Efficiency improvement in an nD systems approach to polynomial optimization

The problem of finding the global minimum of a so-called Minkowski-norm dominated polynomial can be approached by the matrix method of Stetter and Moller, which reformulates it as a large eigenvalue problem. A drawback of this approach is that the matrix involved is usually very large. However, all that is needed for modern iterative eigenproblem solvers is a routine which computes the action of the matrix on a given vector. This paper focuses on improving the efficiency of computing the action of the matrix on a vector. To avoid building the large matrix one can associate the system of first-order conditions with an nD system of difference equations. One way to compute the action of the matrix efficiently is by setting up a corresponding shortest path problem and solving it. It turns out that for large n the shortest path problem has a high computational complexity, and therefore some heuristic procedures are developed for arriving cheaply at suboptimal paths with acceptable performance.

[1]  J. Demmel,et al.  Balancing sparse matrices for computing eigenvalues , 2000 .

[2]  H. M. Möller,et al.  Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems , 1995 .

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

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

[5]  Jean B. Lasserre,et al.  An Explicit Equivalent Positive Semidefinite Program for Nonlinear 0-1 Programs , 2002, SIAM J. Optim..

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

[7]  Gerard L. G. Sleijpen,et al.  A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems , 1996, SIAM J. Matrix Anal. Appl..

[8]  Gerard L. G. Sleijpen,et al.  Jacobi-Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils , 1998, SIAM J. Sci. Comput..

[9]  Michiel Hazewinkel,et al.  Constructive algebra and systems theory , 2006 .

[10]  D. Jibetean,et al.  Algebraic Optimization with Applications in System Theory , 2003 .

[11]  Bernard Hanzon,et al.  REPORT RAPPORT , 2001 .

[12]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[13]  S. Zampieri,et al.  State space realization of 2-D finite-dimensional behaviours , 1993 .

[14]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[15]  S. Attasi.  Modelling and Recursive Estimation for Double Indexed Sequences , 1976 .

[16]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

[17]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

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

[19]  D. Lainiotis,et al.  System identification : advances and case studies , 1976 .

[20]  Fabrice Rouillier,et al.  An environment for Symbolic and Numeric Computation , 2002 .

[21]  Bernard Hanzon,et al.  Model reduction in H2 using matrix solutions of polynomial equations , 1998 .

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