Joshua D. Knowles

Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy

We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its simplest form, is a (11) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. (11)-PAES is intended to be a baseline approach against which more involved methods may be compared. It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. We introduce (1) and () variants of PAES as extensions to the basic algorithm. Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions. Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions. This allows standard statistical analysis to be carried out for comparative purposes. Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks.

The Pareto archived evolution strategy: a new baseline algorithm for Pareto multiobjective optimisation

Most popular evolutionary algorithms for multiobjective optimisation maintain a population of solutions from which individuals are selected for reproduction. In this paper, we introduce a simpler evolution scheme for multiobjective problems, called the Pareto archived evolution strategy (PAES). We argue that PAES may represent the simplest possible non-trivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm is identified as being a (1+1) evolution strategy, using local search from a population of one but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. PAES is intended as a good baseline approach, against which more involved methods may be compared, and may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. The performance of the new algorithm is compared with that of a MOEA based on the niched Pareto GA on a real world application from the telecommunications field. In addition, we include results from experiments carried out on a suite of four test functions, to demonstrate the algorithms general capability.

The Pareto Envelope-Based Selection Algorithm for Multi-objective Optimisation

We introduce a new multiobjective evolutionary algorithm called PESA (the Pareto Envelope-based Selection Algorithm), in which selection and diversity maintenance are controlled via a simple hyper-grid based scheme. PESAs selection method is relatively unusual in comparison with current well known multiobjective evolutionary algorithms, which tend to use counts based on the degree to which solutions dominate others in the population. The diversity maintenance method is similar to that used by certain other methods. The main attraction of PESA is the integration of selection and diversity maintenance, whereby essentially the same technique is used for both tasks. The resulting algorithm is simple to describe, with full pseudocode provided here and real code available from the authors. We compare PESA with two recent strong-performing MOEAs on some multiobjective test problems recently proposed by Deb. We find that PESA emerges as the best method overall on these problems.

An Evolutionary Approach to Multiobjective Clustering

The framework of multiobjective optimization is used to tackle the unsupervised learning problem, data clustering, following a formulation first proposed in the statistics literature. The conceptual advantages of the multiobjective formulation are discussed and an evolutionary approach to the problem is developed. The resulting algorithm, multiobjective clustering with automatic k-determination, is compared with a number of well-established single-objective clustering algorithms, a modern ensemble technique, and two methods of model selection. The experiments demonstrate that the conceptual advantages of multiobjective clustering translate into practical and scalable performance benefits

ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems

This paper concerns multiobjective optimization in scenarios where each solution evaluation is financially and/or temporally expensive. We make use of nine relatively low-dimensional, nonpathological, real-valued functions, such as arise in many applications, and assess the performance of two algorithms after just 100 and 250 (or 260) function evaluations. The results show that NSGA-II, a popular multiobjective evolutionary algorithm, performs well compared with random search, even within the restricted number of evaluations used. A significantly better performance (particularly, in the worst case) is, however, achieved on our test set by an algorithm proposed herein-ParEGO-which is an extension of the single-objective efficient global optimization (EGO) algorithm of Jones et al. ParEGO uses a design-of-experiments inspired initialization procedure and learns a Gaussian processes model of the search landscape, which is updated after every function evaluation. Overall, ParEGO exhibits a promising performance for multiobjective optimization problems where evaluations are expensive or otherwise restricted in number.

M-PAES: a memetic algorithm for multiobjective optimization

A memetic algorithm for tackling multiobjective optimization problems is presented. The algorithm employs the proven local search strategy used in the Pareto archived evolution strategy (PAES) and combines it with the use of a population and recombination. Verification of the new M-PAES (memetic PAES) algorithm is carried out by testing it on a set of multiobjective 0/1 knapsack problems. On each problem instance, a comparison is made between the new memetic algorithm, the (1+1)-PAES local searcher, and the strength Pareto evolutionary algorithm (SPEA) of E. Zitzler and L. Thiele (1998, 1999).

