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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.

Various conic relaxations of quadratic optimization problems in nonnegative variables for combinatorial optimization problems, such as the binary integer quadratic problem, quadratic assignment problem (QAP), and maximum stable set problem have been proposed over the years. The binary and complementarity conditions of the combinatorial optimization problems can be expressed in several ways, each of which results in different conic relaxations. For the completely positive, doubly nonnegative and semidefinite relaxations of the combinatorial optimization problems, we discuss the equivalences and differences among the relaxations by investigating the feasible regions obtained from different representations of the combinatorial condition which we propose as a generalization of the binary and complementarity condition. We also study theoretically the issue of the primal and dual nondegeneracy, the existence of an interior solution and the size of the relaxations, as a result of different representations of the combinatorial condition. These characteristics of the conic relaxations affect the numerical efficiency and stability of the solver used to solve them. We illustrate the theoretical results with numerical experiments on QAP instances solved by SDPT3, SDPNAL+ and the bisection and projection method.

We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time.

In this paper we study the complexity and domination analysis in the context of the \emph{bipartite quadratic assignment problem}. Two variants of the problem, denoted by BQAP1 and BQAP2, are investigated. A formula for calculating the average objective function value $\mathcal{A}$ of all solutions is presented whereas computing the median objective function value is shown to be NP-hard. We show that any heuristic algorithm that produces a solution with objective function value at most $\mathcal{A}$ has the domination ratio at least $\frac{1}{mn}$. Analogous results for the standard \emph{quadratic assignment problem} is an open question. We show that computing a solution whose objective function value is no worse than that of $n^mm^n-{\lceil\frac{n}{\alpha}\rceil}^{\lceil\frac{m}{\alpha}\rceil}{\lceil\frac{m}{\alpha}\rceil}^{\lceil\frac{n}{\alpha}\rceil}$ solutions of BQAP1 or $m^mn^n-{\lceil\frac{m}{\alpha}\rceil}^{\lceil\frac{m}{\alpha}\rceil}{\lceil\frac{n}{\alpha}\rceil}^{\lceil\frac{n}{\alpha}\rceil}$ solutions of BQAP2, is NP-hard for any fixed natural numbers $a$ and $b$ such that $\alpha=\frac{a}{b}>1$. However, a solution with the domination number $\Omega(m^{n-1}n^{m-1}+m^{n+1}n+mn^{m+1})$ for BQAP1 and $\Omega(m^{m-1}n^{n-1}+m^2n^{n}+m^mn^2)$ for BQAP2, can be found in $O(m^3n^3)$ time.

A new location problem is formulated and solved. It is the continuous version of the grey pattern problem which is a special case of the Quadratic Assignment Problem. The problem is a minimization of a convex function subject to non-convex constraints and has infinitely many optimal solutions. We propose several mathematical programming formulations that are suitable for a multi-start heuristic algorithm. In addition to solving these formulations by the Solver in Excel and Mathematica, a special Nelder–Mead algorithm is proposed. This special algorithm provided the best results. One suggested modification may improve the performance of the Nelder–Mead algorithm for other optimization problems as well.

In this paper we study the effect of a decision maker׳s risk attitude on the median and center problems, two well-known location problems, with uncertain demand in the mean–variance framework. We provide a mathematical programming formulation for both problems in the form of quadratic programming and develop solution procedures. In particular, we consider the vertex and absolute median problems separately, and identify a dominant set for the center problem. Glover׳s linearization method is applied to solve the vertex median problem. We also develop a branch and bound algorithm and a heuristic as the linearization technique takes too long for the vertex median problem on large networks. A computational experiment is conducted to compare the performance of the algorithms. We demonstrate the importance of taking into account the volatility and correlation structure when a location decision is made. The closest assignment property is also discussed for these location problems under the mean–variance objective.

We consider the boolean quadratic programming problem with generalized upper bound constraints (BQP-GUB) which subsumes the well-known quadratic semi-assignment problem. BQP-GUB has applications in engineering, production planning and biology. We present various complexity results on the problem along with different metaheuristic algorithms. Results of extensive experimental analysis are presented demonstrating the efficacy of our proposed algorithms.

