A random-based dynamic grouping strategy for large scale multi-objective optimization

Many real-world multi-objective optimization problems have hundreds or even thousands of decision variables, which contrast with the current practice of multi-objective metaheuristics whose performance is typically assessed using benchmark problems with a relatively low number of decision variables (normally, no more than 30). In this paper, we propose a cooperative coevolution framework that is capable of optimizing large scale (in decision variable space) multi-objective optimization problems. We adopt a benchmark that is scalable in the number of decision variables (the ZDT test suite) and compare our proposed algorithm with respect to two state-of-the-art multi-objective evolutionary algorithms (GDE3 and NSGA-II) when using a large number of decision variables (from 200 up to 5000). The results clearly indicate that our proposed approach is effective as well as efficient for solving large scale multi-objective 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

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.

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.

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.

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.

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.

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.

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.

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.

In a real life situation, due to the complexity of the industrial systems and non-linearity of their behavior, it is very difficult to achieve optimum performance of system for desired industrial goals using uncertain, vague and imprecise data. Herein, an approach has been proposed through which the behavior of the system is analyzed in the form of well- known six reliability indices by using triangular fuzzy numbers which allow the consideration of expert opinions, linguistic variables, operating conditions in reliability information. Using their behavior analysis a fuzzy multi-objective optimization problem (FMOOP) has been formulated. Due to the conflicting nature of the multiple objectives, the decision making is difficult and it leads to the Pareto optimal solutions instead of single optimal solutions. Many evolutionary algorithms (EAs) already exist in the literature for solving a multi-objective optimization problem (MOOP), and are termed as multi-objective evolutionary algorithms (MOEAs). Particle swarm optimization (PSO) is one of such MOEA which demonstrates the ability to identity a Pareto-optimal front efficiently. Here, a crisp optimization problem is reformulated from FMOOP by taking into account the preference of decision maker (DM) and then PSO is applied to solve the resulting fuzzified MOOP. The presented approach is applied in order to solve the multi-objective series-parallel system reliability optimization problem for a crystallization unit of a urea plant.

Many real-world optimization problems are characterized by multiple conflicting objectives, which are known as multi-objective optimization problems (MOPs). In the last two decades, evolutionary algorithms have shown promising performance in solving various MOPs, and a large number of multi-objective evolutionary algorithms (MOEAs) have been proposed. In order to determine the most suitable MOEA for a specific MOP, it is usually necessary to perform experiments to compare the performance of multiple candidate MOEAs. In 2017, an evolutionary multi-objective optimization platform was proposed by us, called PlatEMO, which provides the source codes of many state-of-the-art MOEAs and helps researchers perform batch experiments on these MOEAs. In this work, we illustrate the method of using the newest version of PlatEMO to solve MOPs in applications, by means of a case study on the feature selection problem, which is an important and difficult task in machine learning and data mining. This paper details the method of adding the feature selection problem to PlatEMO, and presents the experimental results of eight MOEAs on nine datasets in feature selection.

For large-scale multi-objective optimization, it is particularly challenging for evolutionary algorithms to converge to the Pareto Front. Most existing multi-objective evolutionary algorithms (MOEAs) handle convergence and diversity in a mutually dependent manner during the evolution process. In this case, the performance degradation of one solution may lead to the deterioration of the performance of the other solution. This paper proposes a two-stage multi-objective optimization algorithm based on decision variable clustering (LSMOEA-VT) to solve large-scale optimization problems. In LSMOEA-VT, decision variables are divided into two categories and use dimensionality reduction methods to optimize the variables that affect evolutionary convergence. Following this, we performed an interdependence analysis to break down the convergence variables into multiple subcomponents that are more tractable. Furthermore, a non-dominated dynamic weight aggregation method is used to enhance the diversity of the population. To evaluate the performance of our proposed algorithm, we performed extensive comparative experiments against four optimization algorithms across a diverse set of benchmark problems, including eight multi-objective optimization problems and nine large-scale optimization problems. The experimental results show that our proposed algorithm performs well on some test functions and has certain competitiveness.

There can be a complicated mapping relation between decision variables and objective functions in multi-objective optimization problems (MOPs). It is uncommon that decision variables influence objective functions equally. Decision variables act differently in different objective functions. Hence, often, the mapping relation is unbalanced, which causes some redundancy during the search in a decision space. In response to this scenario, we propose a novel memetic (multi-objective) optimization strategy based on dimension reduction in decision space (DRMOS). DRMOS firstly analyzes the mapping relation between decision variables and objective functions. Then, it reduces the dimension of the search space by dividing the decision space into several subspaces according to the obtained relation. Finally, it improves the population by the memetic local search strategies in these decision subspaces separately. Further, DRMOS has good portability to other multi-objective evolutionary algorithms (MOEAs); that is, it is easily compatible with existing MOEAs. In order to evaluate its performance, we embed DRMOS in several state of the art MOEAs to facilitate our experiments. The results show that DRMOS has the advantage in terms of convergence speed, diversity maintenance, and portability when solving MOPs with an unbalanced mapping relation between decision variables and objective functions.