New results on Hermitian matrix rank-one decomposition

Ai, Wenbao; Huang, Yongwei; Zhang, Shuzhong

doi:10.1007/s10107-009-0304-7

Cited by 185 publications

(127 citation statements)

References 21 publications

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“…In that case, Ai, Huang, and Zhang [1] proved that the joint numerical range g(C m ) is a convex cone. Due to the block square-free property, it is also not pointed at any direction, hence is a subspace.…”

Section: If F ∈ ℜ M D Is a D-th Order Super-symmetric Tensor With Fmentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

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161

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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Section: If F ∈ ℜ M D Is a D-th Order Super-symmetric Tensor With Fmentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

Self Cite

161

177

View full text Add to dashboard Cite

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“…In some special cases, Problem (1) may be convex or otherwise have lower complexity [3], [4]. However, in this work, we will focus on the generic case when Problem (1) is NP-hard.…”

Section: Introductionmentioning

confidence: 99%

A sequential constraint relaxation algorithm for rank-one constrained problems

Cao

Thompson

Poor

2017

2017 25th European Signal Processing Conference (EUSIPCO)

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Abstract-Many optimization problems in communications and signal processing can be formulated as rank-one constrained optimization problems. This has motivated the development of methods to solve such problem in specific scenarios. However, due to the non-convex nature of the rank-one constraint, limited progress has been made in solving generic rank-one constrained optimization problems. In particular, the problem of efficiently finding a locally optimal solution to a generic rankone constrained problem remains open. This paper focuses on solving general rank-one constrained problems via relaxation techniques. However, instead of dropping the rank-one constraint completely as is done in traditional rank-one relaxation methods, a novel algorithm that gradually relaxes the rank-one constraint, termed the sequential rank-one constraint relaxation (SROCR) algorithm, is proposed. Compared with previous algorithms, the SROCR algorithm can solve general rank-one constrained problems, and can find feasible solutions with favorable complexity.

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“…Suppose R ≥ 3. Based on Theorem 2.1 given in [32], we obtain that there is a rank-one decomposition for X F (synthetically denoted as D 3 (X F ,Q 1 ,Q 2 ,Q x )), i.e., X F = (23) and satisfies the optimality conditions (39) and (40) together with the optimal dual solution {y 1 , y 2 , y 3 , y 4 }. Therefore, X ′′ F can be regarded as an optimal rank-one solution of (23).…”

Section: Appendix C Proof Of Theoremmentioning

confidence: 99%

Linear Precoding Designs for Amplify-and-Forward Multiuser Two-Way Relay Systems

Wang

Tao

Huang

2012

IEEE Trans. Wireless Commun.

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Abstract-Two-way relaying can improve spectral efficiency in two-user cooperative communications. It also has great potential in multiuser systems. A major problem of designing a multiuser two-way relay system (MU-TWRS) is transceiver or precoding design to suppress co-channel interference. This paper aims to study linear precoding designs for a cellular MU-TWRS where a multi-antenna base station (BS) conducts bi-directional communications with multiple mobile stations (MSs) via a multi-antenna relay station (RS) with amplify-and-forward relay strategy. The design goal is to optimize uplink performance, including total mean-square error (Total-MSE) and sum rate, while maintaining individual signal-to-interference-plus-noise ratio (SINR) requirement for downlink signals. We show that the BS precoding design with the RS precoder fixed can be converted to a standard second order cone programming (SOCP) and the optimal solution is obtained efficiently. The RS precoding design with the BS precoder fixed, on the other hand, is non-convex and we present an iterative algorithm to find a local optimal solution. Then, the joint BS-RS precoding is obtained by solving the BS precoding and the RS precoding alternately. Comprehensive simulation is conducted to demonstrate the effectiveness of the proposed precoding designs.Index Terms-MIMO precoding, two-way relaying, nonregenerative relay, minimum mean-square-error (MMSE), convex optimization.

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New results on Hermitian matrix rank-one decomposition

Cited by 185 publications

References 21 publications

Maximum Block Improvement and Polynomial Optimization

Maximum Block Improvement and Polynomial Optimization

A sequential constraint relaxation algorithm for rank-one constrained problems

Linear Precoding Designs for Amplify-and-Forward Multiuser Two-Way Relay Systems

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