Multi-objective Test Problems Research Articles

Challenging test problems for multi- and many-objective optimization

In spite of the extensive studies that have been conducted regarding the construction of multi-objective test problems, researchers have mainly focused their interests on designing complicated search spaces, disregarding, in many cases, the design of the Pareto optimal fronts of the problems. In this regard, the work related to scalable multi-objective test problems—i.e., problems that can be formulated for an arbitrary number of objectives—has been much less studied. This paper introduces a new set of continuous and box-constrained multi-objective test problems which are scalable in both the number of objectives and in the number of decision variables. Each test problem included in the proposed test suite has a peculiar Pareto front different from those observed in the existing scalable multi-objective test suites. In addition to different Pareto fronts, the proposed test suite introduces features related to the search space that place obstacles that complicate exploring Pareto optimal solutions. Such features can be easily switched on and off by the user to analyze specific mechanisms of multi-objective evolutionary algorithms (MOEAs). The components used in the proposed test suite can be used as a toolkit to construct new test instances not included in this set of problems. To illustrate the use and difficulties of the proposed test suite, some experiments are presented adopting three MOEAs using selection mechanisms based on Pareto optimality, decomposition, and a performance indicator (hypervolume).

An infeasible interior-point technique to generate the nondominated set for multiobjective optimization problems

In this paper, an infeasible interior-point technique is proposed to generate the nondominated set of nonlinear multi-objective optimization problems with the help of the direction-based cone method. We derive the proposed method for both convex and nonconvex problems. In order to solve the parametric optimization problems of the cone method, the infeasible interior-point method starts with an initial iterate outside the feasible region, and then gradually reduces the primal and dual infeasibility measures and the objective function value across the iterations with the help of a merit function. Estimates of the reduction of primal and dual infeasibility parameters per iteration are given. The convergence analysis of the method and an estimate of the number of iterations to reach an ϵ-precise solution are also provided. We provide the performance of the proposed methods on a variety of convex and nonconvex multi-objective test problems. Performance comparison between the proposed method and popular existing solvers is provided with respect to two performance measures and the corresponding relative efficiency measures. The reduction of a combined infeasibility measure, as the iterations progress, on the test problems is also shown graphically.

Mobile Robot Path Planning with a Novel Multiobjective Slime Mold Algorithm

In the path planning problem of mobile robot in static complex obstacle environment, the original slime mold algorithm (SMA) has some shortcomings, such as initial solution error and easy to fall into local optimum, which leads to low accuracy of robot path planning. A novel Baldwin learning effect mobile robot Path planning with Multi‐Objective Slime Mold Algorithm (BPMOSMA) is proposed to solve the related problems. The Baldwin learning effect is introduced into to guide the development direction of slime mold in the phase of slime mold surrounding food and the k‐nearest neighbor measure is added to prune nondominated solutions to generate a well‐distributed solution set. For comparison purposes, five state‐of‐the‐art multiobjective optimization algorithms are carried on nine multiobjective test problems with kinds of types of Pareto set. The qualitative and quantitative experimental results show that BPMOSMA is very competitive performance on unconstrained multi‐objective optimization problems. BPMOSMA is finally successfully applied in the mobile robot path planning problem.

Counterintuitive Experimental Results in Evolutionary Large-Scale Multiobjective Optimization

Recently, large-scale multiobjective optimization has received increasing attention from the evolutionary multiobjective optimization (EMO) community. This has led to the emergence of a specialized research area called evolutionary large-scale multiobjective optimization (ELMO). In general, it is believed that multiobjective optimization problems become more difficult as the number of decision variables increases. However, the following two counterintuitive observations are obtained from careful examinations of recent ELMO studies. One is that experimental results on some large-scale multiobjective test problems were improved by increasing the number of decision variables. The other is that better results were obtained for some other large-scale multiobjective test problems by conventional EMO algorithms (EMOAs) than state-of-the-art ELMO algorithms (ELMOAs). These observations suggest that ELMOAs have not always been evaluated on appropriate test problems. Moreover, their performance is not always better than the performance of conventional EMOAs. In this letter, we first re-examine the performance of ELMOAs and conventional EMOAs on a wide variety of scalable multiobjective test problems. Then, counterintuitive experimental results are analyzed using the anytime performance evaluation scheme and distributions of randomly generated initial solutions. Based on the analysis, suggestions on how to handle large-scale multiobjective test problems with counterintuitive results are proposed.

