Multi-objective Optimization Test Problems Research Articles

An improved problem transformation algorithm for large-scale multi-objective optimization

Solving Large-Scale Multi-Objective Optimization Problems (LSMOPs) is the major challenge in evolution computation. Due to the large number of decision variables involved, it is not easy for existing multi-objective evolutionary algorithms to search the entire decision variable space with limited computation resources. To address the above problem, a large-scale multi-objective evolutionary algorithm based on problem transformation, called LSMOEA/PT, is presented in this paper. LSMOEA/PT uses an improved problem transformation strategy to transform LSMOPs into small-scale multi-objective problems to accelerate convergence speed and reduce computational resources. Specifically, The transformation strategy selects reference solutions to initialize a weight population by applying the opposite individual method. After optimizing the weight population, a dual line strategy is implemented to combine the weight vectors with the decision variables from the original population, thereby generating a new population. The new population is combined with the original population for environment selection. In LSMOEA/PT, a double learning swarm optimizer is introduced to speed up the convergence of the population. A uniform distribution strategy is conducted to update transformed solutions after the problem transformation process, increasing the diversity of the population. Experimental results show that LSMOEA/PT is competitive with eleven state-of-the-art algorithms on large-scale multi-objective optimization test problems with up to 2000 decision variables.

A problem transformation-based and decomposition-based evolutionary algorithm for large-scale multiobjective optimization

For large-scale multi-objective optimization problems, the search space becomes exceptionally vast, resulting in increased complexity in the search process. The search space usually contains several local optimal individuals, and the difficulty of finding the global optimal individuals increases greatly. A problem transformation-based and decomposition-based large-scale multi-objective evolutionary algorithm is proposed to solve the problem of reducing the dimensionality of the search space and the optimization of populations. Reducing the number of optimization problems in the decision space through problem transformation has lowered the complexity of multi-objective optimization problems and enhanced computational efficiency. In the decision space, employing two direction vectors adaptively guides the generation of promising individuals, thereby preventing the population from falling into local optima. Due to the dimensionality reduction of the decision space and the optimization of the individuals in the objective space, a set of optimal solutions are effectively obtained and uniformly distributed on the approximate Pareto optimal front. Experimental results show that the algorithm is highly competitive with five large-scale multi-objective evolutionary algorithms for large-scale multi-objective optimization test problems with up to 2000 decision variables.

A NEW HYBRIDIZATION FOR IMPROVING THE CONVERGENCE OF THE MOMA-PLUS METHOD

We propose in this article a hybridization of the algorithm of the MOMA-Plus method and that of the Differential Evolution method. This hybridization consists of defining a simplex around an efficient solution generated by MOMA-plus and applying the Differential Evolution algorithm to find a better solution than that obtained by MOMA-plus. The results interpreted through a performance study of the solutions obtained on multiobjective optimization test problems show that this hybridization improves the convergence of the basic MOMA-plus algorithm. Moreover, a better complexity than that of basic MOMA-plus is obtained.

A classification surrogate-assisted multi-objective evolutionary algorithm for expensive optimization

Surrogate-assisted multi-objective evolutionary algorithms (SAMOEAs) have been developed for solving expensive optimization problems. According to the roles that the surrogate models play in SAMOEAs, they can be divided into two categories: prediction-based and classification-based algorithms. Though prediction-based SAMOEAs are the mainstream methods, classification-based ones are gaining their fast developments. In this article, a classification surrogate-assisted multi-objective evolutionary algorithm (CSA-MOEA) is proposed for expensive optimization. The algorithm adopts a classification tree as the surrogate model to predict promising offsprings, which may be non-dominated solutions with good convergence. Then based on two effective infilling strategies, some of these promising individuals are added to the sample archive. By repeating the above steps iteratively, valuable solutions can be obtained. To evaluate the performance of CSA-MOEA, it is compared with several state-of-the-art surrogate-assisted evolutionary algorithms on three sets of multi-objective optimization test problems and an engineering shape optimization problem. The experimental results demonstrate the competitiveness of CSA-MOEA.

