Extensions of Yuan’s Lemma to fourth-order tensor system with applications

Yuan’s lemma is a basic proposition on homogeneous quadratic function system. In this paper, we extend Yuan’s lemma to 4th-order tensor system. We first give two gen- eralized definitions of positive semidefinite of 4th-order tensor, and based on them, two extensions of Yuan’s lemma are proposed. We illustrate the difference between our ex- tensions and … Read more

The extensions of Yuan’s lemma and applications in S-lemma

In this paper we extend a lemma due to Yuan from several aspects. A new proof of Yuan’s lemma is given. A rank-one decomposition of positive semidefinite matrix is further developed. With the extended rank-one de- composition results, we generalize the Yuan’s lemma to general quadratic function systems, interval quadratic function systems and quadratic matrix … Read more

Implementing the ADMM to Big Datasets: A Case Study of LASSO

The alternating direction method of multipliers (ADMM) has been popularly used for a wide range of applications in the literature. When big datasets with high-dimensional variables are considered, subproblems arising from the ADMM must be solved inexactly even though theoretically they may have closed-form solutions. Such a scenario immediately poses mathematical ambiguities such as how … Read more

Modified alternating direction methods for the modified multiple-sets split feasibility problems

Inthispaper, weproposetwonewmultiple-setssplitfeasibilityproblem(MSFP)models, where the MSFP requires to find a point closest to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation will be closest to the intersection of another family of closed convex sets in the image space. This problem arises in image restoration, … Read more

On Solving Biquadratic Optimization via Semidefinite Relaxation

In this paper, we study a class of biquadratic optimization problems. We first relax the original problem to its semidefinite programming (SDP) problem and discuss the approximation ratio between them. Under some conditions, we show that the relaxed problem is tight. Then we consider how to approximately solve the problems in polynomial time. Under several … Read more

Approximation algorithms for trilinear optimization with nonconvex constraints and its extensions

In this paper, we study trilinear optimization problems with nonconvex constraints under some assumptions. We first consider the semidefinite relaxation (SDR) of the original problem. Then motivated by So \cite{So2010}, we reduce the problem to that of determining the $L_2$-diameters of certain convex bodies, which can be approximately solved in deterministic polynomial-time. After the relaxed … Read more