An Analysis of Quality Indicators Using Approximated Optimal Distributions in a 3-D Objective Space

The hypervolume and the inverted generational distance (IGD) have been frequently used for the comparison of evolutionary multiobjective optimization algorithms. For the hypervolume, the relation between the location of a reference point and the optimal distribution of solutions has been studied in the literature. However, such a relation has not been studied for the IGD whereas IGD-based comparison results depend on the specification of reference points. Our intention is to clearly demonstrate the dependency of IGD-based comparison results on reference point specification. First, we explain difficulties of fair comparison in the following two cases: one is the use of all nondominated solutions among obtained solutions by compared algorithms as reference points, and the other is the use of a small number of uniformly sampled reference points. Discussions on these two cases show the necessity of a large number of uniformly sampled reference points on the entire Pareto front. Then, we show a bias of the IGD with such a reference point set through computational experiments. It is shown that the IGD tends to favor a solution set with much smaller diversity than a fully expanded solution set over the entire Pareto front. Finally, we propose a new specification method where reference points are uniformly sampled not only from the Pareto front but also from outside the Pareto front.

In evolutionary multi-objective optimization (EMO) algorithms, diversity maintenance has been mainly discussed in the objective space in order to search for uniformly distributed non-dominated solutions along the entire Pareto front. In this paper, we propose three types of many-objective test problems with multiple Pareto optimal regions in the decision space. One type has multiple equivalent Pareto optimal regions. Another type has different but somewhat similar Pareto optimal regions. The other type has Pareto and local Pareto optimal regions. Our many-objective test problems are generated by placing multiple polygons of the same or similar shapes in a decision space. The ith objective is the minimization of the distance from a solution to the nearest ith vertex over all polygons. Thus the number of objectives is the same as the number of vertices of the polygons. The number of equivalent or similar Pareto regions in the decision space is the same as the number of the polygons.

Most evolutionary multiobjective optimization algorithms are designed to approximate the entire Pareto front. During the last decade, a series of preference-based evolutionary algorithms have been developed, where a part of the Pareto front is approximated by incorporating the preferences of a Decision Maker. However, only a few such algorithms are able to obtain well-distributed solutions covering the complete “region of interest” that is determined by a reference point. In this paper, a preference-based evolutionary algorithm for approximating the region of interest is proposed. It is based on the state-of-the-art genetic algorithm NSGA-II and the CHIM approach introduced in the NBI method which is used to obtain uniformly distributed solutions in the region of interest. The efficiency of the proposed algorithm has been experimentally evaluated and compared to other state-of-the-art multiobjective preference-based evolutionary algorithms by solving a set of multiobjective optimization benchmark problems. It has been shown that the incorporation of the Decision Maker’s preferences and the CHIM approach into the NSGA-II algorithm allows approximating the whole region of interest accurately while maintaining a good distribution of the obtained solutions.

Multi-modal multi-objective optimization is to locate (almost) equivalent Pareto optimal solutions as many as possible. While decomposition-based evolutionary algorithms have good performance for multi-objective optimization, they are likely to perform poorly for multi-modal multi-objective optimization due to the lack of mechanisms to maintain the solution space diversity. To address this issue, this paper proposes a framework to improve the performance of decomposition-based evolutionary algorithms for multi-modal multi-objective optimization. Our framework is based on three operations: assignment, deletion, and addition operations. One or more individuals can be assigned to the same subproblem to handle multiple equivalent solutions. In each iteration, a child is assigned to a subproblem based on its objective vector, i.e., its location in the objective space. The child is compared with its neighbors in the solution space assigned to the same subproblem. The performance of improved versions of six decomposition-based evolutionary algorithms by our framework is evaluated on various test problems regarding the number of objectives, decision variables, and equivalent Pareto optimal solution sets. Results show that the improved versions perform clearly better than their original algorithms.

Many software engineering problems are multi-objective in nature, which has been largely recognized by the Search-based Software Engineering (SBSE) community. In this regard, Pareto- based search algorithms, e.g., Non-dominated Sorting Genetic Algorithm II, have already shown good performance for solving multi-objective optimization problems. These algorithms produce Pareto fronts, where each Pareto front consists of a set of non- dominated solutions. Eventually, a user selects one or more of the solutions from a Pareto front for their specific problems. A key challenge of applying Pareto-based search algorithms is to select appropriate quality indicators, e.g., hypervolume, to assess the quality of Pareto fronts. Based on the results of an extended literature review, we found that the current literature and practice in SBSE lacks a practical guide for selecting quality indicators despite a large number of published SBSE works. In this direction, the paper presents a practical guide for the SBSE community to select quality indicators for assessing Pareto-based search algorithms in different software engineering contexts. The practical guide is derived from the following complementary theoretical and empirical methods: 1) key theoretical foundations of quality indicators; 2) evidence from an extended literature review; and 3) evidence collected from an extensive experiment that was conducted to evaluate eight quality indicators from four different categories with six Pareto-based search algorithms using three real industrial problems from two diverse domains.

Dual-space distribution metric-based evolutionary algorithm for multimodal multi-objective optimization

This paper presents the Pareto-Box problem for modelling evolutionary multi-objective search. The problem is to find the Pareto set of randomly selected points in the unit hypercube. While the Pareto set itself is only comprised of the point 0, this problem allows for a complete analysis of random search and demonstrates the fact that with increasing number of objectives, the probability of finding a dominated vector is decreasing exponentially. Since most nowadays evolutionary multi-objective optimization algorithms rely on the existence of dominated individuals, they show poor performance on this problem. However, the fuzzification of the Pareto-dominance is an example for an approach that does not need dominated individuals, thus it is able to solve the Pareto-Box problem even for a higher number of objectives.