A quadratic assignment problem (QAP), which is a combinatorial optimisation problem, is developed to model the problem of locating facilities with material flows between them. The aim of solving the QAP formulation for a facility layout problem (FLP) is to increase a system’s operating efficiency by reducing material handling costs, which can be measured by interdepartmental distances and flows. The QAP-formulated FLP can be viewed as a discrete optimisation problem, where the quadratic objective function is optimised with respect to discrete decision variables subject to linear equality constraints. The conventional approach for solving this discrete optimisation problem is to use the linearisation of the quadratic objective function whereby additional discrete variables and constraints are introduced. The adoption of the linearisation process can result in a significantly increased number of variables and constraints; solving the resulting problem can therefore be challenging. In this paper, a new appro...

In this paper, we study the problem of minimizing the ratio of two complex indefinite quadratic functions subject to a strictly convex quadratic constraint. First, using the known method due to Dinkelbach, we reformulate the fractional problem as a univariate equation. To find the root of the univariate equation, the generalized Newton method is utilized that requires solving a nonconvex quadratic optimization problem at each iteration. To solve these nonconvex quadratic problems, we present an efficient algorithm by a diagonalization scheme that requires solving a univariate minimization problem at each iteration. Moreover, for the homogeneous case, it requires solving a simple linear optimization problem. Our preliminary numerical experiments on several randomly generated test problems show that, the new approach is much faster in finding the global optimal solution than the known semidefinite relaxation approach, especially when solving large scale problems.

We introduce a new binary quadratic program that subsumes the well known quadratic assignment problem. This problem is shown to be NP-hard when some associated parameters are restricted to simple values but solvable in polynomial time when some other parameter values are restricted. Three different neighborhood structures are introduced. A best solution in all these neighborhoods can be identified in polynomial time even though two of these neighborhoods are of exponential size. Different local search algorithms and their enhancements using tabu search are developed using these neighborhoods. Experimental analysis show that two hybrid algorithms obtained by combining these neighborhoods in specific ways yield results that are superior to the algorithms that use these neighborhoods separately. Extensive computational results and statistical analysis of the resulting data are presented.

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore–Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. We also present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem.

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.

We propose an exact algorithm to find the optimal solution to the grey pattern problem which is a special instance of the quadratic assignment problem. A very effective branch and bound algorithm is constructed. The special structure of the problem is exploited and optimal solutions for many problem instances established. One instance based on 64 facilities, organized in an 8 by 8 pattern, that was extensively researched in the literature was optimally solved in 15 s of computer time while existing papers report solution times of hours.

Pareto Local Search (PLS) is a simple and effective local search method for tackling multi-objective combinatorial optimization problems. It is also a crucial component of many state-of-the-art algorithms for such problems. However, PLS may be not very effective when terminated before completion. In other words, PLS has poor anytime behavior. In this paper, we study the effect that various PLS algorithmic components have on its anytime behavior. We show that the anytime behavior of PLS can be greatly improved by using alternative algorithmic components. We also propose Dynagrid, a dynamic discretization of the objective space that helps PLS to converge faster to a good approximation of the Pareto front and continue to improve it if more time is available. We perform a detailed empirical evaluation of the new proposals on the bi-objective traveling salesman problem and the bi-objective quadratic assignment problem. Our results demonstrate that the new PLS variants not only have significantly better anytime behavior than the original PLS, but also may obtain better results for longer computation time or upon completion., SCOPUS: ar.j, info:eu-repo/semantics/published

We present an integer linear formulation that uses the so-called “distance variables” to solve the quadratic assignment problem (QAP). The formulation performs particularly well for problems with Manhattan distance matrices. It involves $$O(n^2)$$ variables. Valid equalities and inequalities are proposed divided into two families. First, a family of inequalities valid for any quadratic assignment problems, and second, a family valid only for problems with Manhattan distance matrices, for which we exploit metric properties, as well as an algebraic characterization that Mittelman and Peng (SIAM J Opt 2010:20(6), 3408–3426, 2010) recently proved. We numerically tested the lower bound provided by the linear relaxation using instances of the quadratic assignment problem library (QAPLIB) with randomly generated distance matrices, as well as Manhattan distance matrices. Our results are compared with the best known lower bounds. For Manhattan distance matrices, the formulation gives a very competitive lower bound in a short computational time, improving seven best lower bounds of QAPLIB instances for which no optimality proofs exist.

