Metaheuristics for multiobjective optimisation

This is a summary of the author’s PhD thesis supervised by Laetitia Jourdan and El-Ghazali Talbi and defended on 8 December 2009 at the Université Lille 1. The thesis is written in French and is available from . This work deals with the design, implementation and experimental analysis of metaheuristics for solving multiobjective optimisation problems, with a particular interest on hard and large combinatorial problems from the field of logistics. After focusing on a unified view of multiobjective metaheuristics, we propose new cooperative, adaptive and parallel approaches. The performance of these methods are experimented on a scheduling and a routing problem involving two or three objective functions. We finally discuss how to adapt such metaheuristics during the search process in order to handle uncertainty that may occur from many different sources.     

MultiGLODS: global and local multiobjective optimization using direct search

The optimization of multimodal functions is a challenging task, in particular when derivatives are not available for use. Recently, in a directional direct search framework, a clever multistart strategy was proposed for global derivative-free optimization of single objective functions. The goal of the current work is to generalize this approach to the computation of global Pareto fronts for multiobjective multimodal derivative-free optimization problems. The proposed algorithm alternates between initializing new searches, using a multistart strategy, and exploring promising subregions, resorting to directional direct search. Components of the objective function are not aggregated and new points are accepted using the concept of Pareto dominance. The initialized searches are not all conducted until the end, merging when they start to be close to each other. The convergence of the method is analyzed under the common assumptions of directional direct search. Numerical experiments show its ability to generate approximations to the different Pareto fronts of a given problem.     

The efficiency of indicator-based local search for multi-objective combinatorial optimisation problems

In the last few years, a significant number of multi-objective metaheuristics have been proposed in the literature in order to address real-world problems. Local search methods play a major role in many of these metaheuristic procedures. In this paper, we adapt a recent and popular indicator-based selection method proposed by Zitzler and Künzli in 2004, in order to define a population-based multi-objective local search. The proposed algorithm is designed in order to be easily adaptable, parameter independent and to have a high convergence rate. In order to evaluate the capacity of our algorithm to reach these goals, a large part of the paper is dedicated to experiments. Three combinatorial optimisation problems are tested: a flow shop problem, a ring star problem and a nurse scheduling problem. The experiments show that our algorithm can be applied with success to different types of multi-objective optimisation problems and that it outperforms some classical metaheuristics. Furthermore, the parameter sensitivity analysis enables us to provide some useful guidelines about how to set the parameters.     

A hybrid genetic algorithm for multiobjective problems with activity analysis-based local search

The objective of this research was the development of a method that integrated an activity analysis model of profits from production with a biophysical model, and included the capacity for optimization over multiple objectives. We specified a hybrid genetic algorithm using activity analysis as a local search method, and NSGA-II for calculation of the multiple objective Pareto optimal set. We describe a parallel computing approach to computation of the genetic algorithm, and apply the algorithm to evaluation of an input tax to regulate pollution from agricultural production.     

Survey and unification of local search techniques in metaheuristics for multi-objective combinatorial optimisation

Metaheuristics are algorithms that have proven their efficiency on multi-objective combinatorial optimisation problems. They often use local search techniques, either at their core or as intensification mechanisms, to obtain a well-converged and diversified final result. This paper surveys the use of local search techniques in multi-objective metaheuristics and proposes a general structure to describe and unify their underlying components. This structure can instantiate most of the multi-objective local search techniques and algorithms in literature.     

Global search perspectives for multiobjective optimization

Extending the notion of global search to multiobjective optimization is far than straightforward, mainly for the reason that one almost always has to deal with infinite Pareto optima and correspondingly infinite optimal values. Adopting Stephen Smale’s global analysis framework, we highlight the geometrical features of the set of Pareto optima and we are led to consistent notions of global convergence. We formulate then a multiobjective version of a celebrated result by Stephens and Baritompa, about the necessity of generating everywhere dense sample sequences, and describe a globally convergent algorithm in case the Lipschitz constant of the determinant of the Jacobian is known.     

Interactive multiobjective complex search

An interactive procedure based on Box's complex search is used to solve the vector maximization problem. This method has the advantage that the decision maker's underlying value function need not be explicitly specified. Also, the problem may have nonlinear objective functions and nonlinear constraints. Several example problems are presented.     

