A positive definite polynomial Hessian that does not factor

The notion of sos-convexity has recently been proposed as a tractable sufficient condition for convexity of polynomials based on a sum of squares decomposition of the Hessian matrix. A multivariate polynomial p(x) = p(x1,…,xn)is said to be sos-convex if its Hessian H(x) can be factored as H(x) = MT(x)M(x) with a possibly nonsquare polynomial matrix M(x). The problem of deciding sos-convexity of a polynomial can be reduced to the feasibility of a semidefinite program, which can be checked efficiently. Motivated by this computational tractability, it has been speculated whether every convex polynomial must necessarily be sos-convex. In this paper, we answer this question in the negative by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sos-convex.

[1]  Man-Duen Choi.  Positive semidefinite biquadratic forms , 1975 .

[2]  Yeung Sam Hung,et al.  Establishing Convexity of Polynomial Lyapunov Functions and Their Sublevel Sets , 2008, IEEE Transactions on Automatic Control.

[3]  Jean B. Lasserre.  A Convex Positivstellensatz , 2008 .

[4]  Jean B. Lasserre,et al.  Convexity in SemiAlgebraic Geometry and Polynomial Optimization , 2008, SIAM J. Optim..

[5]  Panos M. Pardalos,et al.  Open questions in complexity theory for numerical optimization , 1992, Mathematical programming.

[6]  Pablo A. Parrilo,et al.  Computing sum of squares decompositions with rational coefficients , 2008 .

[7]  M. Kojima.  Sums of Squares Relaxations of Polynomial Semidefinite Programs , 2003 .

[8]  A. Magnani,et al.  Tractable fitting with convex polynomials via sum-of-squares , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[9]  N. Z. Shor.  Class of global minimum bounds of polynomial functions , 1987 .

[10]  J. William Helton,et al.  Semidefinite representation of convex sets , 2007, Math. Program..

[11]  D. J. Newman,et al.  Arithmetic, Geometric Inequality , 1960 .

[12]  Katta G. Murty,et al.  Some NP-complete problems in quadratic and nonlinear programming , 1987, Math. Program..

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

[14]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[15]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[16]  A. Garulli,et al.  Positive Polynomials in Control , 2005 .

[17]  Yurii Nesterov,et al.  Squared Functional Systems and Optimization Problems , 2000 .

[18]  Pablo A. Parrilo,et al.  Explicit SOS decompositions of univariate polynomial matrices and the Kalman-Yakubovich-Popov lemma , 2007, 2007 46th IEEE Conference on Decision and Control.

[19]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

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

[21]  Bruce Reznick,et al.  Real zeros of positive semidefinite forms. I , 1980 .

[22]  B. Reznick,et al.  Sums of squares of real polynomials , 1995 .

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

[24]  Konrad Schmüdgen.  An Example of a Positive Polynomial which is not a Sum of Squares of Polynomials A Positive, but not Strongly Positive Functional , 1979 .

[25]  D. Hilbert.  Über die Darstellung definiter Formen als Summe von Formenquadraten , 1888 .

[26]  B. Guo.  On the difficulty of deciding the convexity of polynomials over simplexes , 1996, Int. J. Comput. Geom. Appl..

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

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