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The Reformulation Linearization Technique (RLT) of Sherali and Adams (Manag Sci 32(10):1274–1290, 1986; SIAM J Discrete Math 3(3):411–430, 1990), when applied to a pure 0–1 quadratic optimization problem with linear constraints (P), constructs a hierarchy of LP (i.e., continuous and linear) models of increasing sizes. These provide monotonically improving continuous bounds on the optimal value of (P) as the level, i.e., the stage in the process, increases. When the level reaches the dimension of the original solution space, the last model provides an LP bound equal to the IP optimum. In practice, unfortunately, the problem size increases so rapidly that for large instances, even computing bounds for RLT models of level k (called RLTk) for small k may be challenging. Their size and their complexity increase drastically with k. To our knowledge, only results for bounds of levels 1, 2, and 3 have been reported in the literature. We are proposing, for certain quadratic problem types, a way of producing stronger bounds than continuous RLT1 bounds in a fraction of the time it would take to compute continuous RLT2 bounds. The approach consists in applying a specific decomposable Lagrangean relaxation to a specially constructed RLT1-type linear 0–1 model. If the overall Lagrangean problem does not have the integrality property, and if it can be solved as a 0–1 rather than a continuous problem, one may be able to obtain 0–1 RLT1 bounds of roughly the same quality as standard continuous RLT2 bounds, but in a fraction of the time and with much smaller storage requirements. If one actually decomposes the Lagrangean relaxation model, this two-step procedure, reformulation plus decomposed Lagrangean relaxation, will produce linear 0–1 Lagrangean subproblems with a dimension no larger than that of the original model. We first present numerical results for the Crossdock Door Assignment Problem, a special case of the Generalized Quadratic Assignment Problem. These show that just solving one Lagrangean relaxation problem in 0–1 variables produces a bound close to or better than the standard continuous RLT2 bound (when available) but much faster, especially for larger instances, even if one does not actually decompose the Lagrangean problem. We then present numerical results for the 0–1 quadratic knapsack problem, for which no RLT2 bounds are available to our knowledge, but we show that solving an initial Lagrangean relaxation of a specific 0–1 RLT1 decomposable model drastically improves the quality of the bounds. In both cases, solving the fully decomposed rather than the decomposable Lagrangean problem to optimality will make it feasible to compute such bounds for instances much too large for computing the standard continuous RLT2 bounds.

Semidefinite optimization is a strong tool in the study of NP-hard combinatorial optimization problems. On the one hand, semidefinite optimization problems are in principle solvable in polynomial time (with fixed precision), on the other hand, their modeling power allows to naturally handle quadratic constraints. Contrary to linear optimization with the efficiency of the Simplex method, the algorithmic treatment of semidefinite problems is much more subtle and also practically quite expensive. This survey-type article is meant as an introduction for a non-expert to this exciting area. The basic concepts are explained on a mostly intuitive level, and pointers to advanced topics are given. We provide a variety of semidefinite optimization models on a selection of graph optimization problems and give a flavour of their practical impact.

In genetic algorithms crossover is the most important operator where pair of chromosomes and crossover site along their common length are selected randomly. Then the information after the crossover site of the parent chromosomes is swapped. On the other hand, mutation operator randomly alters some genes of a chromosome, and thus diversifies the search space. We consider three crossover and ten mutation operators for the genetic algorithms which are then compared for the quadratic assignment problem on some benchmark QAPLIB instances. The experimental study shows the effectiveness of the sequential constructive crossover and the adaptive mutation operators for the problem.

The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programming (MILP) problem. On the one hand, most of these linearizations are tight, but rarely exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation (Eur. J. Oper. Res. 2(3):204–211, 1978) is the smallest of these linearizations, but very weak. In this paper, we analyze how the Kaufman and Broeckx formulation can be tightened to obtain better QAP-MILP formulations. As shown in our numerical experiments, these tightened formulations remain small but computationally effective to solve the QAP by means of general purpose MILP solvers.

The rectangular assignment problem is a generalization of the linear assignment problem (LAP): one wants to assign a number of persons to a smaller number of jobs, minimizing the total corresponding costs. Applications are, e.g., in the fields of object recognition and scheduling. Further, we show how it can be used to solve variants of the LAP, such as the k-cardinality LAP and the LAP with outsourcing by transformation. We introduce a transformation to solve the machine replacement LAP.We describe improvements of a LAP-algorithm for the rectangular problem, in general and slightly adapted for these variants, based on the structure of the corresponding cost matrices. For these problem instances, we compared the best special codes from literature to our codes, which are more general and easier to implement. The improvements lead to more efficient and robust codes, making them competitive on all problem instances and often showing much shorter computing times.

The Multi-Story Space Assignment Problem (MSAP) is an innovative formulation of the multi-story facility assignment problem that allows one to model the location of departments of unequal size within multi-story facilities as a Generalized Quadratic 3-dimensional Assignment Problem (GQ3AP). Not only can the MSAP generate the design of the location of the departments in the facility, the MSAP also includes the evacuation planning for the facility. The formulation, background mathematical development, and computational experience with a branch and bound algorithm for the MSAP are also presented.

In this article, local optimality in multiobjective combinatorial optimization is used as a baseline for the design and analysis of two iterative improvement algorithms. Both algorithms search in a neighborhood that is defined on a collection of sets of feasible solutions and their acceptance criterion is based on outperformance relations. Proofs of the soundness and completeness of these algorithms are given.

We consider a vector linear combinatorial optimization problem in which initial coefficients of objective functions are subject to perturbations. For Pareto and lexicographic principles of efficiency we introduce appropriate measures of the quality of a given feasible solution. These measures correspond to so-called stability and accuracy functions defined earlier for scalar optimization problems. Then we study properties of such functions and calculate the maximum norms of perturbations for which an efficient solution preserves the efficiency.

