Global optimization of a class of nonconvex quadratically constrained quadratic programming problems

In this paper we study a class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Strong duality holds if a redundant constraint is introduced. As an application, a new lower bound is proposed for the quadratic assignment problem.     

Compact linearization for binary quadratic problems

We show that a well-known linearization technique initially proposed for quadratic assignment problems can be generalized to a broader class of quadratic 0–1 mixed-integer problems subject to assignment constraints. The resulting linearized formulation is more compact and tighter than that obtained with a more usual linearization technique. We discuss the application of the compact linearization to three classes of problems in the literature, among which the graph partitioning problem.      

A Heuristic Approach to Quadratic Assignment Problem

A new algorithm for solving quadratic assignment problems is presented. The algorithm, which employs a sequential search technique, constructs a matrix of lower bounds on the costs of locating facilities at different sites. It then improves the elements of this matrix, one by one, by solving a succession of linear assignment problems. After all the elements of the matrix are improved, a feasible assignment is obtained, which results in an improved value for the objective function of the quadratic assignment problem. The procedure is repeated until the desired accuracy in the objective function value is obtained.     

On the Use of Exact and Heuristic Cutting Plane Methods for the Quadratic Assignment Problem

This paper uses the formulation of the quadratic assignment problem as that of minimizing a concave quadratic function over the assignment polytope. Cutting plane procedures are investigated for solving this problem. A lower bound derived on the number of cuts needed for termination indicates that conventional cutting plane procedures would require a huge computational effort for the exact solution of the quadratic assignment problems. However, several heuristics which are derived from the cutting planes produce optimal or good quality solutions early on in the search process. An illustrative example and computational results are presented.     

Random assignment problems

Analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problem’s solutions, feasible region, and optimal values, especially in large-scale cases. A class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems, and many important problems in operations research and computer science can be formulated as assignment problems. This paper presents an overview of the recent results and developments in the area of probabilistic assignment problems, including the linear and multidimensional assignment problems, quadratic assignment problem, etc.     

Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem

We consider semidefinite programming relaxations of the quadratic assignment problem, and show how to exploit group symmetry in the problem data. Thus we are able to compute the best known lower bounds for several instances of quadratic assignment problems from the problem library: (Burkard et al. in J Global Optim 10:291–403, 1997).     

Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization

We prove a sufficient global optimality condition for the problem of minimizing a quadratic function subject to quadratic equality constraints where the variables are allowed to take values –1 and 1. We extend the condition to quadratic problems with matrix variables and orthonormality constraints, and in particular to the quadratic assignment problem.     

Approximating Global Quadratic Optimization with Convex Quadratic Constraints

We consider the problem of approximating the global maximum of a quadratic program (QP) subject to convex non-homogeneous quadratic constraints. We prove an approximation quality bound that is related to a condition number of the convex feasible set; and it is the currently best for approximating certain problems, such as quadratic optimization over the assignment polytope, according to the best of our knowledge.     

The Use of Specially Structured Models for Obtaining Bounds in the Quadratic Assignment Problem

In this paper we study a procedure for finding bounds for the quadratic assignment problem. This procedure may be used as a sub-routine in hybrid procedures for solving this problem. The approach is based upon a data decomposition method, linking the actual data to the data of a special class of assignment problems for which bounds are computationally tractable.     

A heuristic for quadratic Boolean programs with applications to quadratic assignment problems

A heuristic for quadratic Boolean programs is presented. Computational tests with quadratic assignment problems (QAP) showed that it finds very good suboptimal solutions in moderate time and behaves computationally stable. In the appendix a FORTRAN-program for QAP is listed which improves an earlier code published by Burkard and Derigs.     

