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On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming
, 2010
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Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LPbased Approximation Algorithm
"... Abstract. We show that for every positive ε> 0, unless N P ⊂ RP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2 log1−ε n by a reduction from the maximum label cover problem. Then, we present an O ( √ n)approximation algorithm for the probl ..."
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Abstract. We show that for every positive ε> 0, unless N P ⊂ RP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2 log1−ε n by a reduction from the maximum label cover problem. Then, we present an O ( √ n)approximation algorithm for the problem based on rounding of the linear programming relaxation often used in the state of the art exact algorithms. 1
On New Classes of Nonnegative Symmetric Tensors
, 2014
"... Abstract. In this paper we introduce three new classes of nonnegative forms (or equivalently, symmetric tensors) and their extensions. The newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors (order 6 or above). For the special ..."
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Abstract. In this paper we introduce three new classes of nonnegative forms (or equivalently, symmetric tensors) and their extensions. The newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors (order 6 or above). For the special case of quartic forms, they collapse into the set of convex quartic homogeneous polynomial functions. We discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization. Key words. symmetric tensors, nonnegative forms, polynomial and tensor optimization AMS subject classifications. 15A69, 12Y05, 90C26
CONVEX HULL OF THE ORTHOGONAL SIMILARITY SET WITH APPLICATIONS IN QUADRATIC ASSIGNMENT PROBLEMS
"... (Communicated by Duan Li) Abstract. In this paper, we study thoroughly the convex hull of the orthogonal similarity set and give a new representation. When applied in quadratic assignment problems, it motivates two new lower bounds. The first is equivalent to the projected eigenvalue bound, while th ..."
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(Communicated by Duan Li) Abstract. In this paper, we study thoroughly the convex hull of the orthogonal similarity set and give a new representation. When applied in quadratic assignment problems, it motivates two new lower bounds. The first is equivalent to the projected eigenvalue bound, while the second highly outperforms several wellknown lower bounds in literature. 1. Introduction. Consider the quadratic assignment problem (QAP), which is one of the great challenges in combinatorial optimization, see [4, 5, 6, 7, 16, 19] for comprehensive surveys. We focus on the following trace formulation introduced by
Semidefinite Programming Relaxation of Quadratic Assignment Problems based on Nonredundant Matrix Splitting
, 2013
"... Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semidefinite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25, 28]. In this paper, we consider the issue of how to choose an appro ..."
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Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semidefinite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25, 28]. In this paper, we consider the issue of how to choose an appropriate matrix splitting scheme so that the resulting relaxation model is easy to solve and able to provide a strong bound. For this, we first introduce a new notion of the socalled redundant and nonredundant matrix splitting and show that the relaxation based on a nonredundant matrix splitting can provide a stronger bound than a redundant one. Then we propose to follow the minimal trace principle to find a nonredundant matrix splitting via solving an auxiliary semidefinite programming problem (SDP). We show that applying the minimal trace principle directly leads to the socalled orthogonal matrix splitting introduced in [28]. To find other nonredundant matrix splitting schemes whose resulting relaxation models are relatively easy to solve, we elaborate on two splitting schemes based on the socalled onematrix and the summatrix. We analyze the solutions from the auxiliary problems for these two cases and characterize when they can pro