Hesitant adaptive search for global optimisation

Pure adaptive search is a stochastic algorithm which has been analysed in distinct ways for finite and continuous global optimisation. In this paper, motivated by the behaviour of practical algorithms such as simulated annealing, we extend these ideas. We present a unified theory which yields both the finite and continuous results for pure adaptive search. At the same time, we allow our extended algorithm to hesitate before improvement continues. Results are obtained for the expected number of iterations to convergence for such an algorithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

Scatter tabu search for multiobjective clustering problems

We propose a hybrid heuristic procedure based on scatter search and tabu search for the problem of clustering objects to optimize multiple criteria. Our goal is to search for good approximations of the efficient frontier for this class of problems and provide a means for improving decision making in multiple application areas. Our procedure can be viewed as an extension of SSPMO (a scatter search application to nonlinear multiobjective optimization) to which we add new elements and strategies specially suited for combinatorial optimization problems. Clustering problems have been the subject of numerous studies; however, most of the work has focused on single-objective problems. Clustering using multiple criteria and/or multiple data sources has received limited attention in the operational research literature. Our scatter tabu search implementation is general and tackles several problems classes within this area of combinatorial data analysis. We conduct extensive experimentation to show that our method is capable of delivering good approximations of the efficient frontier for improved analysis and decision making.     

Choquet-based optimisation in multiobjective shortest path and spanning tree problems

This paper is devoted to the search of Choquet-optimal solutions in finite graph problems with multiple objectives. The Choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives. We first present a condition on preferences (name hereafter preference for interior points) that characterizes preferences favouring compromise solutions, a natural attitude in various contexts such as multicriteria optimisation, robust optimisation and optimisation with multiple agents. Within Choquet expected utility theory, this condition amounts to using a submodular capacity and a convex utility function. Under these assumptions, we focus on the fast determination of Choquet-optimal paths and spanning trees. After investigating the complexity of these problems, we introduce a lower bound for the Choquet integral, computable in polynomial time. Then, we propose different algorithms using this bound, either based on a controlled enumeration of solutions (ranking approach) or an implicit enumeration scheme (branch and bound). Finally, we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes.     

Solving the musical orchestration problem using multiobjective constrained optimization with a genetic local search approach

In this paper a computational approach of musical orchestration is presented. We consider orchestration as the search of relevant sound combinations within large instruments sample databases and propose two cooperating metaheuristics to solve this problem. Orchestration is seen here as a particular case of finding optimal constrained multisets on a large ensemble with respect to several objectives. We suggest a generic and easily extendible formalization of orchestration as a constrained multiobjective search towards a target timbre, in which several perceptual dimensions are jointly optimized. We introduce Orchidée, a time-efficient evolutionary orchestration algorithm that allows the discovery of optimal solutions and favors the exploration of non-intuitive sound mixtures. We also define a formal framework for global constraints specification and introduce the innovative CDCSolver repair metaheuristic, thanks to which the search is led towards regions fulfilling a set of musical-related requirements. Evaluation of our approach on a wide set of real orchestration problems is also provided.     

Mechanisms for local search

Local search methods for combinatorial optimization make a series of steps, at each stage improving the current solution by moving to a neighbouring solution. This is usually done by considering the neighbouring solutions one at a time and moving to the first one which gives an improvement (a first-improving method). In this paper we consider whether there are circumstances in which some other strategy will have better performance. In exploring this question we begin by giving a theoretical treatment of a simple model with random objective values at each solution point. We carry out some experiments on the Travelling Salesman Problem and the Quadratic Assignment Problem using varying values of a spread parameter, k. The value of k corresponds to the number of improving solutions looked at before making a move. We also make some conjectures relating the overall performance of the local search method to the average number of solutions which are evaluated before a local minimum is reached.     

On dominance-based multiobjective local search: design, implementation and experimental analysis on scheduling and traveling salesman problems

This paper discusses simple local search approaches for approximating the efficient set of multiobjective combinatorial optimization problems. We focus on algorithms defined by a neighborhood structure and a dominance relation that iteratively improve an archive of nondominated solutions. Such methods are referred to as dominance-based multiobjective local search. We first provide a concise overview of existing algorithms, and we propose a model trying to unify them through a fine-grained decomposition. The main problem-independent search components of dominance relation, solution selection, neighborhood exploration and archiving are largely discussed. Then, a number of state-of-the-art and original strategies are experimented on solving a permutation flowshop scheduling problem and a traveling salesman problem, both on a two- and a three-objective formulation. Experimental results and a statistical comparison are reported in the paper, and some directions for future research are highlighted.     

