Sparse SOS Relaxations for Minimizing Functions that are Summations of Small Polynomials

This paper discusses how to find the global minimum of functions that are summations of small polynomials (“small” means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational complexity and lower bounds with prior SOS relaxations. Under certain conditions, we also discuss how to extract the global minimizers from these sparse relaxations. The proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems. Numerical experiments are presented which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems. Lastly, we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization.

[1]  S. J. Benson,et al.  DSDP3: dual scaling algorithm for general positive semidefinite programming. , 2001 .

[2]  D. Kamenetsky.  Symmetry Groups , 2003 .

[3]  Masakazu Muramatsu,et al.  Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity , 2004 .

[4]  B. Peyton,et al.  An Introduction to Chordal Graphs and Clique Trees , 1993 .

[5]  Charles R. Johnson,et al.  Positive definite completions of partial Hermitian matrices , 1984 .

[6]  Andrew J. Sommese,et al.  Solution of Polynomial Systems Derived from Differential Equations , 2005, Computing.

[7]  R. Curto,et al.  The truncated complex -moment problem , 2000 .

[8]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

[9]  Kim-Chuan Toh,et al.  Semidefinite Programming Approaches for Sensor Network Localization With Noisy Distance Measurements , 2006, IEEE Transactions on Automation Science and Engineering.

[10]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[11]  Pablo A. Parrilo,et al.  Minimizing Polynomial Functions , 2001, Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science.

[12]  B. Sturmfels.  SOLVING SYSTEMS OF POLYNOMIAL EQUATIONS , 2002 .

[13]  S. Nash.  Newton-Type Minimization via the Lanczos Method , 1984 .

[14]  J. Lasserre.  A Sum of Squares Approximation of Nonnegative Polynomials , 2004, SIAM Journal on Optimization.

[15]  Markus Schweighofer.  Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares , 2006, SIAM J. Optim..

[16]  Didier Henrion,et al.  GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi , 2003, TOMS.

[17]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[18]  Grigoriy Blekherman.  There are significantly more nonegative polynomials than sums of squares , 2003, math/0309130.

[19]  Olga Taussky-Todd.  SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM , 1996 .

[20]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[21]  B. Reznick.  Some concrete aspects of Hilbert's 17th Problem , 2000 .

[22]  Jean B. Lasserre,et al.  Convergent SDP-Relaxations in Polynomial Optimization with Sparsity , 2006, SIAM J. Optim..

[23]  J. Lasserre,et al.  Detecting global optimality and extracting solutions in GloptiPoly , 2003 .

[24]  A. Fialkow,et al.  THE TRUNCATED COMPLEX K-MOMENT PROBLEM , 2000 .

[25]  James Aspnes,et al.  On the Computational Complexity of Sensor Network Localization , 2004, ALGOSENSORS.

[26]  Pablo A. Parrilo,et al.  Exploiting structure in sum of squares programs , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[27]  Monique Laurent,et al.  Semidefinite Approximations for Global Unconstrained Polynomial Optimization , 2005, SIAM J. Optim..

[28]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[29]  Jiawang Nie,et al.  Sum of squares method for sensor network localization , 2006, Comput. Optim. Appl..

[30]  M. Kojima,et al.  B-411 Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity , 2004 .

[31]  Masakazu Kojima,et al.  Generalized Lagrangian Duals and Sums of Squares Relaxations of Sparse Polynomial Optimization Problems , 2005, SIAM J. Optim..

[32]  James Demmel,et al.  Minimizing Polynomials via Sum of Squares over the Gradient Ideal , 2004, Math. Program..

[33]  B. Reznick.  Extremal PSD forms with few terms , 1978 .

[34]  P. Parrilo,et al.  Symmetry groups, semidefinite programs, and sums of squares , 2002, math/0211450.

[35]  Grigoriy Blekherman.  Volumes of Nonnegative Polynomials, Sums of Squares and Powers of Linear Forms , 2004 .

[36]  Masakazu Kojima,et al.  Sparsity in sums of squares of polynomials , 2005, Math. Program..

[37]  P. Toint,et al.  Testing a class of methods for solving minimization problems with simple bounds on the variables , 1988 .