Theorems of the Alternative for Inequality Systems of Real Polynomials

In this paper, we establish theorems of the alternative for inequality systems of real polynomials. For the real quadratic inequality system, we present two new results on the matrix decomposition, by which we establish two theorems of the alternative for the inequality system of three quadratic polynomials under an assumption that at least one of the involved forms be negative semidefinite. We also extend a theorem of the alternative to the case with a regular cone. For the inequality system of higher degree real polynomials, defined by even order tensors, a theorem of the alternative for the inequality system of two higher degree polynomials is established under suitable assumptions. As a byproduct, we give an equivalence result between two statements involving two higher degree polynomials. Based on this result, we investigate the optimality condition of a class of polynomial optimization problems under suitable assumptions.

[1]  L. Qi,et al.  Higher Order Positive Semidefinite Diffusion Tensor Imaging , 2010, SIAM J. Imaging Sci..

[2]  Shuzhong Zhang,et al.  New results on Hermitian matrix rank-one decomposition , 2011, Math. Program..

[3]  Fei Wang,et al.  Z-eigenvalue methods for a global polynomial optimization problem , 2009, Math. Program..

[4]  F. Giannessi,et al.  On the Theory of Vector Optimization and Variational Inequalities. Image Space Analysis and Separation , 2000 .

[5]  Shuzhong Zhang,et al.  On Cones of Nonnegative Quadratic Functions , 2003, Math. Oper. Res..

[6]  Xin Chen,et al.  A note on quadratic forms , 1999, Math. Program..

[7]  A. Seeger,et al.  Yuan's alternative theorem and the maximization of the minimum eigenvalue function , 1994 .

[8]  G. Chesi.  Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems , 2009 .

[9]  N. Shor.  Nondifferentiable Optimization and Polynomial Problems , 1998 .

[10]  Chen Ling,et al.  Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations , 2009, SIAM J. Optim..

[11]  Anthony Man-Cho So,et al.  A Unified Theorem on Sdp Rank Reduction , 2008, Math. Oper. Res..

[12]  Kok Lay Teo,et al.  Multivariate Polynomial Minimization and Its Application in Signal Processing , 2003, J. Glob. Optim..

[13]  Abdeljelil Baccari,et al.  On the Classical Necessary Second-Order Optimality Conditions in the Presence of Equality and Inequality Constraints , 2005, SIAM J. Optim..

[14]  Ya-Xiang Yuan,et al.  On a subproblem of trust region algorithms for constrained optimization , 1990, Math. Program..

[15]  Tamás Terlaky,et al.  A Survey of the S-Lemma , 2007, SIAM Rev..

[16]  Shuzhong Zhang,et al.  Complex Matrix Decomposition and Quadratic Programming , 2007, Math. Oper. Res..

[17]  J.-B. Hiriart-Urruty,et al.  Permanently Going Back and Forth between the ``Quadratic World'' and the ``Convexity World'' in Optimization , 2002 .

[18]  Paul Pinsler.  Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen , 1936 .

[19]  V. Jeyakumar,et al.  Farkas Lemma: Generalizations , 2009, Encyclopedia of Optimization.

[20]  Jean-Baptiste Hiriart-Urruty,et al.  Conditions for Global Optimality 2 , 1998, J. Glob. Optim..

[21]  Amir Beck,et al.  On the convexity of a class of quadratic mappings and its application to the problem of finding the smallest ball enclosing a given intersection of balls , 2007, J. Glob. Optim..

[22]  Shuzhong Zhang,et al.  New Results on Quadratic Minimization , 2003, SIAM J. Optim..

[23]  Vaithilingam Jeyakumar,et al.  Alternative Theorems for Quadratic Inequality Systems and Global Quadratic Optimization , 2009, SIAM J. Optim..

[24]  Shuzhong Zhang,et al.  On the Low Rank Solutions for Linear Matrix Inequalities , 2008, Math. Oper. Res..

[25]  Zhi-Quan Luo,et al.  A Semidefinite Relaxation Scheme for Multivariate Quartic Polynomial Optimization with Quadratic Constraints , 2010, SIAM J. Optim..

[26]  Boris Polyak.  Convexity of Quadratic Transformations and Its Use in Control and Optimization , 1998 .

[27]  L. Brickman.  ON THE FIELD OF VALUES OF A MATRIX , 1961 .

[28]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[29]  J. J. Moré.  Generalizations of the trust region problem , 1993 .

[30]  Graziano Chesi,et al.  On the role of homogeneous forms in robustness analysis of control systems , 2003 .

[31]  L. L. Dines.  On the mapping of quadratic forms , 1941 .

[32]  Ya-Xiang Yuan,et al.  Optimality Conditions for the Minimization of a Quadratic with Two Quadratic Constraints , 1997, SIAM J. Optim..

[33]  L. Qi,et al.  Positive definiteness of Diffusion Kurtosis Imaging , 2012 .

[34]  F. Giannessi.  Vector Variational Inequalities and Vector Equilibria , 2000 .