Use of cooperative coevolution for solving large scale multiobjective optimization problems

Multi-objective evolutionary algorithms (MOEAs) of the state of the art are created with the only purpose of dealing with the number of objective functions in a multi-objective optimization problem (MOP) and treat the decision variables of a MOP as a whole. However, when dealing with MOPs with a large number of decision variables (more than 100) their efficacy decreases as the number of decision variables of the MOP increases. On the other hand, problem decomposition, in terms of decision variables, has been found to be extremely efficient and effective for solving large scale optimization problems. Nevertheless, most of the currently available approaches for large scale optimization rely on models based on cooperative coevolution or linkage learning methods that use multiple subpopulations or preliminary analysis, respectively, which is computationally expensive (in terms of function evaluations) when used within MOEAs. In this work, we study the effect of what we call operational decomposition, which is a novel framework based on coevolutionary concepts to apply MOEAs’s crossover operator without adding any extra cost. We investigate the improvements that NSGA-III can achieve when combined with our proposed coevolutionary operators. This new scheme is capable of improving efficiency of a MOEA when dealing with large scale MOPs having from 200 up to 1200 decision variables.

A fast sampling based evolutionary algorithm for million-dimensional multiobjective optimization

This paper presents a random-based dynamic grouping strategy (RDG) for cooperative coevolution to deal with large scale multi-objective optimization problems (MOPs) by decomposing the whole dimension into several groups of variables with an equal size. First, a decomposer pool containing different group sizes is designed. Then, a group size is dynamically selected with probability in the evolution process. The probability of each group size in the pool is computed based on the historical performance measured by C-metric, a common metric in multi-objective optimization. Under the selected group size, random grouping is executed to decompose the whole dimension into groups. Through this, both the group size and the group components are dynamic. Finally, combining RDG with a traditional and famous multi-objective evolutionary algorithm (MOEA) named MOEA/D, we develop MOEA/D-RDG to cope with large scale MOPs. The efficacy of the proposed MOEA/D-RDG is verified on two sets of MOPs (UF1-UF10 and WFG1-WFG9) through comparing with two MOEA/D variants.

Many engineering optimization problems are typically multi-objective in their natures and multidisciplinary with a large number of decision variables. Furthermore, Pareto dominance loses its effectiveness in such situations. Thus, developing a robust optimization algorithm undoubtedly becomes a true challenge. This paper proposes a multi-objective orthogonal opposition-based crow search algorithm (M2O-CSA) for solving large-scale multi-objective optimization problems (LSMOPs). In the M2O-CSA, a multi-orthogonal opposition strategy is employed to mitigate the conflicts among the convergence and distribution of solutions. First, two individuals are randomly chosen to undergo the crossover stage and then orthogonal array is presented to obtain nine individuals. Then individuals are used in the opposition stage to improve the diversity of solutions. The effectiveness of the proposed M2O-CSA is investigated by implementing it on different dimensions of multi-objective optimization problems (MOPs). The Pareto front solutions of these MOPs have various characteristics such as convex, non-convex and discrete. It is also applied to solve multi-objective design applications with distinctive features such as four bar truss (FBT) design, welded beam (WB) deign, disk brake (DB) design, and speed reduced (SR) design, where they involve different characteristics. In this context, a new decision making tool based on multi-objective optimization on the basis of ratio analysis (MOORA) technique is employed to help the designer for extracting the operating point as the best compromise or satisfactory solution to execute the candidate engineering design. Simulation results affirm that the proposed M2O-CSA works efficiently and effectively.

Decomposition is an effective and popular strategy used by evolutionary algorithms to solve multiobjective optimization problems (MOPs). It can reduce the difficulty of directly solving MOPs, increase the diversity of the obtained solutions, and facilitate parallel computing. However, with the increase of the number of decision variables, the performance of multiobjective evolutionary algorithms (MOEAs) often deteriorates sharply. The advantages of the decomposition strategy are not fully exploited when solving such large-scale MOPs (LSMOPs). To this end, this paper proposes a parallel MOEA based on two-space decomposition (TSD) to solve LSMOPs. The main idea of the algorithm is to decompose the objective space and decision space into multiple subspaces, each of which is expected to contain some complete Pareto-optimal solutions, and then use multiple populations to conduct parallel searches in these subspaces. Specifically, the objective space decomposition approach adopts the traditional reference vector-based method, whereas the decision space decomposition approach adopts the proposed method based on a diversity design subspace (DDS). The algorithm uses a message passing interface (MPI) to implement its parallel environment. The experimental results demonstrate the effectiveness of the proposed DDS-based method. Compared with the state-of-the-art MOEAs in solving various benchmark and real-world problems, the proposed algorithm exhibits advantages in terms of general performance and computational efficiency.

