Hybrid Population-Based Algorithms for the Bi-Objective Quadratic Assignment Problem

We present variants of an ant colony optimization (MO-ACO) algorithm and of an evolutionary algorithm (SPEA2) for tackling multi-objective combinatorial optimization problems, hybridized with an iterative improvement algorithm and the robust tabu search algorithm. The performance of the resulting hybrid stochastic local search (SLS) algorithms is experimentally investigated for the bi-objective quadratic assignment problem (bQAP) and compared against repeated applications of the underlying local search algorithms for several scalarizations. The experiments consider structured and unstructured bQAP instances with various degrees of correlation between the flow matrices. We do a systematic experimental analysis of the algorithms using outperformance relations and the attainment functions methodology to asses differences in the performance of the algorithms. The experimental results show the usefulness of the hybrid algorithms if the available computation time is not too limited and identify SPEA2 hybridized with very short tabu search runs as the most promising variant. This research was mainly done while Luís Paquete and Thomas Stützle were with the Intellectics Group at the Computer Science Department of Darmstadt University of Technology, Germany     

Applying the pilot method to improve VNS and GRASP metaheuristics for the design of SDH/WDM networks

Variable neighborhood search (VNS) and Greedy randomized adaptive search procedure (GRASP) are among the well studied local search based metaheuristics providing good results for many combinatorial optimization problems throughout the last decade. While they are usually explored in different environments one may encounter quite obvious commonalities. Based on previous successful applications of these two types of metaheuristics on various network design problems in telecommunications, we further enhance these approaches by incorporating ideas from the pilot method. The different heuristics are compared among each other as well as against objective function values obtained from a mathematical programming formulation based on a commercial solver. The problem instances cover a large variety of networks and demand patterns.     

A column generation method for the multiple-choice multi-dimensional knapsack problem

In this paper, we propose to solve large-scale multiple-choice multi-dimensional knapsack problems. We investigate the use of the column generation and effective solution procedures. The method is in the spirit of well-known local search metaheuristics, in which the search process is composed of two complementary stages: (i) a rounding solution stage and (ii) a restricted exact solution procedure. The method is analyzed computationally on a set of problem instances of the literature and compared to the results reached by both Cplex solver and a recent reactive local search. For these instances, most of which cannot be solved to proven optimality in a reasonable runtime, the proposed method improves 21 out of 27.     

Bilevel multiplicative problems: A penalty approach to optimality and a cutting plane based algorithm

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterised by the existence of two optimisation problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimisation problem. In this paper we focus on the class of bilevel problems in which the upper level objective function is linear multiplicative, the lower level one is linear and the common constraint region is a bounded polyhedron. After replacing the lower level problem by its Karush–Kuhn–Tucker conditions, the existence of an extreme point which solves the problem is proved by using a penalty function approach. Besides, an algorithm based on the successive introduction of valid cutting planes is developed obtaining a global optimal solution. Finally, we generalise the problem by including upper level constraints which involve both level variables.     

Landscape analysis and efficient metaheuristics for solving the n-queens problem

The n-queens problem is a classical combinatorial optimization problem which has been proved to be NP-hard. The goal is to place n non-attacking queens on an n×n chessboard. In this paper, two single-solution-based (Local Search (LS) and Tuned Simulated Annealing (SA)) and two population-based metaheuristics (two versions of Scatter Search (SS)) are presented for solving the problem. Since parameters of heuristic and metaheuristic algorithms have great influence on their performance, a TOPSIS-Taguchi based parameter tuning method is proposed, which not only considers the number of fitness function evaluations, but also aims to minimize the runtime of the presented metaheuristics. The performance of the suggested approaches was investigated through computational analyses, which showed that the Local Search method outperformed the other two algorithms in terms of average runtimes and average number of fitness function evaluations. The LS was also compared to the Cooperative PSO (CPSO) and SA algorithms, which are currently the best algorithms in the literature for finding the first solution to the n-queens problem, and the results showed that the average fitness function evaluation of the LS is approximately 21 and 8 times less than that of SA and CPSO, respectively. Also, a fitness analysis of landscape for the n-queens problem was conducted which indicated that the distribution of local optima is uniformly random and scattered over the search space. The landscape is rugged and there is no significant correlation between fitness and distance of solutions, and so a local search heuristic can search a rugged plain landscape effectively and find a solution quickly. As a result, it was statistically and analytically proved that single-solution-based metaheuristics outperform population-based metaheuristics in finding the first solution of the n-queens problem.     

