Marco Laumanns

Performance assessment of multiobjective optimizers: an analysis and review

An important issue in multiobjective optimization is the quantitative comparison of the performance of different algorithms. In the case of multiobjective evolutionary algorithms, the outcome is usually an approximation of the Pareto-optimal set, which is denoted as an approximation set, and therefore the question arises of how to evaluate the quality of approximation sets. Most popular are methods that assign each approximation set a vector of real numbers that reflect different aspects of the quality. Sometimes, pairs of approximation sets are also considered. In this study, we provide a rigorous analysis of the limitations underlying this type of quality assessment. To this end, a mathematical framework is developed which allows one to classify and discuss existing techniques.

Scalable multi-objective optimization test problems

After adequately demonstrating the ability to solve different two-objective optimization problems, multi-objective evolutionary algorithms (MOEAs) must show their efficacy in handling problems having more than two objectives. In this paper, we suggest three different approaches for systematically designing test problems for this purpose. The simplicity of construction, scalability to any number of decision variables and objectives, knowledge of exact shape and location of the resulting Pareto-optimal front, and ability to control difficulties in both converging to the true Pareto-optimal front and maintaining a widely distributed set of solutions are the main features of the suggested test problems. Because of these features, they should be useful in various research activities on MOEAs, such as testing the performance of a new MOEA, comparing different MOEAs, and having a better understanding of the working principles of MOEAs.

Combining convergence and diversity in evolutionary multiobjective optimization

Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to find a number of Pareto-optimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Pareto-optimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept of -dominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modifications to the baseline algorithm are also suggested. The concept of -dominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.

A Tutorial on Evolutionary Multiobjective Optimization

Multiple, often conflicting objectives arise naturally in most real-world optimization scenarios. As evolutionary algorithms possess several characteristics that are desirable for this type of problem, this class of search strategies has been used for multiobjective optimization for more than a decade. Meanwhile evolutionary multiobjective optimization has become established as a separate subdiscipline combining the fields of evolutionary computation and classical multiple criteria decision making.

PISA: a platform and programming language independent interface for search algorithms

This paper introduces an interface specification (PISA) that allows to separate the problem-specific part of an optimizer from the problem-independent part. We propose a view of the general optimization scenario, where the problem representation together with the variation operators is seen as an integral part of the optimization problem and can hence be easily separated from the selection operators. Both parts are implemented as independent programs, that can be provided as ready-to-use packages and arbitrarily combined. This makes it possible to specify and implement representation-independent selection modules, which form the essence of modern multiobjective optimization algorithms. The variation operators, on the other hand, have to be defined in one module together with the optimization problem, facilitating a customized problem description. Besides the specification, the paper contains a correctness proof for the protocol and measured efficiency results.

An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method

Abstract This paper discusses methods for generating or approximating the Pareto set of multiobjective optimization problems by solving a sequence of constrained single-objective problems. The necessity of determining the constraint value a priori is shown to be a serious drawback of the original epsilon-constraint method. We therefore propose a new, adaptive scheme to generate appropriate constraint values during the run. A simple example problem is presented, where the running time (measured by the number of constrained single-objective sub-problems to be solved) of the original epsilon-constraint method is exponential in the problem size (number of decision variables), although the size of the Pareto set grows only linearly. We prove that––independent of the problem or the problem size––the time complexity of the new scheme is O(km−1), where k is the number of Pareto-optimal solutions to be found and m the number of objectives. Simulation results for the example problem as well as for different instances of the multiobjective knapsack problem demonstrate the behavior of the method, and links to reference implementations are provided.

A Spatial Predator-Prey Approach to Multi-objective Optimization: A Preliminary Study

This paper presents a novel evolutionary approach of approximating the shape of the Pareto-optimal set of multi-objective optimization problems. The evolutionary algorithm (EA) uses the predator-prey model from ecology. The prey are the usual individuals of an EA that represent possible solutions to the optimization task. They are placed at vertices of a graph, remain stationary, reproduce, and are chased by predators that traverse the graph. The predators chase the prey only within its current neighborhood and according to one of the optimization criteria. Because there are several predators with different selection criteria, those prey individuals, which perform best with respect to all objectives, are able to produce more descendants than inferior ones. As soon as a vertex for the prey becomes free, it is refilled by descendants from alive parents in the usual way of EA, i.e., by inheriting slightly altered attributes. After a while, the prey concentrate at Pareto-optimal positions. The main objective of this preliminary study is the answer to the question whether the predator-prey approach to multi-objective optimization works at all. The performance of this evolutionary algorithm is examined under several step-size adaptation rules.

A unified model for multi-objective evolutionary algorithms with elitism

Though it has been claimed that elitism could improve evolutionary multi-objective search significantly, a thorough and extensive evaluation of its effects is still missing. Guidelines on how elitism could successfully be incorporated have not yet been developed. This paper presents a unified model of multi-objective evolutionary algorithms, in which arbitrary variation and selection operators can be combined as building blocks, including archiving and re-insertion strategies. The presented model enables most specific multi-objective (evolutionary) algorithm to be formulated as an instance of it, which will be demonstrated by simple examples. We further show how elitism can be quantified by the models parameters and how this allows an easy evaluation of the effect of elitism on different algorithms.

Running time analysis of multiobjective evolutionary algorithms on pseudo-Boolean functions

This paper presents a rigorous running time analysis of evolutionary algorithms on pseudo-Boolean multiobjective optimization problems. We propose and analyze different population-based algorithms, the simple evolutionary multiobjective optimizer (SEMO), and two improved versions, fair evolutionary multiobjective optimizer (FEMO) and greedy evolutionary multiobjective optimizer (GEMO). The analysis is carried out on two biobjective model problems, leading ones trailing zeroes (LOTZ) and count ones count zeroes (COCZ), as well as on the scalable m-objective versions mLOTZ and mCOCZ. Results on the running time of the different population-based algorithms and for an alternative approach, a multistart (1+1)-EA based on the /spl epsi/-constraint method, are derived. The comparison reveals that for many problems, the simple algorithm SEMO is as efficient as this (1+1)-EA. For some problems, the improved variants FEMO and GEMO are provably better. For the analysis, we propose and apply two general tools, an upper bound technique based on a decision space partition and a randomized graph search algorithm, which facilitate the analysis considerably.

On the Effects of Archiving, Elitism, and Density Based Selection in Evolutionary Multi-objective Optimization

This paper studies the influence of what are recognized as key issues in evolutionary multi-objective optimization: archiving (to keep track of the current non-dominated solutions), elitism (to let the archived solutions take part in the search process), and diversity maintenance (through density dependent selection). Many proposed algorithms use these concepts in different ways, but a common framework does not exist yet. Here, we extend a unified model for multiobjective evolutionary algorithms so that each specific method can be expressed as an instance of a generic operator. This model forms the basis for a new type of empirical investigation regarding the effects of certain operators and parameters on the performance of the search process. The experiments of this study indicate that interactions between operators as well as between standard parameters (like the mutation intensity) cannot be neglected. The results lead not only to better insight into the working principle of multi-objective evolutionary algorithms but also to design recommendations that can help possible users in including the essential features into their own algorithms in a modular fashion.