Instead of using traditional (two-parent) crossover operator, multi-parent crossover operator is used in genetic algorithms to improve solution quality for many numerical optimization problems. However, very few literatures are available on multi-parent crossover operator for combinatorial optimization problems, especially, quadratic assignment problem (QAP). This paper proposes a multi-parent extension of the sequential constructive crossover (MPSCX), which is a generalization of the traditional sequential constructive crossover (SCX), for the QAP. Then a multi-parent genetic algorithm (MPGA) using MPSCX is developed. Experimental results on ten QAPLIB instances show that our MPGA significantly improves GA using SCX by up to 1.75 % in average assignment cost with maximum of 2.79 % away from the best known solution value. Finally, the efficiency of our MPGA is compared against MPGA using an existing multi-parent crossover for the problem. Experimental results show that our MPGA is better.

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.

In this study, we consider a bicriteria multiresource generalized assignment problem. Our criteria are the total assignment load and maximum assignment load over all agents. We aim to generate all nondominated objective vectors and the corresponding efficient solutions. We propose several lower and upper bounds and use them in our optimization and heuristic algorithms. The computational results have shown the satisfactory behaviors of our approaches. © 2014 Wiley Periodicals, Inc. Naval Research Logistics, 61: 621–636, 2014

Robust combinatorial optimization problems with cardinality constrained uncertainty may be solved by a finite number of nominal problems. In this paper, we show that the number of nominal problems to be solved can be reduced significantly.

We consider a variant of the multidimensional assignment problem (MAP) with decomposable costs in which the resulting optimal assignment is described as a set of disjoint stars. This problem arises in the context of multi-sensor multi-target tracking problems, where a set of measurements, obtained from a collection of sensors, must be associated to a set of different targets. To solve this problem we study two different formulations. First, we introduce a continuous nonlinear program and its linearization, along with additional valid inequalities that improve the lower bounds. Second, we state the standard MAP formulation as a set partitioning problem, and solve it via branch and price. These approaches were put to test by solving instances ranging from tripartite to 20-partite graphs of 4 to 30 nodes per partition. Computational results show that our approaches are a viable option to solve this problem. A comparative study is presented.

Performance comparisons between algorithms have a long tradition in metaheuristic research. An early example is comparisons between Tabu Search (TS) and Simulated Annealing (SA) algorithms for tackling the Quadratic Assignment Problem (QAP). The results of these comparisons are to a certain extent inconclusive, even when focusing on only these two types of algorithms. While comparisons of SA and TS algorithms were based on rather small-sized instances, here we focus on possible dependencies of the relative performance between SA and TS algorithms on instance size. In fact, our experimental results show that the assertion whether one algorithm is better than the other can depend strongly on QAP instance size even if one focuses on instances with otherwise same characteristics.

The reformulation–linearization technique (RLT), introduced in [Sherali, H. D., Adams. W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430], provides a way to compute a hierarchy of linear programming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs.

In this paper, we study a multi-objective multi-choice assignment problem considering cost and time objectives subject to some realistic constraints including multi-job assignment. We assume that the decision-maker provides multiple aspiration levels regarding both cost and time objectives using discrete choices as well as interval values. To obtain efficient allocation plans, we use multi-choice goal programming methodology to solve the assignment problem. The proposed methodology is illustrated through numerical examples.

In this paper we discuss two particular layout problems, namely the Single-Row Equidistant Facility Layout Problem (SREFLP) and the Single-Row Facility Layout Problem (SRFLP). Our aim is to consolidate the two respective branches in the layout literature. We show that the SREFLP is not only a special case of the Quadratic Assignment Problem but also a special case of the SRFLP. This new connection is relevant as the strongest exact methods for the SRFLP outperform the best approaches specialised to the SREFLP. We describe and compare the exact approaches for the SRFLP, the SREFLP and Linear Arrangement that is again a special case of the SREFLP. In a computational study we showcase that the strongest exact approach for the SRFLP clearly outperforms the strongest exact approach tailored to the SREFLP on medium and large benchmark instances from the literature.

To each instance of the Quadratic Assignment Problem (QAP) a relaxed instance can be associated. Both variances of their solution values can be calculated in polynomial time. The graph isomorphism problem (GIP) can be modeled as a QAP, associating its pair of data matrices with a pair of graphs of the same order and size. We look for invariant edge weight functions for the graphs composing the instances in order to try to find quantitative differences between variances that could be associated with the absence of isomorphism. This technique is sensitive enough to show the effect of a single edge exchange between two regular graphs of up to 3,000 vertices and 300,000 edges with degrees up to 200. Planar graph pairs from a dense family up to 300,000 vertices were also discriminated. We conjecture the existence of functions able to discriminate non-isomorphic pairs for every instance of the problem.