Stochastic local search for large-scale instances of the haplotype inference problem by pure parsimony

Haplotype Inference is a challenging problem in bioinformatics that consists in inferring the basic genetic constitution of diploid organisms on the basis of their genotype. This information allows researchers to perform association studies for the genetic variants involved in diseases and the individual responses to therapeutic agents. A notable approach to the problem is to encode it as a combinatorial problem (under certain hypotheses, such as the pure parsimony criterion) and to solve it using off-the-shelf combinatorial optimization techniques. The main methods applied to Haplotype Inference are either simple greedy heuristic or exact methods (Integer Linear Programming, Semidefinite Programming, SAT and pseudo-boolean encoding) that, at present, are adequate only for moderate size instances. In this paper, we present and discuss an approach based on the combination of local search metaheuristics and a reduction procedure based on an analysis of the problem structure. Some relevant design issues are first described, then a family of local search metaheuristics is defined to tackle the Haplotype Inference. Results on common Haplotype Inference benchmarks show that the approach achieves a good trade-off between solution quality and execution time.     

Design of balanced MBA student teams

In some schools and universities, students must sometimes be divided into several teams in such a way that each team provides a good representation of the classroom population. In this paper, two different ways of measuring the balance among teams are proposed: min-sum and min-max objective functions. For the first function and the L₁-norm used in the space of attributes, an exact solution method based on a set partitioning formulation and on the enumeration of all possible team patterns is presented. For the second objective function, a set partitioning formulation is also considered, but as an approximation. In order to solve large problem instances, we have also developed metaheuristics based on variable neighbourhood search. Models and methods are tested on data from an MBA programme.     

Fixed point optimization algorithm and its application to network bandwidth allocation

A convex optimization problem for a strictly convex objective function over the fixed point set of a nonexpansive mapping includes a network bandwidth allocation problem, which is one of the central issues in modern communication networks. We devised an iterative algorithm, called a fixed point optimization algorithm, for solving the convex optimization problem and conducted a convergence analysis on the algorithm. The analysis guarantees that the algorithm, with slowly diminishing step-size sequences, weakly converges to a unique solution to the problem. Moreover, we apply the proposed algorithm to a network bandwidth allocation problem and show its effectiveness.     

Order acceptance and scheduling with earliness and tardiness penalties

This paper addresses a production scheduling problem in a single-machine environment, where a job can be either early, on time, late, or rejected. In order acceptance and scheduling contexts, it is assumed that the production capacity of a company is overloaded. The problem is therefore to decide which orders to accept and how to sequence their production. In contrast with the existing literature, the considered problem jointly takes into account the following features: earliness and tardiness penalties (which can be linear or quadratic), sequence-dependent setup times and costs, rejection penalties, and the possibility of having idle times. The practical relevance of this new NP-hard problem is discussed and various solution methods are proposed, ranging from a basic greedy algorithm to refined metaheuristics (e.g., tabu search, the adaptive memory algorithm, population-based approaches loosely inspired on ant algorithms). The methods are compared for instances with various structures containing up to 200 jobs. For small linear instances, the metaheuristics are favorably compared with an exact formulation using CPLEX 12.2. Managerial insights and recommendations are finally given.     

Simulated annealing and tabu search for multi-mode resource-constrained project scheduling with positive discounted cash flows and different payment models

In this paper the multi-mode resource-constrained project scheduling problem with discounted cash flows is considered. A project is represented by an activity-on-node (AoN) network. A positive cash flow is associated with each activity. Four different payment models are considered: lump-sum payment at the completion of the project, payments at activities' completion times, payments at equal time intervals and progress payments. The objective is to maximize the net present value of all cash flows of the project. Local search metaheuristics: simulated annealing and tabu search are proposed to solve this strongly NP-hard problem. A comprehensive computational experiment is described, performed on a set of instances based on standard test problems constructed by the ProGen project generator, where, additionally, the activities' cash flows are generated randomly with the uniform distribution. The metaheuristics are computationally compared, the results are analyzed and discussed and some conclusions are given.     

