A new dominance procedure for combinatorial optimization problems

It is known that the effectiveness of the branch and bound algorithms for combinatorial optimization problems can be improved through dominance criteria which allow fathomings of large solution subsets. We describe a new dominance procedure which overcomes some of the drawbacks of the commonly used dominance criteria. An application to the Multiple Knapsack Problem and some computational results are also reported.     

Multi-start methods for combinatorial optimization

Multi-start methods strategically sample the solution space of an optimization problem. The most successful of these methods have two phases that are alternated for a certain number of global iterations. The first phase generates a solution and the second seeks to improve the outcome. Each global iteration produces a solution that is typically a local optimum, and the best overall solution is the output of the algorithm. The interaction between the two phases creates a balance between search diversification (structural variation) and search intensification (improvement), to yield an effective means for generating high-quality solutions. This survey briefly sketches historical developments that have motivated the field, and then focuses on modern contributions that define the current state-of-the-art. We consider two categories of multi-start methods: memory-based and memoryless procedures. The former are based on identifying and recording specific types of information (attributes) to exploit in future constructions. The latter are based on order statistics of sampling and generate unconnected solutions. An interplay between the features of these two categories provides an inviting area for future exploration.     

A proposal for a hybrid meta-strategy for combinatorial optimization problems

In this paper, we propose an algorithm named BDS (Bound-Driven Search) that combines features of exact and approximate methods. The proposed procedure may be seen as a local search algorithm that systematically explores (in a branch-and bound sense) the most promising nodes, thus preventing solutions from being reevaluated. Additionally, it can be regarded as an exact method as it may be able to guarantee that the solution found is optimal. We present the application of this new algorithm to a specific problem domain: the permutation flow shop scheduling problem with makespan objective. The subsequent computational experiments are encouraging, as the algorithm is able to yield exact or near exact solutions to most instances of the problem. Furthermore, the algorithm outperforms one of the best state-of-the-art algorithms for the problem.     

Bulk-Robust combinatorial optimization

Mathematical Programming - We commence an algorithmic study of Bulk-Robustness, a new model of robustness in combinatorial optimization. Unlike most existing models, Bulk-Robust combinatorial...     

Metaheuristic algorithms for combinatorial optimization problems

We present the main results in the author’s Ph.D. thesis (Iori 2004), defended at the University of Bologna in April 2004 and supervised by S. Martello. The thesis is written in English and is available from the author upon request. It proposes exact and metaheuristic algorithms for solving some relevant combinatorial optimization problems, with particular emphasis on scheduling, two-dimensional cutting and packing and capacitated vehicle routing. The performance of each algorithm is tested through extensive computational experiments and comparison with other approaches in the literature.Received: 21 September 2004, AMS classification: 90-08, 90C27, 90C59     

A comparative study of decomposition algorithms for stochastic combinatorial optimization

This paper presents comparative computational results using three decomposition algorithms on a battery of instances drawn from two different applications. In order to preserve the commonalities among the algorithms in our experiments, we have designed a testbed which is used to study instances arising in server location under uncertainty and strategic supply chain planning under uncertainty. Insights related to alternative implementation issues leading to more efficient implementations, benchmarks for serial processing, and scalability of the methods are also presented. The computational experience demonstrates the promising potential of the disjunctive decomposition (D ²) approach towards solving several large-scale problem instances from the two application areas. Furthermore, the study shows that convergence of the D ² methods for stochastic combinatorial optimization (SCO) is in fact attainable since the methods scale well with the number of scenarios.     

Inverse multi-objective combinatorial optimization

Inverse multi-objective combinatorial optimization consists of finding a minimal adjustment of the objective functions coefficients such that a given set of feasible solutions becomes efficient. An algorithm is proposed for rendering a given feasible solution into an efficient one. This is a simplified version of the inverse problem when the cardinality of the set is equal to one. The adjustment is measured by the Chebyshev distance. It is shown how to build an optimal adjustment in linear time based on this distance, and why it is right to perform a binary search for determining the optimal distance. These results led us to develop an approach based on the resolution of mixed-integer linear programs. A second approach based on a branch-and-bound is proposed to handle any distance function that can be linearized. Finally, the initial inverse problem is solved by a cutting plane algorithm.     

Approximative solution methods for multiobjective combinatorial optimization

In this paper we present a review of approximative solution methods, that is, heuristics and metaheuristics designed for the solution of multiobjective combinatorial optimization problems (MOCO). First, we discuss questions related to approximation in this context, such as performance ratios, bounds, and quality measures. We give some examples of heuristics proposed for the solution of MOCO problems. The main part of the paper covers metaheuristics and more precisely non-evolutionary methods. The pioneering methods and their derivatives are described in a unified way. We provide an algorithmic presentation of each of the methods together with examples of applications, extensions, and a bibliographic note. Finally, we outline trends in this area. The research of M. Ehrgott has been partially supported by University of Auckland grant 3602178/9275 and grant Ka 477/27-1 of the Deutsche Forschungsgemeinschaft (DFG).     