This paper reports heuristic and exact solution advances for the Quadratic Assignment Problem (QAP).QAPinstances 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 instances as soon as the problem size approaches 30 to 40. This article presents new QAP instances that are ill conditioned for many metaheuristic-based methods. However, these new instances are shown to be solved relatively well by some exact methods, since problem instances up to a size of 75 have been exactly solved.

The vertex coloring problem has been the subject of extensive research for many years. Driven by application potential as well as computational challenge, a variety of methods have been proposed for this difficult class of problems. Recent successes in the use of the unconstrained quadratic programming (UQP) model as a unified framework for modeling and solving combinatorial optimization problems have motivated a new approach to the vertex coloring problem. In this paper we present a UQP approach to this problem and illustrate its attractiveness with preliminary computational experience.

The cross-entropy method is a relatively new method for combinatorial optimization. The idea of this method came from the simulation field and then was successfully applied to different combinatorial optimization problems. The method consists of an iterative stochastic procedure that makes use of the importance sampling technique. In this paper we prove the asymptotical convergence of some modifications of the cross-entropy method.

In previous work the authors consider the dynamic assignment problem, which involves solving sequences of assignment problems over time in the presence of uncertain information about the future. The algorithm proposed by the authors provides generally high-quality but non-optimal solutions. In this work, though, the authors prove that if the optimal solution to a dynamic assignment problem in one of two problem classes is unique, then the optimal solution is a fixed point under the algorithm.

In this paper, we show how an extended Guided Local Search (GLS) can be applied to the Quadratic Assignment Problem (QAP). GLS is a general, penalty-based meta-heuristic, which sits on top of local search algorithms, to help guide them out of local minima. We present empirical results of applying several extended versions of GLS to the QAP, and show that these extensions can improve the range of parameter settings within which Guided Local Search performs well. Finally, we compare the results of running our extended GLS with some state of the art algorithms for the QAP.

In this paper a particular quadratic minimum program, having a particular d.c. objective function, is studied. Some theoretical properties of the problem are stated and the existence of minimizers is characterized. A solution algorithm, based on the so called “optimal level solutions” approach, is finally proposed.

We consider families of optimization problems with quadratic object function and affine linear constraints, which depend smoothly on one real parameter. For a generic subclass of such problems only three different types of (generalized) critical points occur, whereas in the general case (of nonlinear one-parameter families of constrained optimization problems on Rn) five types are to be distinguished. We clarify the theoretical background of these phenomena and illustrate the underlying mechanism with simple examples.

The assignment of M unique machines to M locations with the objective of minimizing the total machine-to-machine material transportation cost in a flow line may be formulated as a quadratic assignment problem (QAP). Instead of having M unique machines, if an application involves one or more sets of identical machines, the location problem becomes a tertiary assignment problem (TAP). Solving a large problem of this kind is extremely difficult because of its combinatorial nature. When machine-to-machine flow is fixed, the TAP may be specialized to a QAP for which the unique machine problem is a special case. Obtaining an optimum solution to this problem when M is large is also computationally intractable. However, this problem may be solved by identifying sets of identical machines which may be partitioned into individual, "unique" machines. Properties of a special type of matrix called the amoebic matrix are used in the partitioned problems to provide approximate solutions, which are relabeled to prescribe a solution to the original problem. Results are demonstrated along with suggestions for further research.

This paper presents the application of a tabu search algorithm for solving the frequency assignment problem. This problem, known to be NP-hard, is to find an assignment of frequencies for a number of communication links, which satisfy various constraints. We report on our computational experiments in terms of computational efficiency and quality of the solutions obtained for realistic, computer-generated problem instances. The method is efficient, robust and stable and gives solutions which compare more favourably than ones obtained using a genetic algorithm.

In the partial accessibility constrained vehicle routing problem, a route can be covered by two types of vehicles, i.e. truck or truck + trailer. Some customers are accessible by both vehicle types, whereas others solely by trucks. After introducing an integer programming formulation for the problem, we describe a two-phase heuristic method which extends a classical vehicle routing algorithm. Since it is necessary to solve a combinatorial problem that has some similarities with the generalized assignment problem, we propose an enumerative procedure in which bounds are obtained from a Lagrangian relaxation. The routine provides very encouraging results on a set of test problems.

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.

The Generalized Assignment Problem, in the class of NP-hard problems, occurs in a wide range of applications — vehicle packing, computers, and logistics, to name only a few. Previous research has been concentrated on optimization methodologies for the GAP. Because the Generalized Assignment Problem is NP-hard, optimization methods tend to require larger computation times for large-scale problems. This paper presents a new heuristic,Variable-Depth-Search Heuristic (VDSH). We show that on the sets of large test problems, the quality of the solution found by VDSH exceeds that of the leading heuristic by an average of over twenty percent, while maintaining acceptable solution times. On difficult problem instances, VDSH provides solutions having costs 140% less than those found by the leading heuristic. A duality gap analysis of VDSH demonstrates the robustness of our heuristics.

In practical problem situations data are usually inherently unreliable. A mathematical representation of uncertainty leads to stochastic optimization problems. In this paper the complexity of stochastic combinatorial optimization problems is discussed. Surprisingly, certain stochastic versions of NP-hard determinstic combinatorial problems appear to be solvable in polynomial time.

A class of dynamic location problems is introduced. The relationship between a static problem and its corresponding dynamic one is studied. We concentrate on two types of dynamic problems. The first is the global optimization problem, in which one looks for the all-times optimum. The second is the steady-state problem in which one seeks to determine the steady-state behavior of the system if one exists. General approaches to these problems are discussed.