A cut-peak function method for global optimization

A new method is proposed for solving box constrained global optimization problems. The basic idea of the method is described as follows: Constructing a so-called cut-peak function and a choice function for each present minimizer, the original problem of finding a global solution is converted into an auxiliary minimization problem of finding local minimizers of the choice function, whose objective function values are smaller than the previous ones. For a local minimum solution of auxiliary problems this procedure is repeated until no new minimizer with a smaller objective function value could be found for the last minimizer. Construction of auxiliary problems and choice of parameters are relatively simple, so the algorithm is relatively easy to implement, and the results of the numerical tests are satisfactory compared to other methods.     

Crossdocking— Just in Time scheduling: an alternative solution approach

In this paper, a scheduling problem which allows a warehouse to function as a crossdock where transit storage time for cargo is minimized according to Just in Time scheduling is studied. A model that uses the machine scheduling notation to describe the problem is written. As the problem is NP-hard, a solution approach based on a combination of two metaheuristics, Reactive GRASP and Tabu Search (RGTS), is provided. Experiments are carried out to determine the usefulness of this approach. The results obtained from the exact method that uses the ILOG CPLEX 9.1 solver for 16 problem instances and the results obtained from the RGTS metaheuristic scheduling algorithm and two other algorithms proposed by other authors for the same problem instances are discussed. Analysis and comparisons are made.     

Efficient sample sizes in stochastic nonlinear programming

We consider a class of stochastic nonlinear programs for which an approximation to a locally optimal solution is specified in terms of a fractional reduction of the initial cost error. We show that such an approximate solution can be found by approximately solving a sequence of sample average approximations. The key issue in this approach is the determination of the required sequence of sample average approximations as well as the number of iterations to be carried out on each sample average approximation in this sequence. We show that one can express this requirement as an idealized optimization problem whose cost function is the computing work required to obtain the required error reduction. The specification of this idealized optimization problem requires the exact knowledge of a few problems and algorithm parameters. Since the exact values of these parameters are not known, we use estimates, which can be updated as the computation progresses. We illustrate our approach using two numerical examples from structural engineering design.     

Competitive genetic algorithms for the open-shop scheduling problem

For more than two machines, and when preemption is forbidden, the computation of minimum makespan schedules for the open-shop problem is NP-hard. Compared to the flow-shop and the job-shop, the open-shop has free job routes which lead to a much larger solution space, to smaller gaps between the optimal makespan and the lower bounds, and to disappointing results for the algorithms based on the disjunctive graph model. For instance, the best existing branch and bound method cannot solve some 7 ×7 hard instances to optimality, and all published metaheuristics (working by reversing some disjunctions already fixed) do not better than some greedy or steepest-descent heuristics which need a much smaller computational effort. In this context, the intrinsic parallelism of genetic algorithms (GAs) seems well adapted, for detecting global optima disseminated among many quasi-optimal schedules. This paper presents several GAs for the open-shop problem. It is shown that even simple and fast versions can compete with the best known heuristics and metaheuristics, thanks to two key-features: a population in which each individual has a distinct makespan, and a special procedure which reorders every new chromosome. Using problem-specific heuristics, it is possible to design more powerful GAs which give excellent results, even on the hardest benchmarks of the literature: for instance, all hard open instances from Taillard are broken, except one for which the best known solution is improved.     

An efficient algorithm for large scale global optimization of continuous functions

A fast descent algorithm, resorting to a “stretching” function technique and built on one hybrid method (GRSA) which combines simulated annealing (SA) algorithm and gradient based methods for large scale global optimizations, is proposed. Unlike the previously proposed method in which the original objective functions remain unchanged during the whole course of optimization, the new method firstly constructs an auxiliary function on one local minimizer obtained by gradient based methods and then SA is executed on this constructed auxiliary function instead of on the original objective function in order that we can improve the jumping ability of SA algorithm to escape from the currently discovered local minimum to a better one from which the gradient based methods restart a new local search. The above procedure is repeated until a global minimum is detected. In addition, corresponding to the adopted “stretching” technique, a new next trial point generating scheme is designed. It is verified by simulation especially on large scale problems that the convergence speed is greatly accelerated, which is its main difference from many other reported methods that mostly cope with functions with less than 50 variables and does not apply to large scale optimization problems. Furthermore, the new algorithm functions as a global optimization procedure with a high success probability and high solution precision.     

