Comparing local search metaheuristics for the maximum diversity problem

The Maximum Diversity Problem (MDP) requires to extract a subset M of given cardinality from a set N, maximising the sum of the pair-wise diversities between the extracted elements. The MDP has recently been the subject of much research, and several sophisticated heuristics have been proposed to solve it. The present work compares four local search metaheuristics for the MDP, all based on the same Tabu Search procedure, with the aim to identify what additional elements provide the strongest improvement. The four metaheuristics are an Exploring Tabu Search, a Scatter Search, a Variable Neighbourhood Search and a simple Random Restart algorithm. All of them prove competitive with the best algorithms proposed in the literature. Quite surprisingly, the best ones are the simple Random Restart algorithm and a Variable Neighbourhood Search algorithm with an unusual parameter setting, which makes it quite close to random restart. Although this is probably related to the elementary structure of the MDP, it also suggests that, more often than expected, simpler algorithms might be better.     

Financial Applications of a Tabu Search Variable Selection Model

We illustrate how a comparatively new technique, a Tabu search variable selection model [Drezner, Marcoulides and Salhi (1999)], can be applied efficiently within finance when the researcher must select a subset of variables from among the whole set of explanatory variables under consideration. Several types of problems in finance, including corporate and personal bankruptcy prediction, mortgage and credit scoring, and the selection of variables for the Arbitrage Pricing Model, require the researcher to select a subset of variables from a larger set. In order to demonstrate the usefulness of the Tabu search variable selection model, we: (1) illustrate its efficiency in comparison to the main alternative search procedures, such as stepwise regression and the Maximum R 2 procedure, and (2) show how a version of the Tabu search procedure may be implemented when attempting to predict corporate bankruptcy. We accomplish (2) by indicating that a Tabu Search procedure increases the predictability of corporate bankruptcy by up to 10 percentage points in comparison to Altman's (1968) Z-Score model.     

Benders's method revisited

The master problem in Benders's partitioning method is an integer program with a very large number of constraints, each of which is usually generated by solving the integer program with the constraints generated earlier. Computational experience shows that the subset B of those constraints of the master problem that are satisfied with equality at the linear programming optimum often play a crucial role in determining the integer optimum, in the sense that only a few of the remaining inequalities are needed. We characterize this subset B of inequalities. If an optimal basic solution to the linear program is nondegenerate in the continuous variables and has p integer-constrained basic variables, then the corresponding set B contains at most 2^p inequalities, none of which is implied by the others. We give an efficient procedure for generating an appropriate subset of the inequalities in B, which leads to a considerably improved version of Benders's method.     

Minimizing variable selection criteria by Markov chain Monte Carlo

Regression models with a large number of predictors arise in diverse fields of social sciences and natural sciences. For proper interpretation, we often would like to identify a smaller subset of the variables that shows the strongest information. In such a large size of candidate predictors setting, one would encounter a computationally cumbersome search in practice by optimizing some criteria for selecting variables, such as AIC, \(C_{P}\) and BIC, through all possible subsets. In this paper, we present two efficient optimization algorithms vis Markov chain Monte Carlo (MCMC) approach for searching the global optimal subset. Simulated examples as well as one real data set exhibit that our proposed MCMC algorithms did find better solutions than other popular search methods in terms of minimizing a given criterion.     

Solving the p-median problem with pos/neg weights by variable neighborhood search and some results for special cases

The p-median problem with positive and negative weights has been introduced by Burkard and Krarup [Computing 60 (1998) 193]. In this paper we discuss some special cases of this problem on trees and propose a variable neighborhood search procedure for general networks, which is in fact a modification of the one proposed by Hansen and Mladenovic [Locat. Sci. 5 (1997) 207] for the p-median. We also compare the results with those obtained by a Tabu search procedure.     

A hybrid Granular Tabu Search algorithm for the Multi-Depot Vehicle Routing Problem

In this paper, we propose a hybrid Granular Tabu Search algorithm to solve the Multi-Depot Vehicle Routing Problem (MDVRP). We are given on input a set of identical vehicles (each having a capacity and a maximum duration), a set of depots, and a set of customers with deterministic demands and service times. The problem consists of determining the routes to be performed to fulfill the demand of the customers by satisfying, for each route, the associated capacity and maximum duration constraints. The objective is to minimize the sum of the traveling costs related to the performed routes. The proposed algorithm is based on a heuristic framework previously introduced by the authors for the solution of the Capacitated Location Routing Problem (CLRP). The algorithm applies a hybrid Granular Tabu Search procedure, which considers different neighborhoods and diversification strategies, to improve the initial solution obtained by a hybrid procedure. Computational experiments on benchmark instances from the literature show that the proposed algorithm is able to produce, within short computing time, several best solutions obtained by the previously published methods and new best solutions.     

