Indicator-based Evolutionary Algorithms Research Articles

New Findings on Indicator-based Multi-Objective Evolutionary Algorithms: A Brief Summary

Quality indicators (QIs) are functions that assign a real value to a set that represents the Pareto front approximation of a multi-objective optimization problem. In the evolutionary multi-objective optimization community, QIs have been mainly employed in two ways: (1) for the performance assessment of multi-objective evolutionary algorithms (MOEAs), which produce Pareto front approximations, and (2) to be adopted as the backbone of selection mechanisms of MOEAs. Regardless of the continuing advances on QIs and their utilization in MOEAs, there are currently a vast number of open questions in this researcharea. In this doctoral thesis, we have focused on two main research directions: the design of new selection mechanisms based on the competition and cooperation of multiple QIs, aiming to compensate for the weaknesses (in terms of convergence and diversity properties) of individual QIs with the strengths of the others. The second research axis is the generation of new QIs that are compliant with the Pareto dominance relation extended to sets. Such QIs have a direct impact on the type of conclusions that can be drawn about the performance of MOEAs. Our experimental results have shown that the use of multiple QIs either to design new selection mechanisms or to construct new Pareto-compliant QIs is a promising research direction that can improve the capabilities of MOEAs and that allows for a performance assessment of MOEAs with a higher degree of confidence.

Improved Lebesgue Indicator-Based Evolutionary Algorithm: Reducing Hypervolume Computations

One of the major limitations of evolutionary algorithms based on the Lebesgue measure for multi-objective optimization is the computational cost required to approximate the Pareto front of a problem. Nonetheless, the Pareto compliance property of the Lebesgue measure makes it one of the most investigated indicators in the design of indicator-based evolutionary algorithms (IBEAs). The main deficiency of IBEAs that use the Lebesgue measure is their computational cost which increases with the number of objectives of the problem. On this matter, the investigation presented in this paper introduces an evolutionary algorithm based on the Lebesgue measure to deal with box-constrained continuous multi-objective optimization problems. The proposed algorithm implicitly uses the regularity property of continuous multi-objective optimization problems that has suggested effectiveness when solving continuous problems with rough Pareto sets. On the other hand, the survival selection mechanism considers the local property of the Lebesgue measure, thus reducing the computational time in our algorithmic approach. The emerging indicator-based evolutionary algorithm is examined and compared versus three state-of-the-art multi-objective evolutionary algorithms based on the Lebesgue measure. In addition, we validate its performance on a set of artificial test problems with various characteristics, including multimodality, separability, and various Pareto front forms, incorporating concavity, convexity, and discontinuity. For a more exhaustive study, the proposed algorithm is evaluated in three real-world applications having four, five, and seven objective functions whose properties are unknown. We show the high competitiveness of our proposed approach, which, in many cases, improved the state-of-the-art indicator-based evolutionary algorithms on the multi-objective problems adopted in our investigation.

LIBEA: A Lebesgue Indicator-Based Evolutionary Algorithm for multi-objective optimization

Abstract In multi-objective optimization, the hypervolume indicator (also known as S metric or Lebesgue measure) evaluates, at the same time, the approximation and distribution of non-dominated solutions along the Pareto optimal front of a multi-objective optimization problem (MOP). This indicator has become one of the most studied performance measures in the so-called indicator-based evolutionary algorithms (IBEAs). However, IBEAs based on hypervolume are limited by their high computational cost which increases with the number of objectives. As a consequence, several researchers have proposed new strategies to reduce the computational time of IBEAs based on this indicator. In this regard, the present paper introduces a Lebesgue indicator-based evolutionary algorithm (LIBEA) for continuous box-constrained multi-objective optimization problems. In contrast to most of IBEAs based on hypervolume, LIBEA implicitly employs the regularity property of continuous MOPs which has been suggested to solve continuous MOPs with difficult properties. The proposed LIBEA is validated on a set of test functions with different properties regarding separability, multi-modality, and different Pareto front geometries including convexity, concavity, and discontinuity. The performance of LIBEA is compared against three state-of-the-art evolutionary multi-objective algorithms (EMOAs) based on different principles. For a more comprehensive study, we evaluate the performance of our proposed approach and the adopted EMOAs over two real-world problems whose properties are unknown. We show that our proposed approach is highly competitive and that, in many cases, LIBEA significantly improved the state-of-the-art EMOAs on the test problems adopted in our comparative study.