Large-Scale Evolutionary Multiobjective Optimization Assisted by Directed Sampling

It is particularly challenging for evolutionary algorithms to quickly converge to the Pareto front in large-scale multiobjective optimization. To tackle this problem, this article proposes a large-scale multiobjective evolutionary algorithm assisted by some selected individuals generated by directed sampling (DS). At each generation, a set of individuals closer to the ideal point is chosen for performing a DS in the decision space, and those nondominated ones of the sampled solutions are used to assist the reproduction to improve the convergence in evolutionary large-scale multiobjective optimization. In addition, elitist nondominated sorting is adopted complementarily for environmental selection with a reference vector-based method in order to maintain diversity of the population. Our experimental results show that the proposed algorithm is highly competitive on large-scale multiobjective optimization test problems with up to 5000 decision variables compared to five state-of-the-art multiobjective evolutionary algorithms.

Epsilon NSGAIII (Ɛ-NSGAIII) for Multi-objective Problems

The Non-dominated Sorted Genetic Algorithm III (NSGAIII) is known as an optimization algorithm with good efficiency and reliability in solving multi-objective optimization problems. However, in some cases, NSGAIII has been reported to have difficulties in maintaining diversity in the solutions for multi-objective optimization. In this study, we apply epsilon dominance (Ɛ-dominance) archiving to NSGAIII to enhance the algorithm’s efficiency and reliability. Ɛ-dominance NSGAIII (Ɛ-NSGAIII) was tested on standard evolutionary multi-objective optimization test problems. The results show that introducing Ɛ-dominance archiving can enhance the performance of the algorithm. Furthermore, the performance of Ɛ-NSGAIII was also compared to the original NSGAIII, Ɛ-NSGAII, and Ɛ-MOEA for a multi-objective Tour Guide Assignment Problem (TGAP).

Realistic Constrained Multiobjective Optimization Benchmark Problems From Design

Multiobjective optimization is increasingly used in engineering to design new systems and to identify design tradeoffs. Yet, design problems often have objective functions and constraints that are expensive and highly nonlinear. Combinations of these features lead to poor convergence and diversity loss with common algorithms that have not been specifically designed for constrained optimization. Constrained benchmark problems exist, but they do not necessarily represent the challenges of engineering problems. In this article, a framework to design electro-mechanical actuators, called multiobjective design of actuators (MODAct), is presented and 20 constrained multiobjective optimization test problems are derived from the framework with a specific focus on constraints. The full source code is made available to ease its use. The effects of the constraints are analyzed through their impact on the Pareto front as well as on the convergence performance. A constraint landscape analysis approach is followed and extended with three new metrics to characterize the search and objective spaces. The features of MODAct are compared to existing test suites to highlight the differences. In addition, a convergence analysis using NSGA-II, NSGA-III, and C-TAEA on MODAct and existing test suites suggests that the design problems are indeed difficult due to the constraints. In particular, the number of simultaneously violated constraints in newly generated solutions seems key in understanding the convergence challenges. Thus, MODAct offers an efficient framework to analyze and handle constraints in future optimization algorithm design.

A Multiobjective Evolutionary Algorithm for Finding Knee Regions Using Two Localized Dominance Relationships

In preference-based optimization, knee points are considered the naturally preferred tradeoff solutions, especially when the decision maker has little a priori knowledge about the problem to be solved. However, identifying all convex knee regions of a Pareto front remains extremely challenging, in particular in a high-dimensional objective space. This article presents a new evolutionary multiobjective algorithm for locating knee regions using two localized dominance relationships. In the environmental selection, the α-dominance is applied to each subpopulation partitioned by a set of predefined reference vectors, thereby guiding the search toward different potential knee regions while removing possible dominance resistant solutions. A knee-oriented-dominance measure making use of the extreme points is then proposed to detect knee solutions in convex knee regions and discard solutions in concave knee regions. Our experimental results demonstrate that the proposed algorithm outperforms the state-of-the-art knee identification algorithms on a majority of multiobjective optimization test problems having up to eight objectives and a hybrid electric vehicle controller design problem with seven objectives.

A Generator for Multiobjective Test Problems With Difficult-to-Approximate Pareto Front Boundaries

In some real-world applications, it has been found that the performance of multiobjective optimization evolutionary algorithms (MOEAs) may deteriorate when boundary solutions in the Pareto front (PF) are more difficult to approximate than others. Such a problem feature, referred to as difficult-to-approximate (DtA) PF boundaries, is seldom considered in existing multiobjective optimization test problems. To fill this gap and facilitate possible systematic studies, we introduce a new test problem generator. The proposed generator enables the design of test problems with controllable difficulties regarding the feature of DtA PF boundaries. Three representative MOEAs, NSGA-II, SMS-EMOA, and MOEA/D-DRA, are performed on a series of test problems created using the proposed generator. Experimental results indicate that all the three algorithms perform poorly on the new test problems. Meanwhile, a modified variant of MOEA/D-DRA, denoted as MOEA/D-DRA-UT, is validated to be more effective in dealing with these problems. Subsequently, it is concluded that the rational allocation of computational resources between different PF parts is crucial for MOEAs to handle the problems with DtA PF boundaries.

