On the Cooperation of Multiple Indicator-based Multi-Objective Evolutionary Algorithms

Ensemble learning is one of the most employed methods in machine learning. Its main ground is the construction of stronger mechanisms based on the combination of elementary ones. In this paper, we employ AdaBoost, which is one of the most well-known ensemble methods, to generate an ensemble indicator-based density estimator for multi-objective optimization. It combines the search properties of five density estimators, based on the hypervolume, R2, IGD\(^+\), \(\epsilon ^+\), and \(\varDelta _p\) quality indicators. Through the multi-objective evolutionary search process, the proposed ensemble mechanism adapts itself using a learning process that takes the preferences of the underlying quality indicators into account. The proposed method gives rise to the ensemble indicator-based multi-objective evolutionary algorithm (EIB-MOEA) that shows a robust performance on different multi-objective optimization problems when compared with respect to several existing indicator-based multi-objective evolutionary algorithms.

Reference sets generated with uniformly distributed weight vectors on a unit simplex are widely used by several multi-objective evolutionary algorithms (MOEAs). They have been employed to tackle multi-objective optimization problems (MOPs) with four or more objective functions, i.e., the so-called many-objective optimization problems. These MOEAs have shown a good performance on MOPs with regular Pareto front shapes, i.e., simplex-like shapes. However, it has been observed that in many cases, their performance degrades on MOPs with irregular Pareto front shapes. In this paper, we designed a new selection mechanism that aims to promote a Pareto front shape invariant performance of MOEAs that use weight vector-based reference sets. The newly proposed selection mechanism takes advantage of weight vector-based reference sets and seven pair-potential functions. It was embedded into the non-dominated sorting genetic algorithm III (NSGA-III) to increase its performance on MOPs with different Pareto front geometries. We use the DTLZ and DTLZ\(^{-1}\) test problems to perform an empirical study about the usage of these pair-potential functions for this selection mechanism. Our experimental results show that the pair-potential functions can enhance the distribution of solutions obtained by weight vector-based MOEAs on MOPs with irregular Pareto front shapes. Also, the proposed selection mechanism permits maintaining the good performance of these MOEAs on MOPs with regular Pareto front shapes.

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

The goal of the multi-objective optimization algorithm is to quickly and accurately find a set of trade-off solutions. This paper develops a clustering-based competitive multi-objective particle swarm optimizer using the enhanced grid for solving multi-objective optimization problems, named EGC-CMOPSO. The enhanced grid mechanism involved in EGC-CMOPSO is designed to locate superior Pareto optimal solutions. Subsequently, a hierarchical-based clustering is established on the grid for improving the accuracy rate of the grid selection. Due to the adaptive division of clustering centers, EGC-CMOPSO is applicable for solving MOPs with various Pareto front (PF) shapes. Particularly, the inferior solutions are discarded and the leading particles are identified by the comprehensive ranking of particles in each cluster. Finally, the selected leading particles compete against each other, and the winner guides the update of the current particle. The proposed EGC-CMOPSO and the eight latest multi-objective optimization algorithms are performed on 21 test problems. The experimental results validate that the proposed EGC-CMOPSO is capable of handling multi-objective optimization problems (MOPs) and obtaining superior performance on both convergence and diversity.

Although quality indicators play a crucial role in benchmarking evolutionary multi-objective optimization algorithms, their properties are still unclear. One promising approach for understanding quality indicators is the use of the optimal distribution of objective vectors that optimizes each quality indicator. However, it is difficult to obtain the optimal distribution for each quality indicator, especially when its theoretical property is unknown. Thus, optimal distributions for most quality indicators have not been well investigated. To address these issues, first, we propose a problem formulation of finding the optimal distribution for each quality indicator on an arbitrary Pareto front. Then, we approximate the optimal distributions for nine quality indicators using the proposed problem formulation. We analyze the nine quality indicators using their approximated optimal distributions on eight types of Pareto fronts of three-objective problems. Our analysis demonstrates that uniformly-distributed objective vectors over the entire Pareto front are not optimal in many cases. Each quality indicator has its own optimal distribution for each Pareto front. We also examine the consistency among the nine quality indicators.

Clustering-based genetic offspring generation for solving multi-objective optimization problems with intricate Pareto sets

The Pareto set (PS) of real multi-objective optimization problems (MOPs) are often unknown and complex, so, it is significant for multi-objective evolutionary algorithms (MOEAs) to solve complex PS MOPs (CPS_MOPs namely). In this paper, we combined Latin hypercube sampling (LHS) with MOEA, proposed a LHS based MOEA (LHS-MOEA). We suggested two kinds of LHS-MOEA, in which LHS local search and evolutionary operator are combined to handle CPS_MOPs. Through some experiments, the results demonstrate that LHS-MOEA performs much better than the traditional prevalent MOEA — NSGA-II in solving CPS_MOPs.KeywordsMulti-objective Evolutionary AlgorithmLatin Hypercube SamplingComplex Pareto SetLocal SearchEvolutionary Operator

The main objective of this study was to solve a multi-objective optimization on a representative coronary stent platform with the goal of finding new geometric designs with improved biomechanical performance. The following set of metrics, calculated via finite element models, was used to quantify stent performance: vessel injury, radial recoil, bending resistance, longitudinal resistance, radial strength and prolapse index. The multi-objective optimization problem was solved with the aid of surrogate-based algorithms; for comparison and validation purposes, four surrogate-based multi-objective optimization algorithms (EIhv -EGO, Phv -EGO, ParEGO and SMS-EGO) with a limited sample budget were employed and their results compared. The quality of the non-dominated solution sets outputted by each algorithm was assessed against four quality indicators: hypervolume, R2, epsilon and generational distance. Results showed that Phv -EGO was the algorithm that exhibited the best performance in overall terms. Afterwards, the highest quality Pareto front was chosen for an in-depth analysis of the optimization results. The amount of correlation and conflict was quantified for each pair of objective functions. Next, through cluster analysis, one was able to identify families of solutions with similar performance behavior and to discuss the nature of the existent trade-offs between objectives, and the trends between design parameters and solutions in a biomechanical perspective. In the end, a constrained-based design selection was performed with the goal of finding solutions in the Pareto front with equal or better performance in all objectives against a baseline design.

