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.     

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.     

The Wiener maximum quadratic assignment problem

We investigate a special case of the maximum quadratic assignment problem where one matrix is a product matrix and the other matrix is the distance matrix of a one-dimensional point set. We show that this special case, which we call the Wiener maximum quadratic assignment problem, is NP-hard in the ordinary sense and solvable in pseudo-polynomial time.Our approach also yields a polynomial time solution for the following problem from chemical graph theory: find a tree that maximizes the Wiener index among all trees with a prescribed degree sequence. This settles an open problem from the literature.     

Recent advances in the solution of quadratic assignment problems

 The quadratic assignment problem (QAP) is notoriously difficult for exact solution methods. In the past few years a number of long-open QAPs, including those posed by Steinberg (1961), Nugent et al. (1968) and Krarup (1972) were solved to optimality for the first time. The solution of these problems has utilized both new algorithms and novel computing structures. We describe these developments, as well as recent work which is likely to result in the solution of even more difficult instances. Received: February 13, 2003 / Accepted: April 22, 2003 Published online: May 28, 2003 Key Words. quadratic assignment problem – discrete optimization – branch and bound Mathematics Subject Classification (1991): 90C27, 90C09, 90C20     

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.     

A multi-step hybrid method for multi-input partial quadratic eigenvalue assignment with time delay

A hybrid method was given by Ram et al. (2011) [15] for solving the partial quadratic eigenvalue assignment problem of single-input vibratory systems. In this paper, we consider the partial quadratic eigenvalue assignment problem of multi-input vibratory systems. We solve the multi-input partial quadratic eigenvalue assignment problem by a multi-step hybrid method using both the system matrices and the receptance measurements. Our method can assign the partial expected eigenvalues and keep the no spillover property. We also extend our method to the case when there exists time delay between measurements of state and actuation of control. Numerical tests show the effectiveness of our method.     

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.     

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 new exact discrete linear reformulation of the quadratic assignment problem

The quadratic assignment problem (QAP) is a challenging combinatorial problem. The problem is NP-hard and in addition, it is considered practically intractable to solve large QAP instances, to proven optimality, within reasonable time limits. In this paper we present an attractive mixed integer linear programming (MILP) formulation of the QAP. We first introduce a useful non-linear formulation of the problem and then a method of how to reformulate it to a new exact, compact discrete linear model. This reformulation is efficient for QAP instances with few unique elements in the flow or distance matrices. Finally, we present optimal results, obtained with the discrete linear reformulation, for some previously unsolved instances (with the size n = 32 and 64), from the quadratic assignment problem library, QAPLIB.     

Integer programming models and linearizations for the traveling car renter problem

The traveling car renter problem (CaRS) is an extension of the classical traveling salesman problem (TSP) where different cars are available for use during the salesman’s tour. In this study we present three integer programming formulations for CaRS, of which two have quadratic objective functions and the other has quadratic constraints. The first model with a quadratic objective function is grounded on the TSP interpreted as a special case of the quadratic assignment problem in which the assignment variables refer to visitation orders. The second model with a quadratic objective function is based on the Gavish and Grave’s formulation for the TSP. The model with quadratic constraints is based on the Dantzig–Fulkerson–Johnson’s formulation for the TSP. The formulations are linearized and implemented in two solvers. An experiment with 50 instances is reported.     

Lower bounds for the quadratic assignment problem

We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for the quadratic assignment problem.     

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     

A quadratic assignment formulation of the molecular conformation problem

The molecular conformation problem is discussed, and a concave quadratic global minimization approach for solving it is described. This approach is based on a quadratic assignment formulation of a discrete approximation to the original problem.     

A new bound for the quadratic assignment problem based on convex quadratic programming

We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the trade-off between bound quality and computational effort. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001     

Optimizing a polyhedral-semidefinite relaxation of completely positive programs

It has recently been shown (Burer, Math Program 120:479–495, 2009) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs, i.e., the minimization of a linear function over the convex cone of completely positive matrices subject to linear constraints. Such convex programs are NP-hard in general. A basic tractable relaxation is gotten by approximating the completely positive matrices with doubly nonnegative matrices, i.e., matrices which are both nonnegative and positive semidefinite, resulting in a doubly nonnegative program (DNP). Optimizing a DNP, while polynomial, is expensive in practice for interior-point methods. In this paper, we propose a practically efficient decomposition technique, which approximately solves the DNPs while simultaneously producing lower bounds on the original NQP. We illustrate the effectiveness of our approach for solving the basic relaxation of box-constrained NQPs (BoxQPs) and the quadratic assignment problem. For one quadratic assignment instance, a best-known lower bound is obtained. We also incorporate the lower bounds within a branch-and-bound scheme for solving BoxQPs and the quadratic multiple knapsack problem. In particular, to the best of our knowledge, the resulting algorithm for globally solving BoxQPs is the most efficient to date.     

The asymptotic probabilistic behaviour of quadratic sum assignment problems

In this paper a surprising probabilistic behaviour of quadratic sum assignment problems is shown. The relative difference between worst and optimal solution value tends to zero with probability tending to one as the size of the problem goes to infinity. This result suggests that for high dimensional quadratic assignment problems even very simple approximation algorithms can in practice yield good suboptimal solutions.     

A contribution to quadratic assignment problems

We discuss special eases of the quadratic assignment problem (QAP) being polynomially solvable. In particular we give an algebraic condition for the cost; Matrices of a QAP which guarantees that it is equivalent with a linear assignment problem. Based on these results we develop an approximation algorithm for QAPs with non-negative symmetric cost matrices.     

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.     

On the quality of heuristic solutions to a 19 × 19 quadratic assignment problem

Recently Murtagh et al. communicated the best (?) known solution to a 19 × 19 quadratic assignment problem, presented and solved earlier by Elshafei. Their approaches are based on the application of quadratic programming and on the construction of a good initial solution respectively. Their solutions are confronted with one better solution obtained by Lashkari and Jaisingh by means of linear assignment approximations, and many better solutions obtained by the present author through the application of exchange heuristics. The results stress the nonconvex character of the problem, and show the rather misleading nature of the concept: ‘good initial solution’. The quality of initial and final solution prove to be rather weakly (or even negatively!) correlated. A simple, but broad and randomized search results in significantly, better solutions, while simple statistical aids provide an estimate of the quality of these solutions. Finally we make remarks concerning some practical issues.     

The table placement problem: a research challenge at the EWI 2007

The table placement problem consists in deciding how to seat the participants attending a social lunch or dinner so that the total social benefit of the event is maximum. Four different approaches are presented: a linear model, a bin-packing-based-approach, a quadratic assignment problem, and a greedy heuristic. The different formulations are computationally compared over a set of artificial instances and on the real data for the EURO Winter Institute 2007 Gala dinner.