Multiobjective optimization of classifiers by means of 3D convex-hull-based evolutionary algorithms

The receiver operating characteristic (ROC) and detection error tradeoff (DET) curves are frequently used in the machine learning community to analyze the performance of binary classifiers. Recently, the convex-hull-based multiobjective genetic programming algorithm was proposed and successfully applied to maximize the convex hull area for binary classification problems by minimizing false positive rate and maximizing true positive rate at the same time using indicator-based evolutionary algorithms. The area under the ROC curve was used for the performance assessment and to guide the search. Here we extend this research and propose two major advancements: Firstly we formulate the algorithm in detection error tradeoff space, minimizing false positives and false negatives, with the advantage that misclassification cost tradeoff can be assessed directly. Secondly, we add complexity as an objective function, which gives rise to a 3D objective space (as opposed to a 2D previous ROC space). A domain specific performance indicator for 3D Pareto front approximations, the volume above DET surface, is introduced, and used to guide the indicator-based evolutionary algorithm to find optimal approximation sets. We assess the performance of the new algorithm on designed theoretical problems with different geometries of Pareto fronts and DET surfaces, and two application-oriented benchmarks: (1) Designing spam filters with low numbers of false rejects, false accepts, and low computational cost using rule ensembles, and (2) finding sparse neural networks for binary classification of test data from the UCI machine learning benchmark. The results show a high performance of the new algorithm as compared to conventional methods for multicriteria optimization.

Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications

In recent years, indicator-based evolutionary algorithms, allowing to implicitly incorporate user preferences into the search, have become widely used in practice to solve multiobjective optimization problems. When using this type of methods, the optimization goal changes from optimizing a set of objective functions simultaneously to the single-objective optimization goal of finding a set of μ points that maximizes the underlying indicator. Understanding the difference between these two optimization goals is fundamental when applying indicator-based algorithms in practice. On the one hand, a characterization of the inherent optimization goal of different indicators allows the user to choose the indicator that meets her preferences. On the other hand, knowledge about the sets of μ points with optimal indicator values–the so-called optimalμ-distributions–can be used in performance assessment whenever the indicator is used as a performance criterion. However, theoretical studies on indicator-based optimization are sparse.One of the most popular indicators is the weighted hypervolume indicator. It allows to guide the search towards user-defined objective space regions and at the same time has the property of being a refinement of the Pareto dominance relation with the result that maximizing the indicator results in Pareto-optimal solutions only. In previous work, we theoretically investigated the unweighted hypervolume indicator in terms of a characterization of optimal μ-distributions and the influence of the hypervolume’s reference point for general bi-objective optimization problems. In this paper, we generalize those results to the case of the weighted hypervolume indicator. In particular, we present general investigations for finite μ, derive a limit result for μ going to infinity in terms of a density of points and derive lower bounds (possibly infinite) for placing the reference point to guarantee the Pareto front’s extreme points in an optimal μ-distribution. Furthermore, we state conditions about the slope of the front at the extremes such that there is no finite reference point that allows to include the extremes in an optimal μ-distribution—contradicting previous belief that a reference point chosen just above the nadir point or the objective space boundary is sufficient for obtaining the extremes. However, for fronts where there exists a finite reference point allowing to obtain the extremes, we show that for μ to infinity, a reference point that is slightly worse in all objectives than the nadir point is a sufficient choice. Last, we apply the theoretical results to problems of the ZDT, DTLZ, and WFG test problem suites.