An adaptive penalty-based boundary intersection method for many-objective optimization problem

The Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) is a popular algorithm for solving Multi-Objective Problems (MOPs). The main component of MOEA/D is to decompose a MOP into easier sub-problems using a set of weight vectors. The choice of the number of weight vectors significantly impacts the performance of MOEA/D. However, the right choice for this number varies, given different MOPs and search stages. We adaptively change the number of vectors by removing unnecessary vectors and adding new ones in empty areas of the objective space. Our MOEA/D variant uses the Consolidation Ratio to decide when to change the number of vectors and to decide where to add or remove these weighted vectors. We investigate the effects of this adaptive MOEA/D against MOEA/D with a poorly chosen set of vectors, a MOEA/D with fine-tuned vectors and MOEA/D with Adaptive Weight Adjustment on two commonly used benchmark functions. We analyse the algorithms in terms of hypervolume, IGD and entropy performance. Our results show that the proposed method is equivalent to MOEA/D with fine-tuned vectors and superior to MOEA/D with poorly defined vectors. Thus, our adaptive mechanism mitigates problems related to the choice of the number of weight vectors in MOEA/D, increasing the final performance of MOEA/D by filling empty areas of the objective space and avoiding premature stagnation of the search progress.

Multi-Objective Evolutionary Algorithm based on decomposition (MOEA/D) has been proposed for a decade in solving complex multi-objective problems (MOP) by decomposing it into a set of single-objective problems. MOEA/D-AGR is one of the improved algorithm introduced recently to substitute the original replacement scheme in MOEA/D with a new adaptive global replacement (GR) scheme so that the neighborhood replacement size Tr is increased among the generation to achieve the shifting of focus from solution diversity to convergence. However, the new replacement scheme only considers the one-way convergence of the objective solutions among all sub-problems. It is hard to re-achieve the solution diversity once the algorithm reaches another steeper landscape of solution from a flatten one while it is focusing on convergence. This paper proposes a new adaptive GR scheme to prolong the period of Tr increment so that it can fit the re-increment of fitness landscape. To compensate the shorten period for solution convergence, local searching is adapted for those individuals which has stopped improving its solution value by Simulated Annealing (SA) algorithm. In order to suppress the degree of local searching at the early stage of focusing solution diversity, Fuzzy Logic is used here to coordinate the frequency of local searching according to the average change of objective solution values. To demonstrate the performance of the proposed algorithm, several common benchmark MOPs are used in this paper for comparing with the several state-of-the-art MOEA/D algorithms in terms of IGD. The performance investigation found that the performance of the proposed algorithm was generally better than the other compared algorithms.

There are a large number of multi-objective optimization problems in real-world applications, like in games, that need to be solved in real time. In order to meet this pressing need, we suggests a method of parallelizing the multi-objective evolutionary algorithm based on decomposition (MOEA/D). Furthermore, a novel task decomposition strategy and scalarizing method without the ideal point are proposed for meeting the requirements of real-time and precision of the game. By combining the novel scalarizing function and GPU-based CUDA technology with the MOEA/D, a parallel MOEA/D for real-time multi-objective optimization problems is developed, namely P-MOEA/D. Experimental studies on ZDT and DTLZ benchmark problems suggest that the P-MOEA/D algorithm is efficient and fast.

MOEA/D decomposes a multi-objective optimization problem into a number of single objective optimization problems. Each single objective optimization problem is defined by a scalarizing function using a weight vector. In MOEA/D, there are several scalarizing approaches such as the weighted Tchebycheff, the weighted sum, and the PBI (penalty-based boundary intersection). However, these conventional scalarizing approaches face a difficulty to approximate a widely spread Pareto front in some problems. To enhance the spread of Pareto optimal solutions in the objective space and improve the search performance of MOEA/D especially in many-objective optimization problems, in this work we propose the inverted PBI scalarizing approach which is an extension of the conventional PBI. We use many-objective knapsack problems and WFG4 problems with 2-8 objectives, and compare the search performance of NSGA-III and four MOEA/Ds using the weighted Tchebycheff, the weighted sum, the PBI and the inverted PBI. As results, we show that MOEA/D using the inverted PBI achieves higher search performance than other algorithms in problems with many-objectives and the difficulty to obtain a widely spread Pareto front in the objective space.