Cost optimization of a repairable M/G/1 queue with a randomized policy and single vacation

This paper deals with the (p, N)-policy M/G/1 queue with an unreliable server and single vacation. Immediately after all of the customers in the system are served, the server takes single vacation. As soon as N customers are accumulated in the queue, the server is activated for services with probability p or deactivated with probability (1 − p). When the server returns from vacation and the system size exceeds N, the server begins serving the waiting customers. If the number of customers waiting in the queue is less than N when the server returns from vacation, he waits in the system until the system size reaches or exceeds N. It is assumed that the server is subject to break down according to a Poisson process and the repair time obeys a general distribution. This paper derived the system size distribution for the system described above at a stationary point of time. Various system characteristics were also developed. We then constructed a total expected cost function per unit time and applied the Tabu search method to find the minimum cost. Some numerical results are also given for illustrative purposes.     

Nordsieck methods with computationally verified algebraic stability

We describe the search for algebraically stable Nordsieck methods of order p = s and stage order q = p, where s is the number of stages. This search is based on the theoretical criteria for algebraic stability proposed recently by Hill, and Hewitt and Hill, for general linear methods for ordinary differential equations. These criteria, which are expressed in terms of the non-negativity of the eigenvalues of a Hermitian matrix on the unit circle, are then verified computationally for the derived Nordsieck methods of order p ? 2.     

A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem

Over the last decade, many metaheuristics have been applied to the flowshop scheduling problem, ranging from Simulated Annealing or Tabu Search to complex hybrid techniques. Some of these methods provide excellent effectiveness and efficiency at the expense of being utterly complicated. In fact, several published methods require substantial implementation efforts, exploit problem specific speed-up techniques that cannot be applied to slight variations of the original problem, and often re-implementations of these methods by other researchers produce results that are quite different from the original ones. In this work we present a new iterated greedy algorithm that applies two phases iteratively, named destruction, were some jobs are eliminated from the incumbent solution, and construction, where the eliminated jobs are reinserted into the sequence using the well known NEH construction heuristic. Optionally, a local search can be applied after the construction phase. Our iterated greedy algorithm is both very simple to implement and, as shown by experimental results, highly effective when compared to state-of-the-art methods.     

Density Based Problem Space Search for the Capacitated Clustering p-Median Problem

In the Capacitated Clustering Problem (CCP), a given set of n weighted points is to be partitioned into p clusters such that, the total weight of the points in each cluster does not exceed a given cluster capacity. The objective is to find a set of p centers that minimises total scatter of points allocated to them. In this paper a new constructive method, a general framework to improve the performance of greedy constructive heuristics, and a problem space search procedure for the CCP are proposed. The constructive heuristic finds patterns of natural subgrouping in the input data using concept of density of points. Elements of adaptive computation and periodic construction–deconstruction concepts are implemented within the constructive heuristic to develop a general framework for building efficient heuristics. The problem-space search procedure is based on perturbations of input data for which a controlled perturbation strategy, intensification and diversification strategies are developed. The implemented algorithms are compared with existing methods on a standard set of bench-marks and on new sets of large-sized instances. The results illustrate the strengths of our algorithms in terms of solution quality and computational efficiency.     

Mathematical programming approaches for generating p-efficient points

Probabilistically constrained problems, in which the random variables are finitely distributed, are non-convex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we consider a random vector characterized by a finite set of scenarios and generate pLEPs by solving a mixed-integer programming (MIP) problem. We solve this computationally challenging MIP problem with a new mathematical programming framework. It involves solving a series of increasingly tighter outer approximations and employs, as algorithmic techniques, a bundle preprocessing method, strengthening valid inequalities, and a fixing strategy. The method is exact (resp., heuristic) and ensures the generation of pLEPs (resp., quasi pLEPs) if the fixing strategy is not (resp., is) employed, and it can be used to generate multiple pLEPs. To the best of our knowledge, generating a set of pLEPs using an optimization-based approach and developing effective methods for the application of the p-efficiency concept to the random variables described by a finite set of scenarios are novel. We present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques.     

A hybrid method combining continuous tabu search and Nelder–Mead simplex algorithms for the global optimization of multiminima functions

Tabu search (TS) is a metaheuristic, which proved efficient to solve various combinatorial optimization problems. However, few works deal with its application to the global minimization of functions depending on continuous variables. To perform this task, we propose an hybrid method combining tabu search and simplex search (SS). TS allows to cover widely the solution space, to stimulate the search towards solutions far from the current solution, and to avoid the risk of trapping into a local minimum. SS is used to accelerate the convergence towards a minimum. The Nelder–Mead simplex algorithm is a classical very powerful local descent algorithm, making no use of the objective function derivatives. A “simplex” is a geometrical figure consisting, in n-dimensions, of (n+1) points. If any point of a simplex is taken as the origin, the n other points define vector directions that span the n-dimension vector space. Through a sequence of elementary geometric transformations (reflection, contraction and extension), the initial simplex moves, expands or contracts. To select the appropriate transformation, the method only uses the values of the function to be optimized at the vertices of the simplex considered. After each transformation, the current worst vertex is replaced by a better one. Our algorithm called continuous tabu simplex search (CTSS) implemented in two different forms (CTSS_single, CTSS_multiple) is made up of two steps: first, an adaptation of TS to continuous optimization problems, allowing to localize a “promising area”; then, intensification within this promising area, involving SS. The efficiency of CTSS is extensively tested by using analytical test functions of which global and local minima are known. A comparison is proposed with several variants of tabu search, genetic algorithms and simulated annealing. CTSS is applied to the design of a eddy current sensor aimed at non-destructive control.     