The Quadratic Assignment Problem (QAP) is a difficult and important problem studied in the domain of combinatorial optimisation. It is possible to solve QAP instances with 10--20 facilities using exhaustive parallel algorithms within a few days on a cluster machine. However, large QAP instances with more than 100 facilities are not solvable using exhaustive techniques. We have explored a variety of Genetic Algorithm crossover operators for this problem and verified its performance experimentally using well-known instances from the QAPLIB library. By increasing the number of processors, generations and population sizes we have been able to find solutions that are the same as (or very close to) the best reported solutions for large QAP instances in QAPLIB. In order to parallelise the Genetic Algorithm we generate and evolve separate solution pools on each cluster processor, using an island model. This model exchanges 10% of each processor’s solutions at the initial stages of optimisation. We show experiment...

A differential improvement modification to Hybrid Genetic Algorithms is proposed. The general idea is to perform more extensive improvement algorithms on higher quality solutions. Our proposed Differential Improvement (DI) approach is of rather general character. It can be implemented in many different ways. The paradigm remains invariant and can be easily applied to a wider class of optimization problems. Moreover, the DI framework can also be used within other Hybrid metaheuristics like Hybrid Scatter Search algorithms, Particle Swarm Optimization, or Bee Colony Optimization techniques. Extensive experiments show that the new approach enables to improve significantly the performance of Hybrid Genetic Algorithms without adding extra computer time. Additional experiments investigated the trade-off between the number of generations and the number of iterations of the improvement algorithm. These experiments yielded six new best known solutions to benchmark quadratic assignment problems. Many other variants of the proposed algorithm are suggested for future research.

In this paper, we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function. Examples of such problems include those in which the objective function is a product or ratio of two linear functions, parallel machine scheduling problems with the makespan objective, robust versions of weighted multiobjective optimization problems, and assortment optimization problems with logit choice models. The main idea behind our approximation schemes is the construction of an approximate Pareto-optimal frontier of the functions that constitute the given objective. Using this idea, we give the first fully polynomial-time approximation schemes for the max-min resource allocation problem with a fixed number of agents, combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions, or the product of a fixed number of linear functions, and assortment optimization problems with logit choice model.

We establish strong NP-hardness and in-approximability of the so-called representatives selection problem, a tool selection problem in the area of robust optimization. Our results answer a recent question of Dolgui and Kovalev (4OR Q J Oper Res 10:181–192, 2012). Keywords: Combinatorial optimization; Computational complexity; Robust optimization

We prove that, for two simple functions d r l t , solving the online assignment problem with c r l − d r l t as the contribution for assigning resource r to task l at time t gives the optimal solution to the corresponding offline assignment problem (provided the optimal offline solution is unique). We call such functions d r l t optimal discount functions.

This paper is concerned with automated classification of Combinatorial Optimization Problem instances for instance-specific parameter tuning purpose. We propose the CluPaTra Framework, a generic approach to CLUster instances based on similar PAtterns according to search TRAjectories and apply it on parameter tuning. The key idea is to use the search trajectory as a generic feature for clustering problem instances. The advantage of using search trajectory is that it can be obtained from any local-search based algorithm with small additional computation time. We explore and compare two different search trajectory representations, two sequence alignment techniques (to calculate similarities) as well as two well-known clustering methods. We report experiment results on two classical problems: Travelling Salesman Problem and Quadratic Assignment Problem and industrial case study.

New mixed-integer linear programming formulations are presented for the quadratic assignment problem, based on splittings of the coefficient matrices. Computational results are reported for medium-sized problem instances in the QAPLIB collection.

This paper deals with stability analysis in multi-objective combinatorial optimization problems. The stability radius of an efficient solution is defined as the maximal adjustment of the problem parameters such that this solution remains efficient. An algorithm based on inverse optimization is proposed to compute it. The adjustment is limited to the coefficients of the objective functions and measured by the Chebyshev norm. This approach is applied to randomly generated instances of the bi-objective knapsack problem and computational results are reported. Several illustrative examples are analyzed.

Two criteria in a combinatorial problem are often combined in a weighted sum objective using a weighting parameter between 0 and 1. For special problem types, e.g., when one of the criteria is a bottleneck value, efficient algorithms are known that solve for a given value of the weighting parameter. We transform the underlying enumeration method into a parametric algorithm solving simultaneously for all values of the weighting parameter. Efficient implementations are presented for combinatorial problems with criteria as balanced optimization, min–sum with min–max, and min–sum with balanced optimization, considering the spanning tree, the linear assignment and the single machine scheduling problem. Further the new algorithmic scheme can easily incorporate a trade-off of the criteria by means of penalty functions, again without consequences for the algorithm and its complexity order.