The competitive swarm optimizer (CSO) classifies swarm particles into loser and winner particles and then uses the winner particles to efficiently guide the search of the loser particles. This approach has very promising performance in solving large-scale multiobjective optimization problems (LMOPs). However, most studies of CSOs ignore the evolution of the winner particles, although their quality is very important for the final optimization performance. Aiming to fill this research gap, this article proposes a new neural net-enhanced CSO for solving LMOPs, called NN-CSO, which not only guides the loser particles via the original CSO strategy, but also applies our trained neural network (NN) model to evolve winner particles. First, the swarm particles are classified into winner and loser particles by the pairwise competition. Then, the loser particles and winner particles are, respectively, treated as the input and desired output to train the NN model, which tries to learn promising evolutionary dynamics by driving the loser particles toward the winners. Finally, when model training is complete, the winner particles are evolved by the well-trained NN model, while the loser particles are still guided by the winner particles to maintain the search pattern of CSOs. To evaluate the performance of our designed NN-CSO, several LMOPs with up to ten objectives and 1000 decision variables are adopted, and the experimental results show that our designed NN model can significantly improve the performance of CSOs and shows some advantages over several state-of-the-art large-scale multiobjective evolutionary algorithms as well as over model-based evolutionary algorithms.

Addressing the challenge of efficiently handling high-dimensional search spaces in solving large-scale multiobjective optimization problems (LMOPs) becomes an emerging research topic in evolutionary computation. In response, this paper proposes a new evolutionary optimizer with a tactic of autoencoder-based problem transformation (APT). The APT involves creating an autoencoder to learn the relative importance of each variable by competitively reconstructing the dominated and non-dominated solutions. Using the learned importance, all variables are divided into multiple groups without consuming any function evaluations. The number of groups dynamically increases according to the population's evolutionary status. Each variable group has an associated autoencoder, transforming the search space into an adaptable small-scale representation space. Thus, the search process occurs within these dynamic representation spaces, leading to effective production of offspring solutions. To assess the effectiveness of APT, extensive testing is performed on benchmark suites and real-world LMOPs, encompassing variable sizes ranging from 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> . The comparative results demonstrate the advantages of our proposed optimizer in solving these LMOPs with a limited budget of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> function evaluations.

Multi-objective optimization problems with large-scale decision variables, known as LSMOPs, are characterized by their large-scale search space and multiple conflicting objectives to be optimized. They have been involved in many emergent real-world applications, such as feature and instance selection, data clustering, adversarial attack, vehicle routing problem, and others. In this work, we propose a new teaching-learning-based optimization algorithm to effectively tackle large-scale multi-objective optimization problems. The proposed approach ensures both the diversity of solutions, requested for high dimensionality of LSMOPs, and the balance between the exploitation and exploration during the optimization process. The experimental studies conducted on 54 instances of a large-scale multi-objective optimization problems test suite and the comparisons against well-known and state-of-the-art algorithms have shown the superiority of the proposed algorithm in terms of Pareto front approximation and computation time by using limited evaluation budget.

This article is about a comparison of different measures for determining the robustness or reliability of electric machine designs in the presence of inevitable tolerances. The selected criteria shall be suitable for concurrent evaluation in the course of solving state-of-the-art large scale multi-objective optimization problems. In the past, besides particularly customized criteria, mainly gradient based measures, worst case information, or standard deviation based quantities were considered. In this work, the quantile measure is introduced for electric machine design optimization and compared with the existing solutions. The evaluation of a design's robustness is typically examined based on finite element simulations. As for most measures a significant number of parameter combinations and thus computations are required, a surrogate model assisted approach is presented to minimize computational effort and runtime. A test problem is defined and analyzed to illustrate the differences of selected robustness measures. Results reveal the importance of considering robustness in the optimization process. Moreover, a careful choice of appropriate measures has to be taken. Selected designs are compared and conclusions and an outlook on future activities are presented.

Recently, the large-scale optimization problems have become a common research topic in the field of evolutionary computation. It is hard to find optimal solutions when solving large-scale multi-objective optimization problems (LSMOPs), due to the ineffectiveness of existing operators. In the other word, the search ability of most existing MOEAs on solving LSMOPs is still weak. To address this issue, an efficient competitive swarm optimizer with a strong exploration ability, denoted as E-CSO, is presented in this paper, which designs a novel three-particle-based particle updating strategy to improve the search efficiency. The experimental results validate the high efficiency and effectiveness of our proposed approach when solving various LSMOPs.

Competitive swarm optimizer (CSO) based on multidirectional search plays a crucial role in addressing large-scale multiobjective optimization problems (LSMOPs). However, relying solely on uniform or cluster partitioning of the objective space for sampling, along with two search directions constructed with upper and lower boundaries of global variables, sometimes lacks consideration of regional information. This results in an inefficient search and hinders the global convergence of the algorithm. To solve these problems, this study proposes a large-scale multiobjective competitive swarm optimizer algorithm based on regional multidirectional search (AMSLMOEA). Firstly, an adaptive objective space partitioning method based on the evolutionary state of the population is designed to enhance the adaptability of partitioning. Secondly, an individual multidirectional search strategy is introduced. Considering the algorithm’s computational complexity, the strategy selects the optimal individual within each subregion and constructs four-directional search vectors based on the lower limit of the global decision variables and the upper limit of the individual decision variables within the subregion. To validate the effectiveness of AMSLMOEA, the performance is tested on four benchmark function sets. The results demonstrate that AMSLMOEA outperforms the vast majority of the compared algorithms in terms of the IGD and HV metrics.

