Complicated Pareto Sets Research Articles

Multi-objective evolutionary algorithm with evolutionary-status-driven environmental selection

The hardly dominated boundary (HDB) is a common feature of multi-objective optimization problems (MOPs). Previous studies have proposed several multi-objective evolutionary algorithms (MOEAs) to deal with the problem characterized by HDBs. Nevertheless, these methods lack consideration of the evolutionary status, potentially resulting in low efficiency when addressing problems that entail a confluence of HDB and other intricate characteristics. This paper proposes an MOEA with evolutionary-status-driven environmental selection (MOEA-ESD) to address such an issue. Specifically, in each generation, the current and historical information are used to evaluate the evolutionary status. Based on the estimated evolutionary status, the proposed MOEA-ESD algorithm adaptively uses three types of environmental selection: the traditional environmental selection (TES) based on Pareto-dominance and crowding distance, a convergence-first environmental selection (CFES) based on Gaussian mixture model clustering, and a diversity-first environmental selection (DFES) based on outlier detection and extreme solutions. In this way, inferior solutions situated on the HDB can be effectively eliminated, thereby facilitating the convergence of the population while concurrently preserving a commendable degree of diversity. Moreover, a set of new benchmark problems with different objective magnitudes and complicated Pareto sets is developed to enrich the features of HDB-MOPs and to verify the algorithm's performance. Our experimental results on 22 HDB-MOPs show the promising performance of the proposed algorithm. The source code of MOEA-ESD is available at https://github.com/CIAM-Group/EvolutionaryAlgorithm_Codes/tree/main/MOEA-ESD.

Manifold Learning-Inspired Mating Restriction for Evolutionary Multiobjective Optimization With Complicated Pareto Sets.

Under certain smoothness assumptions, the Pareto set of a continuous multiobjective optimization problem is a piecewise continuous manifold in the decision space, which can be derived from the Karush-Kuhn-Tucker condition. Despite that a number of multiobjective evolutionary algorithms (MOEAs) have been proposed, their performance on multiobjective optimization problems with complicated Pareto sets (MOP-cPS) is still unsatisfying. In this article, we adopt the concept of manifold and propose a manifold learning-inspired mating strategy to enhance the diversity maintenance in MOEAs for solving MOP-cPS efficiently. In the proposed strategy, all of the individuals are first clustered into different manifolds according to their distribution in the objective space, and then the mating reproduction is restricted among individuals in the same manifold. Moreover, we embed the proposed mating strategy in three representative MOEAs and compare the embedded MOEAs with their original versions using the assortative genetic operators on a variety of MOP-cPS. The experimental results demonstrate the significant performance improvements benefitting from the proposed mating restriction strategy.

A set of new multi- and many-objective test problems for continuous optimization and a comprehensive experimental evaluation

Multi- and many-objective optimization problems have wide applications in the real world, and they have received growing attention from the evolutionary computation community. To promote the algorithm development in this area, numerous studies have been devoted to designing multi- and many-objective test problems. Most of these studies focus on handling complicated Pareto fronts (PFs), and the impact of the Pareto sets (PSs) has not been well-studied, although complicated PSs are prevalent in the real world. This paper presents a set of scalable test problems according to a new principle, which considers the geometrical properties of both PF and PS. A position function with a spherical form is proposed to introduce non-linear variable dependences to the PS, so as to simulate the variable dependencies in the real-world problems. According to the proposed principle, the first m (i.e., the number of objectives) decision variables are used to form the surface of a unit hypersphere, while the rest variables are designed to optimize a certain distance function. A set of test problems are generated by the proposed principle, which are then used to investigate six representative algorithms. The experimental results indicate that the proposed test problems pose considerable difficulties to existing algorithms, calling for the need for designing new algorithms capable of handling complicated PF and PS simultaneously.

Comparison between MOEA/D and NSGA-III on a set of novel many and multi-objective benchmark problems with challenging difficulties

Abstract Currently, evolutionary multiobjective optimization (EMO) algorithms have been successfully used to find a good approximation of many-objective optimization problems (MaOPs). To measure the performance of EMO algorithms, many benchmark multiobjective test problems have been constructed. Among them, DTLZ and WFG are two representative test suites with the scalability to the number of variables and objectives. It should be pointed out that MaOPs can be more challenging if they are involved with difficult problem features, such as objective scalability, complicated Pareto set, bias, disconnection, or degeneracy. In this paper, a set of ten new test problems with above-mentioned difficulties are constructed. Some experimental results on these test problems found by two popular EMO algorithms, i.e., MOEA/D and NSGA-III, are reported and analyzed. Moreover, the performance of these two EMO algorithms with different population sizes on these test problems are also studied.