Multiple response optimisation: An approach from multiobjective stochastic programming

The multiresponse surface problem is modelled as a multiobjective stochastic optimisation, and diverse solutions are proposed. There are several crucial differences highlighted between this approach and the other proposed solutions. Finally, some particular solutions are applied and described in detail in a numerical example.     

GPU implementation of a multiobjective search algorithm

In this paper we observe the possibility to accelerate a search algorithm for multiobjective optimization problems with help of a graphics processing unit. Besides an implementation we present test results for it and the conclusions that can be drawn from these results.     

A united search particle swarm optimization algorithm for multiobjective scheduling problem

The performance of a scheduling system, in practice, is not evaluated to satisfy a single objective, but to obtain a trade-off schedule regarding multiple objectives. Therefore, in this research, I make use of multiple objective decision-making method, a global criterion approach, to develop a multi-objective scheduling problem model with different due-dates on parallel machines processes, in which consider three performance measures, namely minimum run time of every machine, earlierness time (no tardiness) and process time of every job, simultaneously. According to this special multi-objective scheduling problem, the method of reverse order drawing GATT will be proposed, at the same time, bring forward a united search particle swarm optimization algorithm (USPSOA) solves this multi-objective scheduling problem. The validity and adaptability of the USPSOA is investigated through experimental results.     

Preference modelling by estimating local utility functions for multiobjective optimization

This paper is intended to design goal programming models for capturing the decision maker's (DM's) preference information and for supporting the search for the best compromise solutions in multiobjective optimization. At first, a linear goal programming model is built to estimate piecewise linear local utility functions based on pairwise comparisons of efficient solutions as well as objectives. The interactive step trade-off method (ISTM) is employed to generate a typical subset of efficient solutions of a multiobjective problem. Another general goal programming model is then constructed to embed the estimated utility functions in the original multiobjective problem for utility optimization using ordinary nonlinear programming algorithms. This technique, consisting of the ISTM method and the newly investigated search process, facilitates the identification and elimination of possible inconsistent information which may exist in the DM's preferences. It also provides various ways to carry out post-optimality analysis to test the robustness of the obtained best solutions. A modified nonlinear multiobjective management problem is taken as example to demonstrate the technique.     

Smoothed performance guarantees for local search

We study popular local search and greedy algorithms for standard machine scheduling problems. The performance guarantee of these algorithms is well understood, but the worst-case lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications. To find out how robust these bounds are, we study the algorithms in the framework of smoothed analysis, in which instances are subject to some degree of random noise. While the lower bounds for all scheduling variants with restricted machines are rather robust, we find out that the bounds are fragile for unrestricted machines. In particular, we show that the smoothed performance guarantee of the jump and the lex-jump algorithm are (in contrast to the worst case) independent of the number of machines. They are  \(\Theta (\phi )\) and  \(\Theta (\log \phi )\) , respectively, where  \(1/\phi \) is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is  \(\Theta (\log \phi )\) for routing games on parallel links. Additionally, we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of   \(\Theta (\log \phi )\) .     

On parallel local search for permutations

We investigate some ways in which massively parallel computing devices can be exploited in local search algorithms. We show that the substantial speedups that can be gained from parallel neighbourhood evaluation enables an efficient best improvement local search, and that this in turn enables further speedups through selection and parallel application of a set of independent, improving moves. Our experiments demonstrate a total speedup of up to several hundred times compared to a classical, sequential best improvement search. We also demonstrate how an exchange of good partial solutions between the incumbent and best found solutions improves the efficiency of the Iterated Local Search algorithm.     

Parallel local search

We present a survey of parallel local search algorithms in which we review the concepts that can be used to incorporate parallelism into local search. For this purpose we distinguish between single-walk and multiple-walk parallel local search and between asynchronous and synchronous parallelism. Within the class of single-walk algorithms we differentiate between multiple-step and single-step parallelism. To describe parallel local search we introduce the concepts of hyper neighborhood structures and distributed neighborhood structures. Furthermore, we present templates that capture most of the parallel local search algorithms proposed in the literature. Finally, we discuss some complexity issues related to parallel local search.