On the Low Rank Solutions for Linear Matrix Inequalities

Ai, Wenbao; Huang, Yongwei; Zhang, Shuzhong

doi:10.1287/moor.1080.0331

Cited by 19 publications

(27 citation statements)

References 18 publications

(32 reference statements)

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“…If a feasible point x * is a global minimizer of problem (12), then there exists λ ≥ 0 such that ∇f (x * ) + λ∇g(x * ) = 0, λg(x * ) = 0, and…”

Section: Theorem 32mentioning

confidence: 99%

“…Proof Suppose that x * be a global minimizer of problem (12). Definef (x) := f (x)− f (x * ), then the systemf (x) < 0 and g(x) < 0 has no solution.…”

Section: Theorem 32mentioning

confidence: 99%

“…The problem (12) with assumptions in Theorem 3.2 includes the Z-eigenvalue problem of a symmetric tensor [4,29,31] as a special case. …”

Section: Theorem 32mentioning

confidence: 99%

“…Many techniques and methods for quadratic optimization problems have been developed in the literature [2][3][4][7][8][9][10][11][12][13][14][15][16][17][18]. Among them, matrix decomposition methods have been an important tool for investigating quadratic optimization problems [9,11,12,16,18,19], since they can be used to prove the convexity of the joint range for quadratic forms as well as to analyse rounding solutions of semidefinite programming relaxation problems for quadratic optimization problems [9].…”

mentioning

confidence: 99%

“…Among them, matrix decomposition methods have been an important tool for investigating quadratic optimization problems [9,11,12,16,18,19], since they can be used to prove the convexity of the joint range for quadratic forms as well as to analyse rounding solutions of semidefinite programming relaxation problems for quadratic optimization problems [9]. In addition, the optimality condition is one of the most important issues in studying quadratic optimization problems, while a theorem of the alternative is a key in establishing the optimality condition of the optimization problem.…”

mentioning

confidence: 99%

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Theorems of the Alternative for Inequality Systems of Real Polynomials

Huang²

2012

J Optim Theory Appl

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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.

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“…If a feasible point x * is a global minimizer of problem (12), then there exists λ ≥ 0 such that ∇f (x * ) + λ∇g(x * ) = 0, λg(x * ) = 0, and…”

Section: Theorem 32mentioning

confidence: 99%

“…Proof Suppose that x * be a global minimizer of problem (12). Definef (x) := f (x)− f (x * ), then the systemf (x) < 0 and g(x) < 0 has no solution.…”

Section: Theorem 32mentioning

confidence: 99%

“…The problem (12) with assumptions in Theorem 3.2 includes the Z-eigenvalue problem of a symmetric tensor [4,29,31] as a special case. …”

Section: Theorem 32mentioning

confidence: 99%