Population Diversity Dynamics Analysis for Imbalanced Multi-objective Optimization

Various evolutionary multiobjective optimization (EMO) algorithms have been used in the field of evolutionary fuzzy systems (EFS), because EMO algorithms can easily handle multiple objective functions such as the accuracy maximization and complexity minimization for fuzzy system design. Most EMO algorithms used in EFS are Pareto dominance-based algorithms such as NSGA-II, SPEA2, and PAES. There are a few studies where other types of EMO algorithms are used in EFS. In this paper, we apply a multiobjective evolutionary algorithm based on decomposition called MOEA/D to EFS for fuzzy classifier design. MOEA/D is one of the most well-known decomposition-based EMO algorithms. The key idea is to divide a multiobjective optimization problem into a number of single-objective problems using a set of uniformly distributed weight vectors in a scalarizing function. We propose a new scalarizing function called an accuracy-oriented function (AOF) which is specialized for classifier design. We examine the effects of using AOF in MOEA/D on the search ability of our multiobjective fuzzy genetics-based machine learning (GBML). We also examine the synergy effect of MOEA/D with AOF and parallel distributed implementation of fuzzy GBML on the generalization ability.

Evolutionary algorithms have been widely used to tackle multi-objective optimization problems. Despite many successful applications there is a lack of theoretical knowledge on how and why this type of algorithm works particularly well for multi-objective problems. This special issue presents current advances to the theoretical understanding of evolutionary algorithms for multi-objective optimization. It comprises four very high quality papers which have been selected via a rigorous reviewing process. The first paper “On the Effect of Populations in Evolutionary Multi-Objective Optimisation” by Oliver Giel and Per Kristian Lehre studies the basic question whether a population is provably useful to compute the whole Pareto front of a multi-objective optimization problem. They compare restart strategies of several single-objective approaches based on one individual to a natural multi-objective approach using a population. Their analyses point out the benefits of the multi-objective evolutionary algorithm in a rigorous way. Using illustrative example functions they provide not only proven results but also a good intuition about why population-based approaches can outperform individual-based approaches in multi-objective optimization. The paper “Exploring the Runtime of an Evolutionary Algorithm for the Multi-Objective Shortest Path Problem” by Christian Horoba examines the behavior of evolutionary algorithms for the classical NP-hard multi-objective shortest path problem. The author considers an evolutionary approach based on -dominance and proves that this algorithm constitutes a fully polynomial-time randomized approximation scheme (FPRAS) for the problem. The algorithm achieves this by mimicking a problemspecific algorithm for that problem. This provides general insight into why evolutionary algorithms can perform well for problems that allow for efficient problem-specific algorithms. Hypervolume-based algorithms have become very popular in evolutionary multiobjective optimization. In the paper, “An Efficient Algorithm for Computing Hypervolume Contributions,” Karl Bringmann and Tobias Friedrich present a faster algorithm for determining the individual with the smallest hypervolume contribution in a given population. Furthermore, they study the problem of removing λ > 1 individuals with the smallest hypervolume contribution from a given population and present an elegant and fast method to carry out this task.

In recent years, evolutionary multiobjective optimization (EMO) algorithms have frequently been used for engineering problems with some conflicting objective functions to be simultaneously optimized. EMO algorithms can provide a number of Pareto optimal solutions to users. Two scenarios are considered in the practical use of EMO algorithms. One is that a decision maker selects a single solution from the obtained ones after the EMO process. The other is that a decision maker utilizes the solutions to analyze the relationship between design variables and objective functions of the corresponding problem. In this paper, we apply fuzzy genetics-based machine learning to the second scenario in order to generate if-then rule-based classifiers which represent the relationship between design variables and objective functions. We also utilize this method during the EMO process to pre-screen candidate offspring solutions. The classifier detects non-promising offspring solutions. Then, they are discarded before their fitness evaluation, so that the computation resource is used only for promising solutions. We apply this method to one engineering problem and examine its effect on the search performance of an EMO algorithm.

An efficient combination of multi-objective evolutionary optimization and reliability analysis for reliability-based design optimization of truss structures

An improved multiobjective differential evolution based on Pareto-adaptive -dominance and orthogonal design

Evolutionary Multi-objective Optimization (EMO) is a hot research direction nowadays and one of the state-of-the-art evolutionary multi-objective optimization algorithms ——NSGA-II has gain wide attention and application in many fields. Gene Expression Programming (GEP) has a powerful search capability, but falls into local optimum easily. Based on the transformed GEP, NSGA-II and the virus evolution mechanism, a new multi-objective evolutionary algorithm GEP Virus NSGA-II is proposed. With the infection operation of virus population, the diversity of the host population is increased, and it’s much easier to jump out of the local optimum. And this algorithm has got good experimental results on 9 standard test problems.

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.

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.

The hypervolume indicator has frequently been used for comparing evolutionary multi-objective optimization (EMO) algorithms. A reference point is needed for hypervolume calculation. However, its specification has not been discussed in detail from a viewpoint of fair performance comparison. A slightly worse point than the nadir point is usually used for hypervolume calculation in the EMO community. In this paper, we propose a reference point specification method for fair performance comparison of EMO algorithms. First, we discuss the relation between the reference point specification and the optimal distribution of solutions for hypervolume maximization. It is demonstrated that the optimal distribution of solutions strongly depends on the location of the reference point when a multi-objective problem has an inverted triangular Pareto front. Next, we propose a reference point specification method based on theoretical discussions on the optimal distribution of solutions. The basic idea is to specify the reference point so that a set of well-distributed solutions over the entire linear Pareto front has a large hypervolume and all solutions in such a solution set have similar hypervolume contributions. Then, we examine whether the proposed method can appropriately specify the reference point through computational experiments on various test problems. Finally, we examine the usefulness of the proposed method in a hypervolume-based EMO algorithm. Our discussions and experimental results clearly show that a slightly worse point than the nadir point is not always appropriate for performance comparison of EMO algorithms.