An Individual and Model-Based Offspring Generation Strategy for Evolutionary Multiobjective Optimization

According to the Karush–Kuhn–Tucker condition, the Pareto set (PS) of a continuous $m$ -objective optimization problem is a continuous ( $m-1$ ) $-D$ piecewise manifold. Based on this regularity property, the ratio of the sum of the first ( $m-1$ ) largest eigenvalue of the population’s covariance matrix to the sum of the whole eigenvalue can be employed to illustrate the degree of convergence of the population. This paper proposes a new algorithm, named DE/RM-MEDA, which hybridizes differential evolution (DE) and estimation of distribution algorithm (EDA) for multiobjective optimization problems (MOPs) with the complicated PS. In the proposed algorithm, EDA extracts the population distribution information to sample new trial solutions by establishing a probability model, while DE uses the individual information to create others new individuals through the mutation and crossover operators. At each generation, the number of new solutions generated by the two operators is adjusted by the above-defined ratio. The proposed algorithm is validated on nine tec09 problems. The sensitivity and the scalability have also been experimentally investigated in this paper. The comparison results between DE/RM-MEDA and the other two state-of-the-art evolutionary algorithms, namely NSGA-II-DE and RM-MEDA, show that the proposed algorithm is highly competitive algorithms for solving MOPs with complicated PSs in terms of convergence and diversity metrics.

Variants for the logarithmic-quadratic proximal point scalarization method for multiobjective programming

We consider the proximal point method for solving unconstrained multiobjective programming problems including two families of real convex functions, one of them defined on the positive orthant and used for modifying a variant of the logarithm-quadratic regularization introduced recently and the other for defining a family of scalar representations based on 0-coercive convex functions. We show convergent results, in particular, each limit point of the sequence generated by the method is a weak Pareto solution. Numerical results over fourteen test problems are shown, some of them with complicated pareto sets.

MOEA3D: a MOEA based on dominance and decomposition with probability distribution model

In multi-objective evolutionary optimization, maintaining a good balance between convergence and diversity is particularly crucial to decision makers, especially when tackling problems with complicated Pareto sets. According to the analysis of dominance-based and decomposition-based selection mechanisms in multi-objective evolutionary algorithms, a multi-objective evolutionary algorithm based on the combination of local non-dominated rank and global decomposition is presented. The Gauss distribution model and differential evolution based on history information are employed as evolutionary operators. Various comparative experiments are conducted on 19 unconstraint test MOPs, and our empirical results validate the effectiveness and competitiveness of our proposed algorithm in solving MOPs of different types.

Matching-Based Selection With Incomplete Lists for Decomposition Multiobjective Optimization

The balance between convergence and diversity is a key issue of evolutionary multi-objective optimization. The recently proposed stable matching-based selection provides a new perspective to handle this balance under the framework of decomposition multi-objective optimization. In particular, the stable matching between subproblems and solutions, which achieves an equilibrium between their mutual preferences, implicitly strikes a balance between the convergence and diversity. Nevertheless, the original stable matching model has a high risk of matching a solution with a unfavorable subproblem which finally leads to an imbalanced selection result. In this paper, we propose an adaptive two-level stable matching-based selection for decomposition multi-objective optimization. Specifically, borrowing the idea of stable matching with incomplete lists, we match each solution with one of its favorite subproblems by restricting the length of its preference list during the first-level stable matching. During the second-level stable matching, the remaining subproblems are thereafter matched with their favorite solutions according to the classic stable matching model. In particular, we develop an adaptive mechanism to automatically set the length of preference list for each solution according to its local competitiveness. The performance of our proposed method is validated and compared with several state-of-the-art evolutionary multi-objective optimization algorithms on 62 benchmark problem instances. Empirical results fully demonstrate the competitive performance of our proposed method on problems with complicated Pareto sets and those with more than three objectives.

