Finding global optimum of a non-convex quadratic function is in general a very difficult task even when the feasible set is a polyhedron. We show that when the feasible set of a quadratic problem consists of orthogonal matrices from Rn×k, then we can transform it into a semidefinite program in matrices of order kn which has the same optimal value. This opens new possibilities to get good lower ...

where μ(Pi,Pj) = ∑ ck∈Pi,cl∈Pj σ(ck, cl). We now transform this simplified VCtoS problem into the quadratic assignment (QAP) problem, which is known to be NP-hard [SG76], as follows. Given n facilities denoted by set NF , n locations denoted by set NL, a flow matrix A, where each entry Aij represents the flow of materials moving from facility i to facility j, and a distance matrix D, where each...

This paper reports heuristic and exact solution advances for the Quadratic Assignment Problem (QAP). QAP instances most often discussed in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly found by most heuristic methods. Exact methods are not able to prove optimality for these instan...

The biquadratic assignment problem BiQAP is a generalization of the quadratic assignment problem QAP It is a nonlinear integer program ming problem where the objective function is a fourth degree multivariable polynomial and the feasible domain is the assignment polytope BiQAP problems appear in VLSI synthesis Due to the di culty of this problem only heuristic solution approaches have been prop...

Augmenting genetic algorithms with local search heuristics is a promising approach to the solution of combinatorial optimization problems. In this paper, a genetic local search approach to the quadratic assignment problem (QAP) is presented. New genetic operators for realizing the approach are described, and its performance is tested on various QAP instances containing between 30 and 256 facili...

Tabu search (TS) algorithms are among the most efficient heuristic techniques in combinatorial optimization. Within these algorithms, it is important that the proper policies for maintaining the tabu tenure (tabu list size) are applied. In this paper, we discuss the mechanisms of defining the tabu tenure for the famous combinatorial optimization problem − the quadratic assignment problem (QAP)....

A heuristic technique that combines a genetic algorithm with a Tabu Search algorithm is applied to the Quadratic Assignment Problem (QAP). The hybrid algorithm improves the results obtained through the application of each of these algorithms separately. The QAP is a NP-hard problem and instances of size n > 15 are still considered intractable. The results of our experiments suggest that CHC com...

We describe a branch and bound algorithm for the quadratic assignment problem QAP that uses a convex quadratic programming QP relaxation to obtain a bound at each node The QP subproblems are approximately solved using the Frank Wolfe algorithm which in this case requires the solution of a linear assignment problem on each iteration Our branching strategy makes extensive use of dual information ...

We consider two polytopes. The quadratic assignment polytope QAP(n) is the convex hull of the set of tensors x⊗x, x ∈ Pn, where Pn is the set of n×n permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph Kn we consider appropriate (n 2 ) × (n 2 ) permutation matrix of the edges of Kn. The Young polytope P ((n − 2, 2)) is the conv...

Quadratic assignment problem (QAP) is one of the hardest combinatorial optimization problems which can model many real life problems. Because of its theoretical and practical importance, QAP has attracted attention of many researchers. In this paper, a multi hybrid genetic algorithm for solving QAP is proposed. The key feature of our approach is the hybridization of three metaheuristics, tabu s...