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.

Discovering the conditions under which an optimization algorithm or search heuristic will succeed or fail is critical for understanding the strengths and weaknesses of different algorithms, and for automated algorithm selection. Large scale experimental studies - studying the performance of a variety of optimization algorithms across a large collection of diverse problem instances - provide the resources to derive these conditions. Data mining techniques can be used to learn the relationships between the critical features of the instances and the performance of algorithms. This paper discusses how we can adequately characterize the features of a problem instance that have impact on difficulty in terms of algorithmic performance, and how such features can be defined and measured for various optimization problems. We provide a comprehensive survey of the research field with a focus on six combinatorial optimization problems: assignment, traveling salesman, and knapsack problems, bin-packing, graph coloring, and timetabling. For these problems - which are important abstractions of many real-world problems - we review hardness-revealing features as developed over decades of research, and we discuss the suitability of more problem-independent landscape metrics. We discuss how the features developed for one problem may be transferred to study related problems exhibiting similar structures.

This paper introduces a new variable depth search method for the Quadratic Assignment Problem. The new method considers the cost of edges assignment as the criterion to decide which vertices to exchange during local search moves. It also presents the results of an extensive experimental study that compares the performance of local search and variable depth search algorithms for the Quadratic Assignment Problem. The investigation presented here contributes to a better understanding of the potential of these techniques, which are widely used as intensification tools in more sophisticated heuristic methods, such as evolutionary algorithms. Different algorithms presented in the literature were implemented and compared to the proposed methods. The results of a computational experiment with 161 benchmark instances are reported. Different statistical tests are applied in order to analyze the results provided by the experiments.

In this article, we deal with the inverse linear optimization problem under l2-norm defined as follows: for a given linear optimization problem LP(b), with the right-hand side vector b, and current operating plan x 0, we would like to perturb the right-hand side vector b to so that x 0 is an optimal solution of and is minimum. We prove that this inverse problem is equivalent to an optimization problem which contains some equilibrium constraints that makes it hard to solve. In order to solve this optimization problem, we employ the interior point approach behind LOQO and show that, at each iteration, the search direction is a descent direction for the l2 merit function. Therefore, an e-solution of the inverse problem can be computed using an extension of polynomial-time interior-point methods for linear and quadratic optimization problems. Finally, computational results of applying proposed approach on some generated inverse problems are given.

This work is devoted to the Dynamic Space Allocation Problem, where project duration is divided into a number of consecutive periods, each of them associated with a number of activities. The resources required by the activities have to be available in the corresponding workspaces and those sitting idle during a period have to be stored. This problem contains the Quadratic Assignment Problem (QAP) as a particular case, which puts it in the NP-hard class. In this context, the difficulty of identifying optimal solutions, even for instances of medium size, justifies the use of heuristic techniques. This work proposes a construction and a hybrid algorithm (HGT) based on the GRASP and Tabu search metaheuristics. Comparisons are presented for values obtained by HGT, pure GRASP versions, Tabu search and literature results. Computational results show the proposed methods to be competitive in relation to instances in the literature and to existing techniques.

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.

Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications.

We analyze a special case of the maximum quadratic assignment problem where one matrix is a monotone anti-Monge matrix and the other matrix has a multi-layered structure that is built on top of certain Toeplitz matrices. To demonstrate an application of our main result, we derive a (simple and concise) alternative proof for a recent result on the scheduling problem of maximizing the variance of job completion times.

The x-and-y-axes travelling salesman problem forms a special case of the Euclidean TSP, where all cities are situated on the x-axis and on the y-axis of an orthogonal coordinate system of the Euclidean plane. By carefully analyzing the underlying combinatorial and geometric structures, we show that this problem can be solved in polynomial time. The running time of the resulting algorithm is quadratic in the number of cities.

In this work we propose a multicriteria evaluation scheme for heuristic algorithms based on the classic Condorcet ranking technique. Weights are associated to the ranking of an algorithm among a set being object of comparison. We used five criteria and a function on the set of natural numbers to create a ranking. The discussed comparison involves three well-known problems of combinatorial optimization - Traveling Salesperson Problem (TSP), Capacitated Vehicle Routing Problem (CVRP) and Quadratic Assignment Problem (QAP). The tested instances came from public libraries. Each algorithm was used with essentially the same structure, the same local search was applied and the initial solutions were similarly built. It is important to note that the work does not make proposals involving algorithms: the results for the three problems are shown only to illustrate the operation of the evaluation technique. Four metaheuristics - GRASP, Tabu Search, ILS and VNS - are therefore only used for the comparisons.

