A tensor product matrix approximation problem in quantum physics

Dahl, Geir; Leinaas, Jon Magne; Myrheim, Jan; Ovrum, Eirik

doi:10.1016/j.laa.2006.08.026

Cited by 90 publications

(64 citation statements)

References 12 publications

(24 reference statements)

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“…This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful.…”

mentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

159

177

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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.Key words. block coordinate descent, polynomial optimization problem, tensor form AMS subject classifications. 90C26, 90C30, 15A69, 49M271. Introduction. The optimization models whose objective and constraints are polynomial functions have recently attracted much research attention. This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful. For instance, even the simplest instances of polynomial optimization, such as maximizing a cubic polynomial over a sphere, is NP-hard (Nesterov [42]). However, the problem is so elementary that it can be even attempted in an undergraduate calculus class. For readers interested in polynomial optimization with simple constraints, we refer to De Klerk [12] for a survey on the computational complexity of optimizing various classes of polynomial functions over a simplex, hypercube or sphere. In particular, De Klerk et al.[13] designed a polynomial-time approximation scheme (PTAS) for minimizing polynomials of fixed degree over the simplex.So far, a few results have been obtained for approximation algorithms with guaranteed worst-case performance ratios for higher degree polynomial optimization problems. Luo and Zhang [39] derived a polynomial-time approximation algorithm to optimize a multivariate quartic polynomial over a region defined by quadratic inequalities.

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confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

159

177

View full text Add to dashboard Cite

show abstract

“…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]. It has fundamental importance in quantum science and has attracted much attention since the pioneer work of Einstein, Podolsky, and Rosen [10] and Schrödinger [33].…”

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confidence: 99%

Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations

Ling¹,

Nie²,

Qi³

et al. 2010

SIAM J. Optim.

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].…”

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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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A tensor product matrix approximation problem in quantum physics

Cited by 90 publications

References 12 publications

Maximum Block Improvement and Polynomial Optimization

Maximum Block Improvement and Polynomial Optimization

Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations

The Best Rank-One Approximation Ratio of a Tensor Space

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