Conditions for strong ellipticity and M-eigenvalues

Qi, Liqun; Dai, Hui–Hui; Han, Deren

doi:10.1007/s11464-009-0016-6

Cited by 96 publications

(79 citation statements)

References 24 publications

(29 reference statements)

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“…In this case, the Hessian matrix of the Lagrangian becomes 17) and the second-order necessary optimality condition is…”

Section: General Bi-homogeneous Optimizationmentioning

confidence: 99%

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Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

Bomze

Ling

et al. 2011

J Glob Optim

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Abstract. A so-called Standard Bi-Quadratic Optimization Problem (StBQP) consists in minimizing a bi-quadratic form over the Cartesian product of two simplices (so this is different from a Bi-Standard QP where a quadratic function is minimized over the same set).An application example arises in portfolio selection. In this paper we present a bi-quartic formulation of StBQP, in order to get rid of the sign constraints. We study the first and second-order optimality conditions of the original StBQP and the reformulated bi-quartic problem over the product of two Euclidean spheres. Furthermore, we discuss the one-to-one correspondence between the global/local solutions of StBQP and the global/local solutions of the reformulation. We introduce a continuously differentiable penalty function. Based upon this, the original problem is converted into the problem of locating an unconstrained global minimizer of a (specially structured) polynomial of degree eight.

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“…In this case, the Hessian matrix of the Lagrangian becomes 17) and the second-order necessary optimality condition is…”

Section: General Bi-homogeneous Optimizationmentioning

confidence: 99%

“…The latter problem arises from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics; see [7,8,9,11,17,18,21] and the references therein. A StBQP should also be not confused with a bi-StQP, which is a special case of a multi-StQP, a problem class studied recently in [6,19].…”

Section: Introductionmentioning

confidence: 99%

Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

Bomze

Ling

et al. 2011

J Glob Optim

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“…Problem (1.1) arises from the strong ellipticity condition problem in solid mechanics (for n = m = 3) [16,29,32,34,39] and the entanglement problem in quantum physics. The entanglement problem is to determine whether a quantum state is separable or inseparable (entangled), or to check whether an mn × mn symmetric matrix A 0 can be decomposed as a convex combination of tensor products of n and m dimensional vectors [6].…”

mentioning

confidence: 99%

Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations

Ling¹,

Nie²,

Qi³

et al. 2010

SIAM J. Optim.

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119

120

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Abstract.This paper studies the so-called biquadratic optimization over unit spheres min x∈R n ,y∈R m 1≤i,k≤n, 1≤j,l≤m b ijkl x i y j x k y l , subject to x = 1, y = 1. We show that this problem is NP-hard, and there is no polynomial time algorithm returning a positive relative approximation bound. Then, we present various approximation methods based on semidefinite programming (SDP) relaxations. Our theoretical results are as follows: For general biquadratic forms, we develop a 1 2 max{m,n} 2 -approximation algorithm under a slightly weaker approximation notion; for biquadratic forms that are square-free, we give a relative approximation bound 1 nm ; when min{n, m} is a constant, we present two polynomial time approximation schemes (PTASs) which are based on sum of squares (SOS) relaxation hierarchy and grid sampling of the standard simplex. For practical computational purposes, we propose the first order SOS relaxation, a convex quadratic SDP relaxation, and a simple minimum eigenvalue method and show their error bounds. Some illustrative numerical examples are also included.

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“…In the previous version of this paper, we got a positive lower bound [5], [11], [16], [20], [22], [25], [27].…”

Section: ▯ Letmentioning

confidence: 99%

“…We may also see that ρ B ðAÞ is the largest absolute value of the M -eigenvalues of A, defined as below [20], [25]. Denote A· xyy as a vector in ℜ n , whose ith component is P n j¼1 P p k;l¼1 a ijkl x j y k y l , and denote Axxy · as a vector in ℜ p , whose lth component is P n i;j¼1 P p k¼1 a ijkl x i x j y k .…”

Section: ▯ Letmentioning

confidence: 99%

The Best Rank-One Approximation Ratio of a Tensor Space

Qi¹

2011

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper we define the best rank-one approximation ratio of a tensor space. It turns out that in the finite dimensional case this provides an upper bound for the quotient of the residual of the best rankone approximation of any tensor in that tensor space and the norm of that tensor. This upper bound is strictly less than one, and it gives a convergence rate for the greedy rank-one update algorithm. For finite dimensional general tensor spaces, third order finite dimensional symmetric tensor spaces, and finite biquadratic tensor spaces, we give positive lower bounds for the best rank-one approximation ratio. For finite symmetric tensor spaces and finite dimensional biquadratic tensor spaces, we give upper bounds for this ratio.

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Conditions for strong ellipticity and M-eigenvalues

Cited by 96 publications

References 24 publications

Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations

The Best Rank-One Approximation Ratio of a Tensor Space

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