Theorems of the alternative for inequality systems of real polynomials
type: Item
instance of: scholarly article
title: Theorems of the alternative for inequality systems of real polynomials
zbMATH Open document ID: 1267.90138
author: Shenglong Hu
author: Zhenghai Huang
published in: Journal of Optimization Theory and Applications
publication date: 2012-11-26T00:00:00Z
review text: This paper presents two new results on the matrix decomposition. One of them is given under an assumption proposed by Ai, Huang, and Zhang in the context of the complex space and the other result is given under a new assumption that at least one of the three involved matrices is negative semidefinite. The authors show that several theorems of the alternative are true under the new proposed assumption. The inequality system of higher degree real polynomials defined by even order tensors is also considered in this paper since it appears in many applied science areas. A theorem of the alternative for the inequality system of two higher degree real polynomials is established under suitable assumptions. In addition, as an application of the obtained theorem of the alternative, the authors investigate the optimality condition of a class of polynomial optimization problems under suitable assumptions and show that the extreme \(\mathbb{Z}\)-eigenvalue problem is a special case of this class of polynomial optimization problems.
reviewed by: Samir Kumar Neogy
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 6110391
zbMATH Keywords: theorem of the alternative
zbMATH Keywords: matrix decomposition
zbMATH Keywords: polynomial system
zbMATH Keywords: polynomial optimization problem
zbMATH Keywords: \(\mathbb{Z}\)-eigenvalue problem
MaRDI profile type: Publication
full work available at URL: s10957-012-9993-9
OpenAlex ID: W2100266890
cites work: Homogeneous polynomial forms for robustness analysis of uncertain systems
cites work: Positive definiteness of diffusion kurtosis imaging
cites work: A Semidefinite Relaxation Scheme for Multivariate Quartic Polynomial Optimization with Quadratic Constraints
cites work: Higher Order Positive Semidefinite Diffusion Tensor Imaging
cites work: Q4040221
cites work: A note on quadratic forms
cites work: Optimality Conditions for the Minimization of a Quadratic with Two Quadratic Constraints
cites work: New Results on Quadratic Minimization
cites work: On a subproblem of trust region algorithms for constrained optimization
cites work: New results on Hermitian matrix rank-one decomposition
cites work: On the Low Rank Solutions for Linear Matrix Inequalities
cites work: 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
cites work: Conditions for global optimality. II
cites work: Permanently going back and forth between the ``quadratic world'' and the ``convexity world'' in optimization
cites work: Complex Matrix Decomposition and Quadratic Programming
cites work: Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations
cites work: On Cones of Nonnegative Quadratic Functions
cites work: A Unified Theorem on SDP Rank Reduction
cites work: On the Field of Values of a Matrix
cites work: On the mapping of quadratic forms
cites work: Yuan's alternative theorem and the maximization of the minimum eigenvalue function
cites work: Convexity of quadratic transformations and its use in control and optimization
cites work: A Survey of the S-Lemma
cites work: Alternative Theorems for Quadratic Inequality Systems and Global Quadratic Optimization
cites work: Q5641059
cites work: Eigenvalues of a real supersymmetric tensor
cites work: Multivariate polynomial minimization and its application in signal processing
cites work: Z-eigenvalue methods for a global polynomial optimization problem
cites work: Convex Analysis
cites work: Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen
cites work: Q4762223
cites work: On the Classical Necessary Second-Order Optimality Conditions in the Presence of Equality and Inequality Constraints
DOI: 10.1007/S10957-012-9993-9
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description: scientific article; zbMATH DE number 6110391