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.

Multi-objective optimization problems (MOPs) need to be solved in real world recently. In this paper, a multi-objective particle swarm optimization based on Pareto set and aggregation approach was proposed to deal with MOPs. Firstly, velocities and positions were updated similar to PSO. Then, global-best set was defined in particle swarm optimizer to preserve Pareto-based set obtained by the population. Specifically, a hybrid updating strategy based on Pareto set and aggregation approach was introduced to update the global-best set and local search was carried on global-best set. Thirdly, personal-best positions were updated in decomposition way, and global-best position was selected from global-best set. Finally, ZDT instances and DTLZ instances were selected to evaluate the performance of MULPSO and the results show validity of the proposed algorithm for MOPs.

This work proposes a new method for approximating the Pareto front of a multi-objective simulation optimization problem (MOP) where the explicit forms of the objective functions are not available. The method iteratively approximates each objective function using a metamodeling scheme and employs a weighted sum method to convert the MOP into a set of single objective optimization problems. The weight on each single objective function is adaptively determined by accessing newly introduced points at the current iteration and the non-dominated points so far. A trust region algorithm is applied to the single objective problems to search for the points on the Pareto front. The numerical results show that the proposed algorithm efficiently generates evenly distributed points for various types of Pareto fronts. 1

Multi‐objective evolutionary optimisation algorithms (MOEAs) have become a widely adopted way of solving the multi‐objective optimisation problems (MOPs). The decomposition‐based MOEAs demonstrate a promising performance for solving regular MOPs. However, when handling the irregular MOPs, the decomposition‐based MOEAs cannot offer a convincing performance because no intersection between weight vector and the Pareto Front (PF) may lead to the same optimal solution assigned to the different weight vectors. To solve this problem, this paper proposes an MOEA based on decomposition with the orthogonal experimental design (MOEA/D‐OED) that involves the selection operation, Orthogonal Experimental Design (OED) operation, and adjustment operation. The selection operation is to judge the unpromising weight vectors based on the history data of relative reduction values and convergence degree. The OED method based on the relative reduction function could make an explicit guidance for removing the worthless weight vectors. The adjustment operation brings in an estimation indicator of both diversity and convergence for adding new weight vectors into the interesting regions. To verify the versatility of the proposed MOEA/D‐OED, 26 test problems with various PFs are evaluated in this paper. Empirical results have demonstrated that the proposed MOEA/D‐OED outperforms eight representative MOEAs on MOPs with various types of PFs, showing promising versatility. The proposed algorithm shows highly competitive performance on all the various MOPs.

In evolutionary multi-objective optimization, it has been illuminated that guide search with neighboring solutions improve the quality of new trial solutions and accelerate algorithms convergence by the regularity property of the continuous multi-objective optimization problems (MOPs). Very recently, clustering learning-based mating strategies have been popular for establishing reproduction operators with neighboring solutions. However, the existing mating strategies may be more reasonable with the full consideration and utilization of the regularity property. The current mating restrictions excessively emphasize algorithm convergence and ignore population diversity. In addition, the selected clustering algorithms in mating restrictions are not conducive for solving MOPs, which have complex Pareto sets (PSs) and\or Pareto fronts (PFs). To solve above problems and address both the algorithm convergence and the population diversity of multi-objective evolutionary algorithms (MOEAs), the spectral clustering based multi-source mating selection strategy (SMMS) is designed to detect regularity properties and balance population diversity while accelerating algorithm convergence. Consequently, a spectral clustering based multi-source mating selection multi-objective evolutionary algorithms is proposed, teamed SMMEA. SMMEA is applied to a number of test suites with a complex PS or PF, and compared with six state-of-the-art MOEAs. The results demonstrate that the proposed algorithm outperforms over the other approaches.

Numerical algorithms for generating an almost even approximation of the Pareto front in nonlinear multi-objective optimization problems

We propose a new metaheuristic, HmGWOGA-MO, for solving multiobjective optimization problems operating with a population of solutions. The method is a hybridization of the HmGWOGA method, which is a single objective optimization method, and the ϵ-constraint approach, which is an aggregation technique. The ϵ-constraint technique is one of the best ways to transform a problem with many objective functions into a single objective problem because it works even if the problem has any kind of Pareto optimal front. Previously, the HmGWOGA method was designed to optimize a positive single-objective function without constraints. The obtained solutions are good. That is why, in this current work, we combined have it with the ϵ-constraint approach for the resolution of multiobjective optimization problems. Our new method proceeds by transforming a given multiobjective optimization problem with constraints into an unconstrained optimization of a single objective function. With the HmGWOGA method, five different test problems with varying Pareto fronts have been successfully solved, and the results are compared with those of NSGA-II regarding convergence towards the Pareto front and the distribution of solutions on the Pareto front. This numerical study indicates that HmGWOGA-MO is the best choice for solving a multiobjective optimization problem when convergence is the most important performance parameter.