Fast local search for the maximum independent set problem

Given a graph G=(V,E), the independent set problem is that of finding a maximum-cardinality subset S of V such that no two vertices in S are adjacent. We introduce two fast local search routines for this problem. The first can determine in linear time whether a maximal solution can be improved by replacing a single vertex with two others. The second routine can determine in O(mΔ) time (where Δ is the highest degree in the graph) whether there are two solution vertices than can be replaced by a set of three. We also present a more elaborate heuristic that successfully applies local search to find near-optimum solutions to a wide variety of instances. We test our algorithms on instances from the literature as well as on new ones proposed in this paper.     

Nonparametric tests of independence between random vectors

A nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Möbius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. If the number of vectors is p and there are n observations, the test is defined from a collection of processes R_n,A, where A is a subset of {1,…,p} of cardinality |A|>1, which are asymptotically independent and Gaussian. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed with the bootstrap which is shown to be consistent. Another similar test, with the same asymptotic properties, for the serial independence of a multivariate stationary sequence is also proposed. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure. Moreover, in singular cases described in Section 4, the test inherits useful invariance properties from the general affine invariance property.     

Guided construction search metaheuristics for the capacitated p-median problem with single source constraint

In the capacitated p-median problem with single source constraint, also known as the capacitated clustering problem, a given set of n weighted points is to be partitioned into p clusters such that the total weight of the points in each cluster does not exceed a given cluster capacity. The objective is to find a set of p centres that minimizes the total scatter of points allocated to these clusters. In this paper, a (λ, μ)-interchange neighbourhood based on the concept of λ-interchange of points restricted to μ-adjacent clusters is proposed. Structural properties of centres are identified and exploited to derive special data structures for their efficient evaluations. Different search and selection strategies including the variable neighbourhood search descent with respect to μ-nearest points are investigated. The most efficient strategies are then embedded in a guided construction search metaheuristic framework based either on a periodic local search procedure or a greedy random adaptive search procedure to solve the problem. Computational experience is reported on a standard set of benchmarks. The computational experience demonstrates the competitive performance of the proposed algorithms when compared to the best-known procedures in the literature in terms of solution quality and computational requirement.     

Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods.     

Integrating heuristic information into exact methods: The case of the vertex p-centre problem

We solve the vertex p-centre problem optimally using an exact method that considers both upper and lower bounds as part of its search engine. Tight upper bounds are generated quickly via an efficient three-level heuristic, which are then used to derive potential ‘lower bounds’ accordingly. These two pieces of information when used together make our chosen exact method more efficient at obtaining optimal solutions relatively quickly. The proposed implementation produced excellent results when tested on the OR Library data set. This integrated approach can be adopted for those exact methods that consider both upper and lower bounds within their search engine and hence provide a wider spectrum of applicability in other hard combinatorial problems.     

Global optimization algorithm for a generalized linear multiplicative programming

This article present a branch and bound algorithm for globally solving generalized linear multiplicative programming problems with coefficients. The main computation involve solving a sequence of linear relaxation programming problems, and the algorithm economizes the required computations by conducting the branch and bound search in R ^p , rather than in R ⁿ , where p is the number of rank and n is the dimension of decision variables. The proposed algorithm will be convergent to the global optimal solution by means of the subsequent solutions of the series of linear relaxation programming problems. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.     

Greedy random adaptive memory programming search for the capacitated clustering problem

In the capacitated clustering problem (CCP), a given set of n weighted points is to be partitioned into p clusters such that, the total weight of the points in each cluster does not exceed a given cluster capacity. The objective is to find a set of p centers that minimises the total scatter of points allocated to these centers. In this paper, we propose a merger of Greedy Random Adaptive Search Procedure (GRASP) and Adaptive Memory Programming (AMP) into a new GRAMPS framework for the CCP. A learning process is kept in charge of tracking information on the best components in an elite set of GRAMPS solutions. The information are strategically combined with problem-domain data to restart the construction search phase. At early stage of constructions, priorities are given to problem-domain data and progressively shifted towards generated information as the learning increases. GRAMPS is implemented with an efficient local search descent based on a restricted λ-interchange neighbourhood. Extensive experiments are reported on on a standard set of bench-marks from the literature and on a new set of large instances. The results show that GRAMPS has an efficient learning mechanism and is competitive with the existing methods in the literature.