A review and evaluation of multi and many-objective optimization: Methods and algorithms

Most optimization problems naturally have several objectives, usually in conflict with each other. The problems with two or three objective functions are referred to as Multi-Objective Problems (MOP). However, many real-world applications often involve four or more objectives, which are commonly recognized as many-objective optimization problems (MaOP). Multi and many-objective algorithms have a great application in engineering science. This study addresses a complete and updated review of the literature for multi and many-objective problems and discusses 32 more important algorithms in detail. Afterward, the ZDT and DLTZ benchmark problems for multi-objective test problems are reviewed. All methods have been studied under recent state-of-the-art quality measures. Moreover, we discuss the historical roots of multi-objective optimization, the motivation to use evolutionary algorithms, and the most popular techniques currently in use.

Using Dominated Solutions at Edges to the Diversity and the Uniformity of Non-dominated Solution Distributions in NSGA-II

This paper proposes a method for improving the diversity of the Pareto front and the uniformity of non-dominated solution distributions in a fast elitist non-dominated sorting genetic algorithm (NSGA-II), which is an evolutionary multi-objective optimization algorithm. Conventional NSGA-II has excellent convergence to the Pareto front, but it has been reported that for some test cases, it does not produce a more diverse solution distribution than the strength Pareto evolutionary algorithm 2 (SPEA2). In addition, selection using the crowding distance may cause a bias in the selected solution distribution. To avoid this problem, we propose a method that archives dominated solutions that may be effective in improving diversity in the conventional search process when used for genetic operations, and mates these archived solutions with non-dominated solutions at the edge of rank 1 for each objective function. We experimentally compare this approach with the conventional method on the typical ZDT suite of multi-objective test problems and a two-objective constrained knapsack problem. By evaluating the performance based on Pareto front diagrams, the number of non-dominated solutions, the maximum spread and hypervolume values, we show that the proposed method is effective at improving the diversity at both ends of the Pareto optimal front and the solution distribution.

Integral Global Optimality Conditions and an Algorithm for Multiobjective Problems

In this work, we propose integral global optimality conditions for multiobjective problems not necessarily differentiable. The integral characterization, already known for single objective problems, are extended to multiobjective problems by weighted sum and Chebyshev weighted scalarizations. Using this last scalarization, we propose an algorithm for obtaining an approximation of the weak Pareto front whose effectiveness is illustrated by solving a collection of multiobjective test problems.

Expensive Multiobjective Optimization Algorithm Based on Equivariate Component Analysis

In order to reduce the cost of candidate solution evaluation in the process of solving expensive optimization problems, an expensive multi-objective optimization algorithm based on equivalence component analysis was proposed to study the influence of decision space equivalence components on the prediction accuracy of agent models. Based on the analysis of the equivalence of decision space attributes, a limit learning network based on the equivalence components was constructed for Pareto dominance prediction among candidate solutions. A multi-objective test problem with equivalent components was selected in Pareto dominance prediction experiments, the results of which showed that the algorithm can effectively improve the accuracy of Pareto dominance prediction among candidate solutions. Successively the candidate solutions were scored with multiple ELM (Extreme Learning Machine) models, selected for evaluation and updated, and integrated into the Pareto-based multi-objective evolutionary algorithm. Through comparative experiments on the test problem, the method could achieve a better Pareto approximation solution under the limitation of a limited number of evaluations, and the goal of reducing the cost of expensive multi-objective optimization calculations.