In this paper, we propose a framework to accelerate the computational efficiency of evolutionary algorithms on large-scale multiobjective optimization. The main idea is to track the Pareto optimal set (PS) directly via problem reformulation. To begin with, the algorithm obtains a set of reference directions in the decision space and associates them with a set of weight variables for locating the PS. Afterwards, the original large-scale multiobjective optimization problem is reformulated into a low-dimensional single-objective optimization problem. In the reformulated problem, the decision space is reconstructed by the weight variables and the objective space is reduced by an indicator function. Thanks to the low dimensionality of the weight variables and reduced objective space, a set of quasi-optimal solutions can be obtained efficiently. Finally, a multiobjective evolutionary algorithm is used to spread the quasi-optimal solutions over the approximate Pareto optimal front evenly. Experiments have been conducted on a variety of large-scale multiobjective problems with up to 5000 decision variables. Four different types of representative algorithms are embedded into the proposed framework and compared with their original versions, respectively. Furthermore, the proposed framework has been compared with two state-of-the-art algorithms for large-scale multiobjective optimization. The experimental results have demonstrated the significant improvement benefited from the framework in terms of its performance and computational efficiency in large-scale multiobjective optimization.

Large-scale multi-objective optimization problems are prevalent in practical applications. Consequently, researchers have proposed numerous large-scale multi-objective evolutionary algorithms (LMOEAs) to address these problems. Among them, neural network (NN)-based LMOEAs have garnered significant attention owing to their superior search efficiency. Regrettably, existing NN-based LMOEAs are limited by a single source of training data and pre-fixed network topology, resulting in insufficient generalization and weak adaptability. We propose a heterogeneous operator mechanism and stochastic configuration network-enhanced LMOEA (HMSCN-LMOEA) to overcome these issues. Specifically, a heterogeneous operator mechanism is introduced to address the limited data sources in NN model training, thereby effectively enhancing the quality of the training set. A stochastic configuration network (SCN) model with dynamic structural adaptability is designed to replace the NN model with a pre-fixed network topology, thereby enhancing the algorithm’s adaptability. Finally, to evaluate the algorithm’s performance, four large-scale multi-objective optimization problems—UF, WFG, LSMOP, and ZCAT—are adopted. Experimental results based on IGD, IGD+, Spacing, and HV metrics demonstrate the significant advantages and competitiveness of HMSCN-LMOEA over state-of-the-art LMOEAs. Furthermore, HMSCN-LMOEA is employed to solve the cloud task scheduling problem across different task scales, and the experimental results show that it achieves the top rank in 75% of the evaluated scenarios, further demonstrating its promising potential.

Large-scale optimisation is of great importance as most real-world optimisation problems involve a large number of decision variables. Cooperative co-evolution (CC) approaches are successful in this field as they can tackle thousands of decision variables. CC methods consist of decomposition, optimisation of decomposed sub-components and cooperative combination based on the `divide-and-conquer' mechanism. Sub-components are optimised cooperatively before the merge. Most of existing CCs optimise different sub-components using the same optimisation algorithm. In this study we propose an adaptive approach based on multi-optimiser cooperative co-evolution for large-scale optimisation problems. As different sub-components may have different characteristics and different contributions towards the global fitness function, they should to be handled differently considering the diversity and the quality of each sub-component. To achieve this effect our proposed approach incorporates four different optimisers adaptively according to the relative contribution. In each optimisation cycle, an adaptive selection strategy is used to allocate the most suited optimiser to each sub-component using quality-diversity measure and the historical performance of the algorithms. The performance of the proposed approach has been evaluated on large-scale optimisation benchmark problems. The results confirmed the effectiveness of the proposed adaptive approach as it can achieve comparable or better performance on almost of test cases compared to state-of-the-art methods.

When solving multi-objective optimization problems (MOPs) with big data, traditional multi-objective evolutionary algorithms (MOEAs) meet challenges because they demand high computational costs that cannot satisfy the demands of online data processing involving optimization. The gradient heuristic optimization methods show great potential in solving large scale numerical optimization problems with acceptable computational costs. However, some intrinsic limitations make them unsuitable for searching for the Pareto fronts. It is believed that the combination of these two types of methods can deal with big MOPs with less computational cost. The main contribution of this paper is that a multi-objective memetic algorithm based on decomposition for big optimization problems (MOMA/D-BigOpt) is proposed and a gradient-based local search operator is embedded in MOMA/D-BigOpt. In the experiments, MOMA/D-BigOpt is tested on the multi-objective big optimization problems with thousands of variables. We also combine the local search operator with other widely used MOEAs to verify its effectiveness. The experimental results show that the proposed algorithm outperforms MOEAs without the gradient heuristic local search operator.