Box-inequalities for quadratic assignment polytopes

Linear Programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years. Their quality has turned out to be quite good in practice. Investigations of the polytopes underlying the corresponding integer linear programming formulations (the non-symmetric and the symmetric quadratic assignment polytope) have been started during the last decade [34, 31, 21, 22]. They have lead to basic knowledge on these polytopes concerning questions like their dimensions, affine hulls, and trivial facets. However, no large class of (facet-defining) inequalities that could be used in cutting plane procedures had been found. We present in this paper the first such class of inequalities, the box inequalities, which have an interesting origin in some well-known hypermetric inequalities for the cut polytope. Computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds. The most effective ones among the new inequalities turn out to be indeed facet-defining for both the non-symmetric as well as for the symmetric quadratic assignment polytope. Received: April 17, 2000 / Accepted: July 3, 2001?Published online September 3, 2001     

Improving port performance in developing countries—A case study

Exchange algorithms are an important class of heuristics for hard combinatorial optimization problems as, e.g., salesman problems or quadratic assignment problems. In Kirkpatrick's and Cerny's exchange algorithms for the travelling salesman problem and placement problem they propose to perform an exchange not only if the objective function value decreases by this exchange, but also in certain cases if the objective function value increases. An exchange increasing the objective function value is performed stochastically depending on the size of the increment.Computational tests with quadratic assignment problems revealed an excellent behaviour in such an approach. Suboptimal solutions differing 1–2% from the best known solutions are obtained by a simple program in short time. By starting this program several times with different starting values all known minimal objective function values were reached. Thus this approach is well suited also for smaller computers and leads in short time to acceptable solutions.     

Solving nonlinear integer programs with large-scale optimization software

This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.     

Solving nonlinear integer programs with large-scale optimization software

This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.     

An efficient implementation of the robust tabu search heuristic for sparse quadratic assignment problems

We propose and develop an efficient implementation of the robust tabu search heuristic for sparse quadratic assignment problems. The traditional implementation of the heuristic applicable to all quadratic assignment problems is of O(N²) complexity per iteration for problems of size N. Using multiple priority queues to determine the next best move instead of scanning all possible moves, and using adjacency lists to minimize the operations needed to determine the cost of moves, we reduce the asymptotic (N → ∞) complexity per iteration to O(N log N). For practical sized problems, the complexity is O(N).     

A low-rank bilinear programming approach for sub-optimal solution of the quadratic assignment problem

This paper is concerned with a new approach for solving quadratic assignment problems (QAP). We first reformulate QAP as a concave quadratic programming problem and apply an outer approximation algorithm. In addition, an improvement routine is incorporated in the final stage of the algorithm. Computational experiments on a set of standard data demonstrate that this algorithm can yield favorable results with a relatively low computational effort.     

The formulation and numerical method for partial quadratic eigenvalue assignment problems

In this paper, we consider the minimum norm and robust partial quadratic eigenvalue assignment problems (PQEVAP). A complete theory on the existence of solutions for the PQEVAP is established. It is shown that solving the PQEVAP is essentially solving an eigenvalue assignment for a linear system of a much lower order, and the minimum norm and robust PQEVAPs are then concerning the minimum norm and robust eigenvalue assignment problems associated with this linear system. Based on this theory, an algorithm for solving the minimum norm and robust PQEVAPs is proposed, and its efficient behaviors are illustrated by some numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Solving the Generalized Machine Assignment Problem in Group Technology

Many existing solution methodologies for machine assignment problems in group technology do not consider factors such as part demand, operation sequence and cost of intercellular moves. We formulate a 0-1 quadratic programming model that takes into account these factors in machine assignment. Two approaches are proposed to solve this problem. The first is an A*-based approach that generates optimal solutions. The second is a heuristic approach developed to solve problems with large number of machines and/or parts. The heuristic approach is shown to be efficient in producing good solutions in a computational study.     

A GRASP for the biquadratic assignment problem

The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment problem (QAP). It is a nonlinear integer programming 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 difficulty of this problem, only heuristic solution approaches have been proposed. In this paper, we propose a new heuristic for the BiQAP, a greedy randomized adaptive search procedure (GRASP). Computational results on instances described in the literature indicate that this procedure consistently finds better solutions than previously described algorithms.     

Fuzzy weighted equilibrium multi-job assignment problem and genetic algorithm

In this paper, the equilibrium optimization problem is proposed and the assignment problem is extended to the equilibrium multi-job assignment problem, equilibrium multi-job quadratic assignment problem and the minimum cost and equilibrium multi-job assignment problem. Furthermore, the mathematical models of the equilibrium multi-job assignment problem and the equilibrium multi-job quadratic assignment problem with fuzzy parameters are formulated. Finally, a genetic algorithm is designed for solving the proposed programming models and some numerical examples are given to verify the efficiency of the designed algorithm.