A MAX-MIN Ant System for the University Course Timetabling Problem

We consider a simplification of a typical university course timetabling problem involving three types of hard and three types of soft constraints. A MAX-MIN Ant System, which makes use of a separate local search routine, is proposed for tackling this problem. We devise an appropriate construction graph and pheromone matrix representation after considering alternatives. The resulting algorithm is tested over a set of eleven instances from three classes of the problem. The results demonstrate that the ant system is able to construct significantly better timetables than an algorithm that iterates the local search procedure from random starting solutions.

Ant-Based Clustering and Topographic Mapping

Ant-based clustering and sorting is a nature-inspired heuristic first introduced as a model for explaining two types of emergent behavior observed in real ant colonies. More recently, it has been applied in a data-mining context to perform both clustering and topographic mapping. Early work demonstrated some promising characteristics of the heuristic but did not extend to a rigorous investigation of its capabilities. We describe an improved version, called ATTA, incorporating adaptive, heterogeneous ants, a time-dependent transporting activity, and a method (for clustering applications) that transforms the spatial embedding produced by the algorithm into an explicit partitioning. ATTA is then subjected to the most rigorous experimental evaluation of an ant-based clustering and sorting algorithm undertaken to date: we compare its performance with standard techniques for clustering and topographic mapping using a set of analytical evaluation functions and a range of synthetic and real data collections. Our results demonstrate the ability of ant-based clustering and sorting to automatically identify the number of clusters inherent in a data collection, and to produce high quality solutions; indeed, we show that it is particularly robust for clusters of differing sizes and for overlapping clusters. The results obtained for topographic mapping are, however, disappointing. We provide evidence that the solutions generated by the ant algorithm are barely topology-preserving, and we explain in detail why results havein spite of thisbeen misinterpreted (much more positively) in previous research.

Quality Assessment of Pareto Set Approximations

This chapter reviews methods for the assessment and comparison of Pareto set approximations. Existing set quality measures from the literature are critically evaluated based on a number of orthogonal criteria, including invariance to scaling, monotonicity and computational effort. Statistical aspects of quality assessment are also considered in the chapter. Three main methods for the statistical treatment of Pareto set approximations deriving from stochastic generating methods are reviewed. The dominance ranking method is a generalization to partially-ordered sets of a standard non-parametric statistical test, allowing collections of Pareto set approximations from two or more stochastic optimizers to be directly compared statistically. The quality indicator method -- the dominant method in the literature -- maps each Pareto set approximation to a number, and performs statistics on the resulting distribution(s) of numbers. The attainment function method estimates the probability of attaining each goal in the objective space, and looks for significant differences between these probability density functions for different optimizers. All three methods are valid approaches to quality assessment, but give different information. We explain the scope and drawbacks of each approach and also consider some more advanced topics, including multiple testing issues, and using combinations of indicators. The chapter should be of interest to anyone concerned with generating and analysing Pareto set approximations.

Memetic Algorithms for Multiobjective Optimization: Issues, Methods and Prospects

The concept of optimization—finding the extrema of a function that maps candidate’ solutions’ to scalar values of ‘quality’—is an extremely general and useful idea that can be, and is, applied to innumerable problems in science, industry, and commerce. However, the vast majority of ‘real’ optimization problems, whatever their origins, comprise more than one objective; that is to say, ‘quality’ is actually a vector, which may be composed of such distinct attributes as cost, performance, profit, environmental impact, and so forth, which are often in mutual conflict. Until relatively recently this uncomfortable truth has been (wilfully?) overlooked in the sciences dealing with optimization, but now, increasingly, the idea of multiobjective optimization is taking over the centre ground. Multiobjective optimization takes seriously the fact that in most problems the different components that describe the quality of a candidate solution cannot be lumped together into one representative, overall measure, at least not easily, and not before some understanding of the possible ‘tradeoffs’ available has been established. Hence a multiobjective optimization algorithm is one which deals directly with a vector objective function and seeks to find multiple solutions offering different, optimal tradeoffs of the multiple objectives. This approach raises several unique issues in optimization algorithm design, which we consider in this article, with a particular focus on memetic algorithms (MAs). We summarize much of the relevant literature, attempting to be inclusive of relatively unexplored ideas, highlight the most important considerations for the design of multiobjective MAs, and finally outline our visions for future research in this exciting area.