Using following heroes operation in multi-objective differential evolution for fast convergence

In this study, a new hybrid variant of multi-objective differential evolution is developed to improve the extent and the speed of convergence by combining the ability of DE/rand/1 strategy (NSDE1) to retain only better solutions in the selection operation with the ability of Following Heroes operation from adaptive social evolution algorithm to rapidly propagate the effect of the best solutions to the entire population. Such an algorithm is quite useful for fast convergence, particularly in solving computationally intensive multi-objective optimization problems for which it is highly desirable to obtain the better quality optimal solutions in a limited number of function calculations. The performance of the algorithm is first analyzed on thirty multi-objective test problems from ZDT, DTLZ, and WFG test suites for a limited number of function calculations, and the results are compared with four well-established multi-objective optimization algorithms: real-coded elitist non-dominated sorting genetic algorithm (RNSGA-II), non-dominated sorting particle swarm optimization (NSPSO), adaptive social evolution (ASE) and non-dominated sorting differential evolution with DE/rand/1 strategy (NSDE1) and found to be superior. The analysis is further extended to model-based algorithms, Pareto efficient global optimization (ParEGO), kriging for expensive evaluation Pareto (KEEP), and surrogate optimization of computationally expensive multi-objective problems (SOCEMO) which are commonly used for computationally intensive problems. These involve an additional computational cost of evolving the surrogate models. The performance of the proposed algorithm is found to be better than these, though, it has a relatively simple algorithmic structure and does not involve any additional computational cost. The algorithm is then evaluated on computationally intensive multi-objective optimization of bulk polymerization of methyl methacrylate. The numerical results obtained for this problem show better convergence for a limited number of function calculations.

A heredity-based adaptive variation operator for reinitialization in dynamic multi-objective problems

A reinitialization approach is an effective way of generalizing a static multi-objective optimization method to a dynamic one. It is usually comprised of a prediction operator for predicting the approximate location(s) of the optimal solution(s) and a variation operator for enhancing the diversity of the reinitialized solution(s) after a change. While many recent studies have focused on prediction methods, the importance of the variation operator has usually been overlooked. This study systematically explores the effects of the accuracy of the prediction method employed as well as the frequency and severity of the change on the optimal strength of the variation used for reinitialization. Subsequently, it introduces an adaptive variation operator for dynamic multi-objective optimization which can learn the optimal variation strength on-the-fly. To develop this method, firstly, a heredity measure for evolutionary algorithms is formulated to quantify the contribution of each reinitialized solution to the optimization process by measuring the presence of its traits in the final population. Some carefully designed descriptive simulations are performed to explore the capability of the proposed method to learn the optimal variation strength and its sensitivity to the change severity, initial variation strength, and accuracy of the employed prediction method. Finally, the performance of this variation operator on 42 dynamic multi-objective test problems is compared with those of five other popular ones, with numerical comparisons revealing its superior learning capability.

A new multi-objective optimization algorithm combined with opposition-based learning

The optimization problems are divided into a single objective and multi-objective. Single objective optimization has only one objective function; whereas, multi-objective optimization has multiple objective functions that generate the Pareto set; therefore, solving a multi-objective problem is a challenging problem. This paper presents a new multi-objective optimization method (called MWDEO) based on improved whale optimization algorithm (WOA) by combining the differential evolution (DE) algorithm and the opposition-based learning (OBL). The MWDEO uses the WOA to perform a global exploration, whereas DE is used to exploit the search space; while the OBL is applied to improve the exploration and exploitation by generating the opposite values. The proposed algorithm is evaluated using 32 multi-objective test problems besides a set of benchmark problems of CEC’2017. The experimental results are compared with nine state-of-the-art multi-objective methods. The analysis of the results showed that the proposed MWDEO outperformed all other algorithms in most of the test problems which indicates that the proposed MWDEO is competitive and effective in solving different types of multi-objective problems.

Performances Assessment of MOMA-Plus Method on Multiobjective Optimization Problems

This work is devoted to evaluate the performances of the MOMA-plus method in solving multiobjective optimization problems. This assessment is doing on the complexity of its algorithm, the convergence and the diversity of solutions in relation to the Pareto front. All these parameters were evaluated on non-linear multiobjective test problems and obtained solutions are compared with those provided by the NSGA-II method. This comparative study made it possible tohighlight the performances of MOMA-plus method for solving non-linear multiobjective problems.