Benchmark Problems and Performance Indicators for Search of Knee Points in Multiobjective Optimization

In multiobjective optimization, it is nontrivial for decision makers to articulate preferences without a priori knowledge, which is particularly true when the number of objectives becomes large. Depending on the shape of the Pareto front, optimal solutions such as knee points may be of interest. Although several multi- and many-objective optimization test suites have been proposed, little work has been reported focusing on designing multiobjective problems whose Pareto front contains complex knee regions. Likewise, few performance indicators dedicated to evaluate an algorithm's ability of accurately locating all knee points in high-dimensional objective space have been suggested. This paper proposes a set of multiobjective optimization test problems whose Pareto front consists of complex knee regions, aiming to assess the capability of evolutionary algorithms to accurately identify all knee points. Various features related to knee points have been taken into account in designing the test problems, including symmetry, differentiability, and degeneration. These features are also combined with other challenges in solving the optimization problems, such as multimodality, linkage between decision variables, nonuniformity, and scalability of the Pareto front. The proposed test problems are scalable to both decision and objective spaces. Accordingly, new performance indicators are suggested for evaluating the capability of optimization algorithms in locating the knee points. The proposed test problems, together with the performance indicators, offer a new means to develop and assess preference-based evolutionary algorithms for solving multi- and many-objective optimization problems.

On Benchmark Problems and Metrics for Decision Space Performance Analysis in Multi-Objective Optimization

A number of benchmark problems exist for evaluating multi-objective evolutionary algorithms (MOEAs) in the objective space. However, the decision space performance analysis is a recent and relatively less explored topic in evolutionary multi-objective optimization research. Among other implications, such analysis can lead to designing more realistic test problems, gaining better understanding about optimal and robust design areas, and design and evaluation of knowledge-based optimization algorithms. This paper complements the existing research in this area and proposes a new method to generate multi-objective optimization test problems with clustered Pareto sets in hyper-rectangular defined areas of decision space. The test problem is parametrized to control number of decision variables, number and position of optimal areas in the decision space and modality of fitness landscape. Three leading MOEAs, including NSGA-II, NSGA-III, and MOEA/D, are evaluated on a number of problem instances with varying characteristics. A new metric is proposed that measures the performance of algorithms in terms of their coverage of the optimal areas in the decision space. The empirical analysis presented in this research shows that the decision space performance may not necessarily be reflective of the objective space performance and that all algorithms are sensitive to population size parameter for the new test problems.

A Multiobjective Evolutionary Algorithm Using Gaussian Process-Based Inverse Modeling

To approximate the Pareto front, most existing multiobjective evolutionary algorithms store the nondominated solutions found so far in the population or in an external archive during the search. Such algorithms often require a high degree of diversity of the stored solutions and only a limited number of solutions can be achieved. By contrast, model-based algorithms can alleviate the requirement on solution diversity and in principle, as many solutions as needed can be generated. This paper proposes a new model-based method for representing and searching nondominated solutions. The main idea is to construct Gaussian process-based inverse models that map all found nondominated solutions from the objective space to the decision space. These inverse models are then used to create offspring by sampling the objective space. To facilitate inverse modeling, the multivariate inverse function is decomposed into a group of univariate functions, where the number of inverse models is reduced using a random grouping technique. Extensive empirical simulations demonstrate that the proposed algorithm exhibits robust search performance on a variety of medium to high dimensional multiobjective optimization test problems. Additional nondominated solutions are generated a posteriori using the constructed models to increase the density of solutions in the preferred regions at a low computational cost.

Multi-objective genetic algorithms: problem difficulties and construction of test problems.

In this paper, we study the problem features that may cause a multi-objective genetic algorithm (GA) difficulty in converging to the true Pareto-optimal front. Identification of such features helps us develop difficult test problems for multi-objective optimization. Multi-objective test problems are constructed from single-objective optimization problems, thereby allowing known difficult features of single-objective problems (such as multi-modality, isolation, or deception) to be directly transferred to the corresponding multi-objective problem. In addition, test problems having features specific to multi-objective optimization are also constructed. More importantly, these difficult test problems will enable researchers to test their algorithms for specific aspects of multi-objective optimization.