Evolutionary algorithm has gained a worldwide popularity among multi-objective optimization. The paper proposes a self-adaptive evolutionary algorithm (called SEA) for multi-objective optimization. In the SEA, the probability of crossover and mutation, P cand P m, are varied depending on the fitness values of the solutions. Fitness assignment of SEA realizes the twin goals of maintaining diversity in the population and guiding the population to the true Pareto Front; fitness value of individual not only depends on improved density estimation but also depends on non-dominated rank. The density estimation can keep diversity in all instances including when scalars of all objectives are much different from each other. SEA is compared against the Non-dominated Sorting Genetic Algorithm (NSGA-II) on a set of test problems introduced by the MOEA community. Simulated results show that SEA is as effective as NSGA-II in most of test functions, but when scalar of objectives are much different from each other, SEA has better distribution of non-dominated solutions.

This paper investigates a curious case of informed initialization technique to solve difficult multi-objective optimization MOP problems. The initial population was injected with non-exact i.e. approximated nadir objective vectors, which are the boundary solutions of a Pareto optimal front PF. The algorithm then successively improves those boundary solutions and utilizes them to generate non-dominated solutions targeted to the vicinity of the PF along the way. The proposed technique was ported to a standard Evolutionary Multi-objective Optimization EMO algorithm and tested on a wide variety of benchmark MOP problems. The experimental results suggest that the proposed approach is very helpful in achieving extremely fast convergence, especially if an experimenter's goal is to find a set of well distributed trade-off solutions within a fix-budgeted solution evaluations SEs. The proposed approach also ensures a more focused exploration of the underlying search space.

Multi-objective evolutionary optimization is widely utilized in industrial design. Despite the success of multi-objective evolutionary optimization algorithms in addressing complex optimization problems, research focusing on input disturbances remains limited. In many manufacturing processes, design parameters are vulnerable to random input disturbances, resulting in products that often perform less effectively than anticipated. To address this issue, we propose a novel robust multi-objective evolutionary optimization algorithm based on the concept of survival rate. The algorithm comprises two stages: the evolutionary optimization stage and the construction stage of the robust optimal front. In the former stage, we introduce the survival rate as a new optimization objective. Subsequently, we seek a robust optimal front that concurrently addresses convergence and robustness by employing a non-dominated sorting approach. Furthermore, we propose a precise sampling method and a random grouping mechanism to accurately recover solutions resilient to real noise while ensuring population’s diversity. In the latter stage, we introduce a performance measure that integrates both robustness and convergence to guide the construction of the robust optimal front. Experimental results demonstrate the superiority of the proposed algorithm in terms of both convergence and robustness compared to existing approaches under noisy conditions.

Multi-clustering via evolutionary multi-objective optimization

Evolutionary multi-objective optimization algorithms are generally employed to generate Pareto optimal solutions by exploring the search space. To enhance the performance, exploration by global search can be complemented with exploitation by combining it with local search. In this paper, we address the issues in integrating local search with global search such as: how to select individuals for local search; how deep the local search is performed; how to combine multiple objectives into single objective for local search. We introduce a Preferential Local Search mechanism to fine tune the global optimal solutions further and an adaptive weight mechanism for combining multi-objectives together. These ideas have been integrated into NSGA-II to arrive at a new memetic algorithm for solving multi-objective optimization problems. The proposed algorithm has been applied on a set of constrained and unconstrained multi-objective benchmark test suite. The performance was analyzed by computing different metrics such as Generational distance, Spread, Max spread, and HyperVolume Ratio for the test suite functions. Statistical test applied on the results obtained suggests that the proposed algorithm outperforms the state-of-art multi-objective algorithms like NSGA-II and SPEA2. To study the performance of our algorithm on a real-world application, Economic Emission Load Dispatch was also taken up for validation. The performance was studied with the help of measures such as Hypervolume and Set Coverage Metrics. Experimental results substantiate that our algorithm has the capability to solve real-world problems like Economic Emission Load Dispatch and is able to produce better solutions, when compared with NSGA-II, SPEA2, and traditional memetic algorithms with fixed local search steps.

An important issue in hypervolume-based evolutionary multi-objective optimization (EMO) algorithms is the specification of a reference point for hypervolume calculation. However, its appropriate specification has not been carefully studied in the literature. Some recent studies have pointed out the importance and difficulty of the reference point specification. Its appropriate specification depends on problem characteristics such as the Pareto front shape and the number of objectives. In this paper, the difficulty of the reference point specification in hypervolume-based EMO algorithms is circumvented by using two reference points. Instead of using only a single reference point, we propose a new hypervolume-based EMO algorithm that can effectively utilize two reference points cooperatively. Experimental results show that the proposed algorithm has good and robust performance on a wide range of test problems. In comparison to hypervolume-based EMO algorithms with only a single reference point, the proposed algorithm can find a wider and more uniformly distributed solution set. On a recently proposed real-world problem suite, the proposed algorithm shows competitive performance in comparison to state-of-the-art algorithms.