An Ensemble Framework of Evolutionary Algorithm for Constrained Multi-Objective Optimization

In the real-world, symmetry or asymmetry widely exists in various problems. Some of them can be formulated as constrained multi-objective optimization problems (CMOPs). During the past few years, handling CMOPs by evolutionary algorithms has become more popular. Lots of constrained multi-objective optimization evolutionary algorithms (CMOEAs) have been proposed. Whereas different CMOEAs may be more suitable for different CMOPs, it is difficult to choose the best one for a CMOP at hand. In this paper, we propose an ensemble framework of CMOEAs that aims to achieve better versatility on handling diverse CMOPs. In the proposed framework, the hypervolume indicator is used to evaluate the performance of CMOEAs, and a decreasing mechanism is devised to delete the poorly performed CMOEAs and to gradually determine the most suitable CMOEA. A new CMOEA, namely ECMOEA, is developed based on the framework and three state-of-the-art CMOEAs. Experimental results on five benchmarks with totally 52 instances demonstrate the effectiveness of our approach. In addition, the superiority of ECMOEA is verified through comparisons to seven state-of-the-art CMOEAs. Moreover, the effectiveness of ECMOEA on the real-world problems is also evaluated for eight instances.

Optimal Number of Pressure Sensors for Real-Time Monitoring of Distribution Networks by Using the Hypervolume Indicator

This article proposes a novel methodology to determine the optimal number of pressure sensors for the real-time monitoring of water distribution networks based on a quality hypervolume indicator. The proposed methodology solves the optimization problem for different numbers of pressure sensors, assesses the gain of installing each set of sensors by means of the hypervolume indicator and determines the optimal number of sensors by the variation of the hypervolume indicator. The methodology was applied to a real case study. Several robustness analyses were carried out. The results demonstrate that the methodology is hardly influenced by the method parameters and that a reasonable estimation of the optimal number of sensors can be easily achieved.

A Novel Method for Calculating the Parametric Hypervolume Indicator

This paper presents a new methodology for calculating the hypervolume indicator (HVI)for multi-objective and parametric data. Existing multi-objective HVI calculation techniques cannot be directly used for parametric data because designers do not have preferences for parameters like they do for objectives. The novel method presented herein allows for the consideration of both objectives and parameters through the newly introduced hypercone heuristic (HCH). This heuristic relaxes the strict rules of parametric Pareto dominance for a more practical dominance assessment when comparing designs of differing parameter values without violating Pareto dominance rules. A parametric HVI (pHVI) enhances a design engineer’s toolkit by enabling both online and offline evaluation of parametric optimization results. The pHVI measure allows designers to compare solution sets, detect optimization convergence, and to better inform optimization procedures in a parametric context. Results show that the HCH-based pHVI yields a similar quality measure to the existing technique based on a support vector domain description (SVDD) in a fraction of the computational time. Furthermore, the novel HCH-based pHVI technique satisfies multi-objective HVI properties allowing previous applications of the HVI to be applied to multi-objective parametric optimization. This contribution enables the field of parametric optimization, and thus parametric design, to benefit from prior and future advances in the multi-objective optimization domain involving the HVI.

The Hypervolume Indicator

The hypervolume indicator is one of the most used set-quality indicators for the assessment of stochastic multiobjective optimizers, as well as for selection in evolutionary multiobjective optimization algorithms. Its theoretical properties justify its wide acceptance, particularly the strict monotonicity with respect to set dominance, which is still unique of hypervolume-based indicators. This article discusses the computation of hypervolume-related problems, highlighting the relations between them, providing an overview of the paradigms and techniques used, a description of the main algorithms for each problem, and a rundown of the fastest algorithms regarding asymptotic complexity and runtime. By providing a complete overview of the computational problems associated to the hypervolume indicator, this article serves as the starting point for the development of new algorithms and supports users in the identification of the most appropriate implementations available for each problem.

Hypervolume Subset Selection with Small Subsets

The hypervolume subset selection problem (HSSP) aims at approximating a set of [Formula: see text] multidimensional points in [Formula: see text] with an optimal subset of a given size. The size [Formula: see text] of the subset is a parameter of the problem, and an approximation is considered best when it maximizes the hypervolume indicator. This problem has proved popular in recent years as a procedure for multiobjective evolutionary algorithms. Efficient algorithms are known for planar points ([Formula: see text]), but there are hardly any results on HSSP in larger dimensions ([Formula: see text]). So far, most algorithms in higher dimensions essentially enumerate all possible subsets to determine the optimal one, and most of the effort has been directed toward improving the efficiency of hypervolume computation. We propose efficient algorithms for the selection problem in dimension 3 when either [Formula: see text] or [Formula: see text] is small, and extend our techniques to arbitrary dimensions for [Formula: see text].