Waki and Sunyoung Kim and Masakazu Kojima and Masakazu Muramatsu},
    title = {Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity},
    journal = {SIAM Journal on Optimization},
    year = {2006},
    volume = {17},
    pages = {218--242}
}

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Abstract

Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations. Key words.

Citations

773	Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming - Goemans, Williamson - 1995
224	MEH^S -A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices - Version 4.0 - Karypis, Kumar - 1998
220	Global optimization with polynomials and the problem of moments - Lasserre - 2001
155	Semidefinite programming relaxations for semialgebraic problems - Parrilo
111	Positive polynomials on compact semi-algebraic sets - Putinar - 1993
75	GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi - Henrion, Lasserre - 2003
61	A comparison of the SheraliAdams, Lovász-Schrijver, and Lasserre relaxations for 0–1 programming - Laurent
51	Newton-type minimization via the Lanczos method - NASH - 1984
47	Über die Darstellung definiter Formen als Summe von Formenquadraten - Hilbert
44	Exploiting sparsity in semidefinite programming via matrix completion I: general framework - Fukuda, Kojima, et al. - 1999
33	Sums of squares of real polynomials - Choi, Lam, et al. - 1995
33	Detecting global optimality and extracting solutions in GloptiPoly - Henrion, Lasserre - 2005
30	An explicit equivalent positive semidefinite program for nonlinear 0-1 programs - Lasserre
29	Semidefinite programming relaxation for nonconvex quadratic programs - FUJIE, KOJIMA - 1997
26	Testing a class of methods for solving minimization problems with simple bounds on the variables - Conn, Gould, et al.
22	Extremal PSD forms with few terms - Reznick - 1978
21	Generalized lagrangian duals and sums of squares relaxations of sparse polynomial optimization problems - Kim, Kojima, et al. - 2005
16	C.A.: The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space - Ewald, Larman, et al. - 1970
16	Sums of squares relaxations of polynomial semidefinite programs - Kojima - 2003
15	Parrilo, “SOSTOOLS: Sum of squares optimization toolbox for Matlab,” Available from http://www.cds.caltech.edu/sostools - Prajna, Papachristodoulou, et al.
13	Handbook of Test Problems for Local and Global Optimization - Floudas, Pardalos, et al. - 1999
13	A general framework for convex relaxation of polynomial optimization problems over cones - Kojima, Kim, et al.
12	Implementation and evaluation - Yamashita, Fujisawa, et al. - 2003
10	An Extension of Sums of Squares Relaxations to Polynomial Optimization Problems over Symmetric Cones - Kojima, Muramatsu
9	Sum of squares relaxations for polynomial semidefinite programming - Hol, Scherer - 2004
8	An efficient trust region method for unconstrained discretetime optimal control problems - Coleman, Liao - 1995
6	A new second-order cone programming relaxation for max-cut problems - Muramatsu, Suzuki - 2001
3	The reference manual for SPOOLES, release 2.3: An object oriented software library for solving linear systems of equations,” Boeing Shared Services Group - Ashcraft, Pierce, et al. - 1999
3	SeDuMi 1.02, a MATLAB toolbox for optimizatin over symmetric cones - Strum - 1999
2	A convergence hierarchy of LMI relaxations for polynomial matrix inequalities”, Laboratoire d’Analyse et d’Architecture des Syst‘emes, Centre National de la Recherche Scientifique, 7 Avenue du Colonel Roche, 31 077 Toulouse, cedex 4 - Henrion, Lasserre - 2004
2	Sparsity in sums of squares of polynomials”, Research Report B-391, Dept - Kojima, Kim, et al. - 2003
2	Optimizationof polynomials on compact semialgebraic sets”, Working paper - Schweighofer - 2004
1	An introduction to chordal graphs and clieque trees,” In - Blair, Peyton - 1993