Performances Assessment of MOMA-Plus Method on Multiobjective Optimization Problems

This work is devoted to evaluate the performances of the MOMA-plus method in solving multiobjective optimization problems. This assessment is doing on the complexity of its algorithm, the convergence and the diversity of solutions in relation to the Pareto front. All these parameters were evaluated on non-linear multiobjective test problems and obtained solutions are compared with those provided by the NSGA-II method. This comparative study made it possible tohighlight the performances of MOMA-plus method for solving non-linear multiobjective problems.

Multi-Objective Particle Swarm Optimization Based on Gaussian Sampling

This paper proposes a multi-objective particle swarm optimization algorithm based on Gaussian sampling (GS-MOPSO) to locate multiple Pareto optimal solutions for solving multimodal multi-objective problems. In the proposed method, the Gaussian sampling mechanism is used to form multiple neighborhoods by learning from optimal information of particles. And particles search their own neighborhoods to obtain more optimal solutions in the decision space. Moreover, an external archive maintenance strategy is proposed which allows the algorithm to maintain an archive containing better distribution and diversity of solutions. Meanwhile, nine new multimodal multi-objective test problems are designed to evaluate the performance of algorithms. The performance of GS-MOPSO is compared with twelve state-of-the-art multi-objective optimization algorithms on forty test problems. The experimental results show that the proposed algorithm is able to handle the multimodal multi-objective problems in terms of finding more and well-distributed Pareto solutions. In addition, the effectiveness of the proposed algorithm is further demonstrated in a real-world problem.

Variable-Length Pareto Optimization via Decomposition-Based Evolutionary Multiobjective Algorithm

Optimization problems with variable-length decision space are a class of challenging optimization problems derived from some real-world applications, such as the composite laminate stacking problem and the sensor coverage problem. Unlike other optimization problems, the solutions in these problems might be represented as the vectors with different variable size (i.e., dimensionality). So far, some research efforts have been done on the use of evolutionary algorithms (EAs) for solving single objective variable-length optimization problems. In fact, the variable-length problem difficulty can also exist in multiobjective optimization. However, such challenging problems have not yet gained much attention in the area of evolutionary multiobjective optimization. To facilitate the research on the variable-length Pareto optimization, we first suggest a systematic toolkit for constructing benchmark multiobjective test problems with variable-length feature in this paper. Then, we also propose a variable-length multiobjective EA based on a two-level decomposition strategy, which decomposes a multiobjective optimization problem in terms of the penalty boundary intersection search directions and the dimensionality of variables. The performance of our proposed algorithm and the other three state-of-the-art algorithms on these problems are compared. To further show the effectiveness of our proposed algorithm, some experimental results on a bi-objective laminate stacking optimization problem are also reported and analyzed.

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.

A niching indicator-based multi-modal many-objective optimizer

Multi-modal multi-objective optimization is to locate (almost) equivalent Pareto optimal solutions as many as possible. Some evolutionary algorithms for multi-modal multi-objective optimization have been proposed in the literature. However, there is no efficient method for multi-modal many-objective optimization, where the number of objectives is more than three. To address this issue, this paper proposes a niching indicator-based multi-modal multi- and many-objective optimization algorithm. In the proposed method, the fitness calculation is performed among a child and its closest individuals in the solution space to maintain the diversity. The performance of the proposed method is evaluated on multi-modal multi-objective test problems with up to 15 objectives. Results show that the proposed method can handle a large number of objectives and find a good approximation of multiple equivalent Pareto optimal solutions. The results also show that the proposed method performs significantly better than eight multi-objective evolutionary algorithms.

Difficulty Adjustable and Scalable Constrained Multiobjective Test Problem Toolkit.