An instance of the quadratic assignment problem (QAP) with cost matrix Q is said to be linearizable if there exists an instance of the linear assignment problem (LAP) with cost matrix C such that for each assignment, the QAP and LAP objective function values are identical. Several sufficiency conditions are known that guarantee linearizability of a QAP. However, no polynomial time algorithm is known to test if a given instance of QAP is linearizable. In this paper, we give a necessary and sufficient condition for an instance of a QAP to be linearizable and develop an O(n4) algorithm to solve the corresponding linearization problem, where n is the size of the QAP.

In this paper, we describe an implementation of the iterated tabu search (ITS) algorithm for the quadratic assignment problem (QAP), which is one of the well-known problems in combinatorial optimization. The medium- and large-scale QAPs are not, to this date, practically solvable to optimality, therefore heuristic algorithms are widely used. In the proposed ITS approach, intensification and diversification mechanisms are combined in a proper way. The goal of intensification is to search for good solutions in the neighbourhood of a given solution, while diversification is responsible for escaping from local optima and moving towards new regions of the search space. In particular, the following enhancements were implemented: new formula for fast evaluation of the objective function and efficient data structure; extended intensification mechanisms (including randomized tabu criterion, combination of tabu search and local search, dynamic tabu list maintaining); enhanced diversification strategy using periodic tabu tenure and special mutation procedure. The ITS algorithm is tested on the different instances taken from the QAP library QAPLIB. The results from the experiments demonstrate promising efficiency of the proposed algorithm, especially for the random QAP instances.

We show how to express the problem of searching for D-optimal matrices as a Linear and Quadratic Integer Optimization problem. We also focus our attention in the case where the size of the circulant submatrices that are used to construct a D-optimal matrix is a multiple of 3. In this particular case, we describe some additional combinatorial and number-theoretic characteristics that a solution of the D-optimal matrix problem must possess. We give some solutions for some quite challenging D-optimal matrix problems that can be used as benchmarks to test the efficiency of Linear and Quadratic Integer Optimization algorithms.

One of the most important variants of the standard linear assignment problem is the bottleneck assignment problem. In this paper we give a method by which one can find all of the asymptotic moments of a random bottleneck assignment problem in which costs (independent and identically distributed) are chosen from a wide variety of continuous distributions. Our method is obtained by determining the asymptotic moments of the time to first complete matching in a random bipartite graph process and then transforming those, via a Maclaurin series expansion for the inverse cumulative distribution function, into the desired moments for the bottleneck assignment problem. Our results improve on the previous best-known expression for the expected value of a random bottleneck assignment problem, yield the first results on moments other than the expected value, and produce the first results on the moments for the time to first complete matching in a random bipartite graph process.

Mukherjee and Basu proposed a new method for solving fuzzy assignment problems. In this paper, some fuzzy assignment problems and fuzzy travelling salesman problems are chosen which cannot be solved by using the fore-mentioned method. Two new methods are proposed for solving such type of fuzzy assignment problems and fuzzy travelling salesman problems. The fuzzy assignment problems and fuzzy travelling salesman problems which can be solved by using the existing method, can also be solved by using the proposed methods. But, there exist certain fuzzy assignment problems and fuzzy travelling salesman problems which can be solved only by using the proposed methods. To illustrate the proposed methods, a fuzzy assignment problem and a fuzzy travelling salesman problem is solved. The proposed methods are easy to understand and apply to find optimal solution of fuzzy assignment problems and fuzzy travelling salesman problems occurring in real life situations.

Classical combinatorial optimization problems can be generalized in a natural way by considering a related problem relative to a given partition of the nodes of the graph into node sets. In the literature one can find generalized problems such as: generalized minimum spanning tree, generalized traveling salesman problem, generalized Steiner tree problem, generalized vehicle routing problem, etc. These generalized problems typically belong to the class of NP-complete problems; they are harder than the classical ones, and nowadays are intensively studied due to their interesting properties and applications in the real world. Because of the complexity of finding the optimal or near-optimal solution in case of the generalized combinatorial optimization problems, great effort has been made, by many researchers, to develop efficient ways of their transformation into classical corresponding variants. We present in this paper an efficient way of transforming the generalized vehicle routing problem into the vehicle routing problem, and a new integer programming formulation of the problem.