A ring-hierarchy-based evolutionary algorithm for multimodal multi-objective optimization

So far, many performance indicators have been proposed to compare different evolutionary multiobjective optimization algorithms (MOEAs). Among them, the inverted generational distance (IGD) is one of the most commonly used, mainly because it can measure a population’s convergence, diversity, and evenness. However, the effectiveness of IGD highly depends on the quality of the reference set. That is to say, all the reference points should be as close to the Pareto front (PF) as possible and evenly distributed to become ready for a fair performance evaluation. Currently, it is still challenging to generate well-configured reference sets, even if the PF can be given analytically. Therefore, biased reference sets might be a significant source of systematic error. However, in most MOEA literature, biased reference sets are utilized in experiments without an error estimation, which may make the experimental results unconvincing. In this paper, we propose an exact IGD (eIGD) for continuous PF, which is derived from the original IGD under an additional assumption that the reference set is perfect, i.e., the PF itself is directly utilized as an infinite-sized reference set. Therefore, the IGD values produced by biased reference sets can be compared with eIGD so that systematic error can be quantitatively evaluated and analyzed.KeywordsMultiobjective optimizationEvolutionary computationPerformance indicatorReference setDifferential geometry

This study compared the performances of three state-of-theart evolutionary multi-objective optimization algorithms: the Non-Dominated Sorted Genetic Algorithm II (NSGA-II), the Epsilon-Dominance Non-Dominated Sorted Genetic Algorithm II (e -NSGA-II) and the Epsilon-Dominance Multi-Objective Evolutionary Algorithm ( e MOEA), on a constrained biobjective crop-mix planning problem (BCP). The BCP test case objectives were to: (i) maximize crop production and (ii) minimize planting area. The BCP problem was enumerated to provide the true Pareto-optimal solution set to facilitate rigorous testing of the evolutionary multi-objective optimization algorithms. The performances of the three algorithms were assessed and compared using three performance metrics (additive epsilon indicator, inverted generational distance and spacing). For the comparative study, a graphical comparison with respect to true Pareto front of optimal crop planning problem was provided. Results of the analyses indicated that the e MOEA greatly outperforms the NSGAII and the e -NSGA-II.

Evolutionary multi-objective algorithms (EMOA) using performance indicators for the selection of individuals have turned out to be a successful technique for multi-objective problems. Especially, the selection based on the $\mathcal{S}$-metric, as implemented in the SMS-EMOA, seems to be effective. A special feature of this EMOA is the greedy (μ + 1) selection. Based on a pathological example for a population of size two and a discrete Pareto front it has been proven that a (μ + 1)- (or 1-greedy) EMOA may fail in finding a population maximizing the $\mathcal{S}$-metric. This work investigates the performance of (μ + 1)-EMOA with small fixed-size populations on Pareto fronts of innumerable size. We prove that an optimal distribution of points can always be achieved on linear Pareto fronts. Empirical studies support the conjecture that this also holds for convex and concave Pareto fronts, but not for continuous shapes in general. Furthermore, the pathological example is generalized to a continuous objective space and it is demonstrated that also (μ + k )-EMOA are not able to robustly detect the globally optimal distribution.

A MOEA/D with adaptive weight subspace for regular and irregular multi-objective optimization problems

In recent years, several indicator-based multi-objective evolutionary algorithms (IB-MOEAs) have been proposed. Each IB-MOEA presents different search preferences depending on the quality indicator (QI) that it uses in its selection mechanism. However, due to these search biases, IB-MOEAs behave differently on each multi-objective optimization problem, producing Pareto front approximations whose characteristics are related to the QI on which they are based. In this paper, we propose a novel algorithm based on the island model that aims to take advantage of the cooperation of individual IB-MOEAs based on the indicators hypervolume, R2, IGD+, +, and Δ p with the aim of improving both convergence and distribution of the Pareto fronts produced. Our experimental results, taking into account seven quality indicators, empirically show that the cooperation of several IB-MOEAs is better than using panmictic versions of them. Additionally, we also show that the performance of our proposal does not depend on the Pareto front shape of the problem being solved.

The model-based evolutionary algorithms are divided into three groups: estimation of distribution algorithms, inverse modeling, and surrogate modeling. Existing inverse modeling is mainly applied to solve multi-objective optimization problems and is not suitable for many-objective optimization problems. Some inversed-model techniques, such as the inversed-model of multi-objective evolutionary algorithm, constructed from the Pareto front (PF) to the Pareto solution on nondominated solutions using a random grouping method and Gaussian process, were introduced. However, some of the most efficient inverse models might be eliminated during this procedure. Also, there are challenges, such as the presence of many local PFs and developing poor solutions when the population has no evident regularity. This paper proposes inverse modeling using random forest regression and uniform reference points that map all nondominated solutions from the objective space to the decision space to solve many-objective optimization problems. The proposed algorithm is evaluated using the benchmark test suite for evolutionary algorithms. The results show an improvement in diversity and convergence performance (quality indicators).

Runtime analysis of a multi-objective evolutionary algorithm for obtaining finite approximations of Pareto fronts