A modified PBI approach for multi-objective optimization with complex Pareto fronts

A decomposition-based many-objective evolutionary algorithm updating weights when required

The stochastic nature of intermittent energy resources has brought significant challenges to the optimal operation of the hybrid energy systems. This article proposes a probabilistic multiobjective evolutionary algorithm based on decomposition (MOEA/D) method with two-step risk-based decision-making strategy to tackle this problem. A scenario-based technique is first utilized to generate a stochastic model of the hybrid energy system. Those scenarios divide the feasible domain into several regions. Then, based on the MOEA/D framework, a probabilistic penalty-based boundary intersection (PBI) with gradient descent differential evolution (GDDE) algorithm is proposed to search the optimal scheme from these regions under different uncertainty budgets. To ensure reliable and low risk operation of the hybrid energy system, the Markov inequality is employed to deduce a proper interval of the uncertainty budget. Further, a fuzzy grid technique is proposed to choose the best scheme for real-world applications. The experimental results confirm that the probabilistic adjustable parameters can properly control the uncertainty budget and lower the risk probability. Further, it is also shown that the proposed MOEA/D-GDDE can significantly enhance the optimization efficiency.

Decomposition-based evolutionary algorithm with dual adjustments for many-objective optimization problems

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) uses crossover operator which often either breaks the building blocks or mix them ineffectively. Multi-objective estimation of distribution algorithm based on decomposition (MEDA/D) evolves a probability vector for each sub-problem to guide the search instead of using crossover operator.However, since the number of the weight vectors in the neighborhood of each weight vector is relatively small and MEDA/D does not provide a way to maintain diversity, the performance of MEDA/D is limited. To overcome the drawbacks of MEDA/D, we proposed a new reproduction operator. This operator could promote diversity. We introduced it into MOEA/D framework and the new algorithm is called s-MEDA/D. We also prove that the parameter newly introduced has physical significance and the reproduction operator is not susceptible to the scale of the problem. The s-MEDA/D was tested on nine instances of the 0/1 multi-objective knapsack problem. Empirical evaluation suggests that the proposed algorithm is effective and efficient.

In this paper, we systematically study the influence of the main design elements of scalarizing function-based multiobjective evolutionary algorithms (MOEAs) on the performance of these algorithms. Such algorithms proved to be very successful in multiple computational experiments and practical applications. Well-known examples of this class of MOEAs are Jaszkiewicz’s multiobjecitve genetic local search and multiobjective evolutionary algorithm based on decomposition (MOEA/D). The two algorithms share the same common structure and differ in two aspects, i.e., the selection of parents for recombination and the selection of weight vectors of scalarizing functions. Using three different multiobjective combinatorial optimization problems, i.e., the multiobjective symmetric traveling salesperson problem, the traveling salesperson problem with profits, and the multiobjective set covering problem, we show that the design element with the highest influence on the performance is the choice of a mechanism for parents selection, while the selection of weight vectors, either random or evenly distributed, has practically negligible influence if the number of evenly distributed weight vectors is sufficiently large.

It is difficult to maintain the balance between convergence and diversity for many-objective optimization problems (MaOPs) in the algorithms of evolutionary multi-objective (EMO). EMO algorithms are useful technology to solve the multi-objective optimization problems (MOPs). However, with the larger of optimization objectives, enough Pareto selection pressure will be loosed, and results in the performance of the algorithms are significantly reduced. The decomposition-based EMO developed for MaOPs have been shown to be effective, and the BBO algorithm is a low-complexity algorithm. In this paper, a hybrid decomposition-based BBO/Complex algorithm (HDB/BBO) for MaOPs is proposed. First, a set of uniformly distributed weight vectors and K-means aggregate method is introduced for decomposing MaOPs into several subsystems. Then, inferior migrated islands will not be chosen unless they pass the Metropolis criterion twice during the within-subsystem migration and cross-subsystem migration. The penalty-based boundary intersection (PBI) distance to calculate neighbor islands distance for balancing the algorithm of convergence and diversity. Finally, after mutation and clear duplication, a uniform distribution Pareto set can be obtained. Experimental results on both DTLZ and WFG benchmarks problems demonstrate the superiority of the proposed algorithm in comparison with three state-of-the-art algorithms in terms of both convergence and diversity.

A collaborative decomposition-based evolutionary algorithm integrating normal and penalty-based boundary intersection methods for many-objective optimization

PBI function based evolutionary algorithm with precise penalty parameter for unconstrained many-objective optimization

A multiobjective evolutionary algorithm based on decomposition with normal boundary intersection for traffic grooming in optical networks

Adaptive decomposition-based evolutionary algorithm for many-objective optimization with two-stage dual-density judgment