Objective Extraction for Many-Objective Optimization Problems: Algorithm and Test Problems

For many-objective optimization problems (MaOPs), in which the number of objectives is greater than three, the performance of most existing evolutionary multi-objective optimization algorithms generally deteriorates over the number of objectives. As some MaOPs may have redundant or correlated objectives, it is desirable to reduce the number of the objectives in such circumstances. However, the Pareto solution of the reduced MaOP obtained by most of the existing objective reduction methods, based on objective selection, may not be the Pareto solution of the original MaOP. In this paper, we propose an objective extraction method (OEM) for MaOPs. It formulates the reduced objective as a linear combination of the original objectives to maximize the conflict between the reduced objectives. Subsequently, the Pareto solution of the reduced MaOP obtained by the proposed algorithm is that of the original MaOP, and the proposed algorithm can thus preserve the dominance structure as much as possible. Moreover, we propose a novel framework that features both simple and complicated Pareto set shapes for many-objective test problems with an arbitrary number of essential objectives. Within this framework, we can control the importance of essential objectives. As there is no direct performance metric for the objective reduction algorithms on the benchmarks, we present a new metric that features simplicity and usability for the objective reduction algorithms. We compare the proposed OEM with three objective reduction methods, i.e., REDGA, L-PCA, and NL-MVU-PCA, on the proposed test problems and benchmark DTLZ5 with different numbers of objectives and essential objectives. Our numerical studies show the effectiveness and robustness of the proposed approach.

Multi-objective artificial bee algorithm based on decomposition by PBI method

This paper proposes a multi-objective artificial bee colony (MOABC) algorithm based on decomposition by penalty-based boundary intersection (PBI) method. It decomposes a MOP into a number of single-objective problems. The objective of every single-objective problem is based on the distance and angle from the current solution to their own ideal point so as to obtain the good convergence and distribution of the optimal solutions. In this way, the aggregate function is proposed by PBI method. Then the three phases of Artificial Bee Colony (ABC) algorithm are adopted, which are the employed bees sending, the unemployed bees following, and the scout bees converting. Among these phases, the improvement degree of every aggregate function is proposed as the fitness function, which could overcome the two shortcomings in the weighted sum function usually applied in the conventional MOABC. Besides, Boltzmann selection mechanism is used to obtain the probability of unemployed bees following the employed bees so that the selection pressure of unemployed bees in the optimizing process could be adjusted dynamically. The algorithm is validated on CEC2009 problems and the problems with complicated Pareto set shapes in terms of four indicators: IGD, HV, SPR, and EPS. Experimental results show that our proposed algorithm can perform better than other state-of-the-art algorithms in the convergence and diversity, and can be considered as a promising alternative to solve MOPs.

Using an outward selective pressure for improving the search quality of the MOEA/D algorithm

In optimization problems it is often necessary to perform an optimization based on more than one objective. The goal of the multiobjective optimization is usually to find an approximation of the Pareto front which contains solutions that represent the best possible trade-offs between the objectives. In a multiobjective scenario it is important to both improve the solutions in terms of the objectives and to find a wide variety of available options. Evolutionary algorithms are often used for multiobjective optimization because they maintain a population of individuals which directly represent a set of solutions of the optimization problem. multiobjective evolutionary algorithm based on decomposition (MOEA/D) is one of the most effective multiobjective algorithms currently used in the literature. This paper investigates several methods which increase the selective pressure to the outside of the Pareto front in the case of the MOEA/D algorithm. Experiments show that by applying greater selective pressure in the outwards direction the quality of results obtained by the algorithm can be increased. The proposed methods were tested on nine test instances with complicated Pareto sets. In the tests the new methods outperformed the original MOEA/D algorithm.

Interrelationship-Based Selection for Decomposition Multiobjective Optimization.

Multiobjective evolutionary algorithm based on decomposition (MOEA/D), which bridges the traditional optimization techniques and population-based methods, has become an increasingly popular framework for evolutionary multiobjective optimization. It decomposes a multiobjective optimization problem (MOP) into a number of optimization subproblems. Each subproblem is handled by an agent in a collaborative manner. The selection of MOEA/D is a process of choosing solutions by agents. In particular, each agent has two requirements on its selected solution: one is the convergence toward the efficient front, the other is the distinction with the other agents' choices. This paper suggests addressing these two requirements by defining mutual-preferences between subproblems and solutions. Afterwards, a simple yet effective method is proposed to build an interrelationship between subproblems and solutions, based on their mutual-preferences. At each generation, this interrelationship is used as a guideline to select the elite solutions to survive as the next parents. By considering the mutual-preferences between subproblems and solutions (i.e., the two requirements of each agent), the selection operator is able to balance the convergence and diversity of the search process. Comprehensive experiments are conducted on several MOP test instances with complicated Pareto sets. Empirical results demonstrate the effectiveness and competitiveness of our proposed algorithm.