Mixed uncertainty sets for robust combinatorial optimization

In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support. In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a “true” model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-of-sample performance, the combination of multiple sets can give better results than each set on its own.

     

Generating hard instances for robust combinatorial optimization

While research in robust optimization has attracted considerable interest over the last decades, its algorithmic development has been hindered by several factors. One of them is a missing set of benchmark instances that make algorithm performance better comparable, and makes reproducing instances unnecessary. Such a benchmark set should contain hard instances in particular, but so far, the standard approach to produce instances has been to sample values randomly from a uniform distribution.In this paper we introduce a new method to produce hard instances for min-max combinatorial optimization problems, which is based on an optimization model itself. Our approach does not make any assumptions on the problem structure and can thus be applied to any combinatorial problem. Using the Selection and Traveling Salesman problems as examples, we show that it is possible to produce instances which are up to 500 times harder to solve for a mixed-integer programming solver than the current state-of-the-art instances.     

Multi-armed bandit-based hyper-heuristics for combinatorial optimization problems

There are significant research opportunities in the integration of Machine Learning (ML) methods and Combinatorial Optimization Problems (COPs). In this work, we focus on metaheuristics to solve COPs that have an important learning component. These algorithms must explore a solution space and learn from the information they obtain in order to find high-quality solutions. Among the metaheuristics, we study Hyper-Heuristics (HHs), algorithms that, given a number of low-level heuristics, iteratively select and apply heuristics to a solution. The HH we consider has a Markov model to produce sequences of low-level heuristics, which we combine with a Multi-Armed Bandit Problem (MAB)-based method to learn its parameters. This work proposes several improvements to the HH metaheuristic that yields a better learning for solving problem instances. Specifically, this is the first work in HHs to present Exponential Weights for Exploration and Exploitation (EXP3) as a learning method, an algorithm that is able to deal with adversarial settings. We also present a case study for the Vehicle Routing Problem with Time Windows (VRPTW), for which we include a list of low-level heuristics that have been proposed in the literature. We show that our algorithms can handle a large and diverse list of heuristics, illustrating that they can be easily configured to solve COPs of different nature. The computational results indicate that our algorithms are competitive methods for the VRPTW (2.16% gap on average with respect to the best known solutions), demonstrating the potential of these algorithms to solve COPs. Finally, we show how algorithms can even detect low-level heuristics that do not contribute to finding better solutions to the problem.     

Genetic local search for multi-objective combinatorial optimization

The paper presents a new genetic local search (GLS) algorithm for multi-objective combinatorial optimization (MOCO). The goal of the algorithm is to generate in a short time a set of approximately efficient solutions that will allow the decision maker to choose a good compromise solution. In each iteration, the algorithm draws at random a utility function and constructs a temporary population composed of a number of best solutions among the prior generated solutions. Then, a pair of solutions selected at random from the temporary population is recombined. Local search procedure is applied to each offspring. Results of the presented experiment indicate that the algorithm outperforms other multi-objective methods based on GLS and a Pareto ranking-based multi-objective genetic algorithm (GA) on travelling salesperson problem (TSP).     

Parallel processing for difficult combinatorial optimization problems

In identifying the general algorithmic problems most frequently encountered in designing and analyzing parallel algorithms (compatibility with machine architecture, choice of suitable shared or distributed data structures, compromise between communication and processing, and load balancing), we present recent research which has been carried out into parallelization of exact search methods such as Branch and Bound. We cover the main problems encountered with such a parallelization and present some theoretical and practical achievements in this field. The parallelization of heuristic search methods is shown to raise the same kind of issues.     

Exponential irreducible neighborhoods for combinatorial optimization problems

This paper deals with irreducible augmentation vectors associated with three combinatorial optimization problems: the TSP, the ATSP, and the SOP. We study families of irreducible vectors of exponential size, derived from alternating cycles, where optimizing a linear function over each of these families can be done in polynomial time. A family of irreducible vectors induces an irreducible neighborhood; an algorithm for optimizing over this family is known as a local search heuristic. Irreducible neighborhoods do not only serve as a tool to improve feasible solutions, but do play an important role in an exact primal algorithm; such families are the primal counterpart of a families of facet inducing inequalities.     

A unified pre-training and adaptation framework for combinatorial optimization on graphs

Combinatorial optimization(CO) on graphs is a classic topic that has been extensively studied across many scientific and industrial fields. Recently, solving CO problems on graphs through learning methods has attracted great attention. Advanced deep learning methods, e.g., graph neural networks(GNNs), have been used to effectively assist the process of solving COs. However, current frameworks based on GNNs are mainly designed for certain CO problems, thereby failing to consider their transferabl...     

A note on the primal-dual column generation method for combinatorial optimization

In this paper, we address the primal-dual column generation technique, which relies on well-centred suboptimal solutions of the restricted master problems. We summarize new theoretical developments and present computational results for two classical combinatorial optimization problems, in which this technique has not been tested before. The results show that the primal-dual column generation technique usually leads to substantial reductions in the number of iterations and CPU time when compared to two other well-established approaches: the classical column generation technique and the analytic centre cutting plane method.