Carlos M. Fonseca

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.

On the Performance Assessment and Comparison of Stochastic Multiobjective Optimizers

This work proposes a quantitative, non-parametric interpretation of statistical performance of stochastic multiobjective optimizers, including, but not limited to, genetic algorithms. It is shown that, according to this interpretation, typical performance can be defined in terms analogous to the notion of median for ordinal data, as can other measures analogous to other quantiles.

Multiobjective optimization and multiple constraint handling with evolutionary algorithms. II. Application example

For part I see ibid., 26-37. The evolutionary approach to multiple function optimization formulated in the first part of the paper is applied to the optimization of the low-pressure spool speed governor of a Pegasus gas turbine engine. This study illustrates how a technique such as the multiobjective genetic algorithm can be applied, and exemplifies how design requirements can be refined as the algorithm runs. Several objective functions and associated goals express design concerns in direct form, i.e., as the designer would state them. While such a designer-oriented formulation is very attractive, its practical usefulness depends heavily on the ability to search and optimize cost surfaces in a class much broader than usual, as already provided to a large extent by the genetic algorithm (GA). The two instances of the problem studied demonstrate the need for preference articulation in cases where many and highly competing objectives lead to a nondominated set too large for a finite population to sample effectively. It is shown that only a very small portion of the nondominated set is of practical relevance, which further substantiates the need to supply preference information to the GA. The paper concludes with a discussion of the results.

An Improved Dimension-Sweep Algorithm for the Hypervolume Indicator

This paper presents a recursive, dimension-sweep algorithm for computing the hypervolume indicator of the quality of a set of n non-dominated points in d > 2 dimensions. It improves upon the existing HSO (Hypervolume by Slicing Objectives) algorithm by pruning the recursion tree to avoid repeated dominance checks and the recalculation of partial hypervolumes. Additionally, it incorporates a recent result for the three-dimensional special case. The proposed algorithm achieves O(nd−2log n) time and linear space complexity in the worst-case, but experimental results show that the pruning techniques used may reduce the time complexity exponent even further.

On the Complexity of Computing the Hypervolume Indicator

The goal of multiobjective optimization is to find a set of best compromise solutions for typically conflicting objectives. Due to the complex nature of most real-life problems, only an approximation to such an optimal set can be obtained within reasonable (computing) time. To compare such approximations, and thereby the performance of multiobjective optimizers providing them, unary quality measures are usually applied. Among these, the hypervolume indicator (or S-metric) is of particular relevance due to its favorable properties. Moreover, this indicator has been successfully integrated into stochastic optimizers, such as evolutionary algorithms, where it serves as a guidance criterion for finding good approximations to the Pareto front. Recent results show that computing the hypervolume indicator can be seen as solving a specialized version of Klees measure problem. In general, Klees measure problem can be solved with O(n logn + nd/2logn) comparisons for an input instance of size n in d dimensions; as of this writing, it is unknown whether a lower bound higher than Omega(n log n) can be proven. In this paper, we derive a lower bound of Omega(n log n) for the complexity of computing the hypervolume indicator in any number of dimensions d > 1 by reducing the so-called uniformgap problem to it. For the 3-D case, we also present a matching upper bound of O(n log n) comparisons that is obtained by extending an algorithm for finding the maxima of a point set.

Inferential Performance Assessment of Stochastic Optimisers and the Attainment Function

The performance of stochastic optimisers can be assessed experimentally on given problems by performing multiple optimisation runs, and analysing the results. Since an optimiser may be viewed as an estimator for the (Pareto) minimum of a (vector) function, stochastic optimiser performance is discussed in the light of the criteria applicable to more usual statistical estimators. Multiobjective optimisers are shown to deviate considerably from standard point estimators, and to require special statistical methodology. The attainment function is formulated, and related results from random closed-set theory are presented, which cast the attainment function as a mean-like measure for the outcomes of multiobjective optimisers. Finally, a covariance-measure is defined, which should bring additional insight into the stochastic behaviour of multiobjective optimisers. Computational issues and directions for further work are discussed at the end of the paper.

Identifying the structure of nonlinear dynamic systems using multiobjective genetic programming

A method for identifying the structure of nonlinear polynomial dynamic models is presented. This approach uses an evolutionary algorithm, genetic programming, in a multiobjective fashion to generate global models which describe the dynamic behavior of the nonlinear system under investigation. The validation stage of system identification is simultaneously evaluated using the multiobjective tool, in order to direct the identification process to a set of global models of the system.

Genetic Algorithms in Control Systems Engineering

Abstract Recent research into the mechanisms of evolution and genetics has shown how biological systems have managed to develop some very powerful methods of optimising and adapting themselves to meet new environmental challenges. For applications in control systems engineering, many of the characteristics exhibited by genetic algorithms are particularly appropriate. They can be used as an optimization tool or as the basis of adaptive systems. The versatile and robust qualities of these algorithms are reviewed and their relevance for control systems is highlighted. Applications are described and implementation issues are addressed, including parallelization. Prospective future directions are identified

Exploring the performance of stochastic multiobjective optimisers with the second-order attainment function

The attainment function has been proposed as a measure of the statistical performance of stochastic multiobjective optimisers which encompasses both the quality of individual non-dominated solutions in objective space and their spread along the trade-off surface. It has also been related to results from random closed-set theory, and cast as a mean-like, first-order moment measure of the outcomes of multiobjective optimisers. In this work, the use of more informative, second-order moment measures for the evaluation and comparison of multiobjective optimiser performance is explored experimentally, with emphasis on the interpretability of the results.

Assessing the performance of multiobjective genetic algorithms for optimization of a batch process scheduling problem

Scheduling optimization problems provide much potential for innovative solutions by genetic algorithms. The complexities, constraints and practicalities of the scheduling process motivate the development of genetic algorithm (GA) techniques to allow innovative and flexible scheduling solutions. Multiobjective genetic algorithms (MOGAs) extend the standard evolutionary-based genetic algorithm optimization technique to allow individual treatment of several objectives simultaneously. This allows the user to attempt to optimize several conflicting objectives, and to explore the trade-offs, conflicts and constraints inherent in this process. The area of MOGA performance assessment and comparison is a relatively new field, as much research concentrates on applications rather than the theory. However, the theoretical exploration of MOGA performance can have tangible effects on the development of highly practical applications, such as the process plant scheduling system under development in this work. By assessing and comparing the strengths, variations and limitations of the developing MOGA using a quantitative method, a highly efficient MOGA can develop to suit the application. The user can also gain insight into behaviour the application itself. In this work, four MOGAs are implemented to solve a process scheduling optimization problem; using two and five objectives, and two schedule building rules.