MOEA/D-SQA: a multi-objective memetic algorithm based on decomposition

A multi-objective memetic algorithm based on decomposition is proposed in this article, in which a simplified quadratic approximation (SQA) is employed as a local search operator for enhancing the performance of a multi-objective evolutionary algorithm based on decomposition (MOEA/D). The SQA is used for a fast local search and the MOEA/D is used as the global optimizer. The multi-objective memetic algorithm based on decomposition, i.e. a hybrid of the MOEA/D with the SQA (MOEA/D-SQA), is designed to balance local versus global search strategies so as to obtain a set of diverse non-dominated solutions as quickly as possible. The emphasis of this article is placed on demonstrating how this local search scheme can improve the performance of MOEA/D for multi-objective optimization. MOEA/D-SQA has been tested on a wide set of benchmark problems with complicated Pareto set shapes. Experimental results indicate that the proposed approach performs better than MOEA/D. In addition, the results obtained are very competitive when comparing MOEA/D-SQA with other state-of-the-art techniques.

A modification to MOEA/D-DE for multiobjective optimization problems with complicated Pareto sets

This paper presents an algorithm, named the uniform design multiobjective differential algorithm based on decomposition (UMODE/D), for optimizing multiobjective problems. The algorithm is a modification to the new version of MOEA/D based on differential evolution (DE), i.e., MOEA/D-DE proposed by Li and Zhang (2009) [20]. Its distinguishing features include: (1) The uniform design method is applied to generate the aggregation coefficient vectors so that the decomposed scalar optimization subproblems are uniformly distributed, and therefore the algorithm could explore uniformly the region of interest from the initial iteration; (2) The simplified quadratic approximation with three best points is employed to improve the local search ability and the accuracy of the minimum scalar aggregation function value. UMODE/D is compared with the original MOEA/D-DE and NSGA-II by solving a wide set of problems with complicated Pareto set shapes in this paper. Experimental results indicate that UMODE/D significantly outperforms MOEA/D-DE and NSGA-II on these test problems. Two sets of experiments are carried out to illustrate the efficiency of the uniform design method and the simplified quadratic approximation separately. In addition, UMODE/D is tested on CEC 2009 problems and combinatorial knapsack problems. Experimental results show that the proposed algorithm performs better than the further improved MOEA/D for almost all the CEC 2009 problems, and the results obtained are very competitive when comparing UMODE/D with some other algorithms on these multiobjective knapsack problems.

MOEA/D + uniform design: A new version of MOEA/D for optimization problems with many objectives

To extend multiobjective evolutionary algorithm based on decomposition (MOEA/D) in higher dimensional objective spaces, this paper proposes a new version of MOEA/D with uniform design, named the uniform design multiobjective evolutionary algorithm based on decomposition (UMOEA/D), and compares the proposed algorithm with MOEA/D and NSGA-II on some scalable test problems with three to five objectives. UMOEA/D adopts the uniform design method to set the aggregation coefficient vectors of the subproblems. Compared with MOEA/D, distribution of the coefficient vectors is more uniform over the design space, and the population size neither increases nonlinearly with the number of objectives nor considers a formulaic setting. The experimental results indicate that UMOEA/D outperforms MOEA/D and NSGA-II on almost all these many-objective test instances, especially on problems with higher dimensional objectives and complicated Pareto set shapes. Experimental results also show that UMOEA/D runs faster than NSGA-II for the problems used in this paper. In additional, the results obtained are very competitive when comparing UMOEA/D with some other algorithm on the multiobjective knapsack problems.

Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II

Partly due to lack of test problems, the impact of the Pareto set (PS) shapes on the performance of evolutionary algorithms has not yet attracted much attention. This paper introduces a general class of continuous multiobjective optimization test instances with arbitrary prescribed PS shapes, which could be used for studying the ability of multiobjective evolutionary algorithms for dealing with complicated PS shapes. It also proposes a new version of MOEA/D based on differential evolution (DE), i.e., MOEA/D-DE, and compares the proposed algorithm with NSGA-II with the same reproduction operators on the test instances introduced in this paper. The experimental results indicate that MOEA/D could significantly outperform NSGA-II on these test instances. It suggests that decomposition based multiobjective evolutionary algorithms are very promising in dealing with complicated PS shapes.