Multiobjective evolutionary algorithms (MOEAs) have progressed significantly in recent decades, but most of them are designed to solve unconstrained multiobjective optimization problems. In fact, many real-world multiobjective problems contain a number of constraints. To promote research on constrained multiobjective optimization, we first propose a problem classification scheme with three primary types of difficulty, which reflect various types of challenges presented by real-world optimization problems, in order to characterize the constraint functions in constrained multiobjective optimization problems (CMOPs). These are feasibility-hardness, convergence-hardness, and diversity-hardness. We then develop a general toolkit to construct difficulty adjustable and scalable CMOPs (DAS-CMOPs, or DAS-CMaOPs when the number of objectives is greater than three) with three types of parameterized constraint functions developed to capture the three proposed types of difficulty. In fact, the combination of the three primary constraint functions with different parameters allows the construction of a large variety of CMOPs, with difficulty that can be defined by a triplet, with each of its parameters specifying the level of one of the types of primary difficulty. Furthermore, the number of objectives in this toolkit can be scaled beyond three. Based on this toolkit, we suggest nine difficulty adjustable and scalable CMOPs and nine CMaOPs, to be called DAS-CMOP1-9 and DAS-CMaOP1-9, respectively. To evaluate the proposed test problems, two popular CMOEAs-MOEA/D-CDP (MOEA/D with constraint dominance principle) and NSGA-II-CDP (NSGA-II with constraint dominance principle) and two popular constrained many-objective evolutionary algorithms (CMaOEAs)-C-MOEA/DD and C-NSGA-III-are used to compare performance on DAS-CMOP1-9 and DAS-CMaOP1-9 with a variety of difficulty triplets, respectively. The experimental results reveal that mechanisms in MOEA/D-CDP may be more effective in solving convergence-hard DAS-CMOPs, while mechanisms of NSGA-II-CDP may be more effective in solving DAS-CMOPs with simultaneous diversity-, feasibility-, and convergence-hardness. Mechanisms in C-NSGA-III may be more effective in solving feasibility-hard CMaOPs, while mechanisms of C-MOEA/DD may be more effective in solving CMaOPs with convergence-hardness. In addition, none of them can solve these problems efficiently, which stimulates us to continue to develop new CMOEAs and CMaOEAs to solve the suggested DAS-CMOPs and DAS-CMaOPs.

On Scalable Multiobjective Test Problems With Hardly Dominated Boundaries

The DTLZ1-DTLZ4 problems are by far one of the most commonly used test problems in the validation and comparison of multiobjective optimization evolutionary algorithms (MOEAs). However, very recently, it has been pointed out that they have the following two special characteristics: 1) the regularly oriented Pareto front shape and 2) the single distance function. As a modification of them, this paper presents a new set of test problems mDTLZ1-mDTLZ4 to avoid the two special characteristics. Using these new test problems, we investigate the performance of three representative multiobjective evolutionary algorithms NSGA-II, MOEA/D-Tch, and SMS-EMOA. Experimental results indicate that the performance of NSGA-II and MOEA/D-Tch deteriorates on mDTLZ1-mDTLZ4. Subsequently, our analysis reveals that there exist the hardly dominated boundaries in each of mDTLZ1-mDTLZ4, which hinder the approximation of Pareto dominance-based algorithms and Tchebycheff-decomposition-based algorithms. Furthermore, we summarize that the hardly dominated boundary should be an often encountered problem feature in multiobjective optimization. Last but not least, we point out and validate some coping strategies for dominance-based algorithms and decomposition-based algorithms to overcome the challenges caused by the hardly dominated boundary.

A novel scalable test problem suite for multimodal multiobjective optimization

Abstract This paper proposes a novel scalable multimodal multiobjective test problem suite. The proposed test problems have various properties, such as presence of local Pareto optimal set (PS), scalable number of PSs, nonuniformly distributed PSs, discrete Pareto front (PF), and scalable number of variables and objectives. All of the test problems proposed in this paper are continuous optimization problems. Therefore, they can be used to measure different capacities of multimodal multiobjective continuous optimization algorithms. Moreover, a landscape visualization method for multiobjective problems is proposed to show the properties of the multimodal multiobjective test problems. Based on the landscapes, the characteristics of these problems are analyzed and characterized. Furthermore, the existing multimodal multiobjective optimization algorithms and several popular multiobjective algorithms are tested and compared with the novel test problem suite. Then, a discussion on the desired properties of multimodal multiobjective optimization algorithms and future works on multimodal multiobjective optimization are presented.