Minimizing submodular functions over families of sets

We consider the problem of characterizing the minimum of a submodular function when the minimization is restricted to a family of subsets. We show that, for many interesting cases, there exist two elementsa andb of the groundset such that the problem is equivalent to the problem of minimizing the submodular function over the sets containinga but notb. This leads to a polynomial-time algorithm for minimizing a submodular function over these families of sets. Our results apply, for example, to the families of odd cardinality subsets or even cardinality subsets separating two given vertices, or to the complement of a lattice family of subsets. We also derive that the second smallest value of a submodular function over a lattice family can be computed in polynomial-time. These results generalize and unify several known results.Research partially supported by NSF contract 9302476-CCR, Air Force contract F49620-92-J-0125 and DARPA contract N00014-92-J-1799.     

A quadratic approximation method for minimizing a class of quasidifferentiable functions

Summary An unconstrained nonlinear programming problem with nondifferentiabilities is considered. The nondifferentiabilities arise from terms of the form max [f ₁(x), ...,f _n (x)], which may enter nonlinearly in the objective function. Local convex polyhedral upper approximations to the objective function are introduced. These approximations are used in an iterative method for solving the problem. The algorithm proceeds by solving quadratic programming subproblems to generate search directions. Approximate line searches ensure global convergence of the method to stationary points. The algorithm is conceptually simple and easy to implement. It generalizes efficient variable metric methods for minimax calculations.     

Design and analysis of hybrid metaheuristics for the Reliability p-Median Problem

In the p-Median Problem, it is assumed that, once the facilities are opened, they may not fail. In practice some of the facilities may become unavailable due to several factors. In the Reliability p-Median Problem some of the facilities may not be operative during certain periods. The objective now is to find facility locations that are both inexpensive and also reliable. We present different configurations of two hybrid metaheuristics to solve the problem, a genetic algorithm and a scatter search approach. We have carried out an extensive computational experiment to study the performance of the algorithms and compare its efficiency solving well-known benchmark instances.     

Global convergence of a modified Fletcher–Reeves conjugate gradient method with Armijo-type line search

In this paper, we are concerned with the conjugate gradient methods for solving unconstrained optimization problems. It is well-known that the direction generated by a conjugate gradient method may not be a descent direction of the objective function. In this paper, we take a little modification to the Fletcher–Reeves (FR) method such that the direction generated by the modified method provides a descent direction for the objective function. This property depends neither on the line search used, nor on the convexity of the objective function. Moreover, the modified method reduces to the standard FR method if line search is exact. Under mild conditions, we prove that the modified method with Armijo-type line search is globally convergent even if the objective function is nonconvex. We also present some numerical results to show the efficiency of the proposed method.Supported by the 973 project (2004CB719402) and the NSF foundation (10471036) of China.     

Hybrid multi-objective Bayesian estimation of distribution algorithm: a comparative analysis for the multi-objective knapsack problem

Nowadays, a number of metaheuristics have been developed for efficiently solving multi-objective optimization problems. Estimation of distribution algorithms are a special class of metaheuristic that intensively apply probabilistic modeling and, as well as local search methods, are widely used to make the search more efficient. In this paper, we apply a Hybrid Multi-objective Bayesian Estimation of Distribution Algorithm (HMOBEDA) in multi and many objective scenarios by modeling the joint probability of decision variables, objectives, and the configuration parameters of an embedded local search (LS). We analyze the benefits of the online configuration of LS parameters by comparing the proposed approach with LS off-line versions using instances of the multi-objective knapsack problem with two to five and eight objectives. HMOBEDA is also compared with five advanced evolutionary methods using the same instances. Results show that HMOBEDA outperforms the other approaches including those with off-line configuration. HMOBEDA not only provides the best value for hypervolume indicator and IGD metric in most of the cases, but it also computes a very diverse solutions set close to the estimated Pareto front.