Proportion Of Non-dominated Solutions Research Articles

Evolutionary many-Objective algorithm based on fractional dominance relation and improved objective space decomposition strategy

Abstract For many-objective optimization problems (MaOPs), the proportion of non-dominated solutions in a population scales up sharply with the increase in the number of objectives. Besides, for an MaOP with a fixed number of objectives, the proportion of non-dominated solutions may also grow to a high level with the progressing of the evolutionary process, sometimes even reaching 100%. Thus, a great challenge has been posed to traditional Pareto-dominance-based many-objective evolutionary algorithms (MaOEAs). To address this issue, a new fractional dominance relation is proposed to distinguish non-dominated solutions for strengthening convergence. To be special, the number of objectives on which one solution is better than the other one is considered in the fractional dominance relationship. At the same time, the objective space decomposition approach is improved with a subspace selection mechanism to maintain the population diversity. Then, two new MaOEAs referred to as FDEA-I and FDEA-II are proposed on the basis of fractional dominance relation and the improved objective space decomposition strategy. The two algorithms first use fractional dominance relation to retain some solutions with promising performance, and then the improved objective space decomposition strategy is used to maintain diversity for the obtained population. Finally, to evaluate the performance of FDEA-I and FDEA-II, extensive experiments are conducted to compare them with six state-of-the-art MaOEAs on 72 many-objective benchmark instances taken from WFG and MaF test suites. The results show that FDEA-I and FDEA-II perform significantly better than the six comparative algorithms.

Hyperplane Assisted Evolutionary Algorithm for Many-Objective Optimization Problems

In many-objective optimization problems (MaOPs), forming sound tradeoffs between convergence and diversity for the environmental selection of evolutionary algorithms is a laborious task. In particular, strengthening the selection pressure of population toward the Pareto-optimal front becomes more challenging, since the proportion of nondominated solutions in the population scales up sharply with the increase of the number of objectives. To address these issues, this paper first defines the nondominated solutions exhibiting evident tendencies toward the Pareto-optimal front as prominent solutions, using the hyperplane formed by their neighboring solutions, to further distinguish among nondominated solutions. Then, a novel environmental selection strategy is proposed with two criteria in mind: 1) if the number of nondominated solutions is larger than the population size, all the prominent solutions are first identified to strengthen the selection pressure. Subsequently, a part of the other nondominated solutions are selected to balance convergence and diversity and 2) otherwise, all the nondominated solutions are selected; then a part of the dominated solutions are selected according to the predefined reference vectors. Moreover, based on the definition of prominent solutions and the new selection strategy, we propose a hyperplane assisted evolutionary algorithm, referred here as hpaEA, for solving MaOPs. To demonstrate the performance of hpaEA, extensive experiments are conducted to compare it with five state-of-the-art many-objective evolutionary algorithms on 36 many-objective benchmark instances. The experimental results show the superiority of hpaEA which significantly outperforms the compared algorithms on 20 out of 36 benchmark instances.

A knee-point-based evolutionary algorithm using weighted subpopulation for many-objective optimization

Abstract Among many-objective optimization problems (MaOPs), the proportion of nondominated solutions is too large to distinguish among different solutions, which is a great obstacle in the process of solving MaOPs. Thus, this paper proposes an algorithm which uses a weighted subpopulation knee point. The weight is used to divide the whole population into a number of subpopulations, and the knee point of each subpopulation guides other solutions to search. Additionally, the convergence of the knee point approach can be exploited, and the subpopulation-based approach improves performance by improving the diversity of the evolutionary algorithm. Therefore, these advantages can make the algorithm suitable for solving MaOPs. Experimental results show that the proposed algorithm performs better on most test problems than six other state-of-the-art many-objective evolutionary algorithms.

An investigation into many-objective optimization on combinatorial problems: Analyzing the pickup and delivery problem

Abstract Many-objective optimization focuses on solving optimization problems with four or more objectives. Effort has been made mainly on studying continuous problems, with interesting results and for which several proposals have appeared. An important result states that the problem does not necessarily becomes more difficult while more objectives are considered. Nevertheless, combinatorial problems have not received an appropriate attention, making this an open research area. This investigation takes this subject on by studying a many-objective combinatorial problem, particularly, the pickup and delivery problem (PDP), which is an important combinatorial optimization problem in the transportation industry and consists of finding a collection of routes with minimum cost. Traditionally, cost has been associated with the number of routes and the total travel distance, however, in many applications, some other objectives emerge, for example, travel time, workload imbalance, and uncollected profit. If we consider all these objectives equally important, PDP can be tackled as a many-objective problem. This study is concerned with the study of: (i) the performance of four representative multi-objective evolutionary algorithms on PDP varying the number of objectives, (ii) the properties of the many-objective PDP regarding scalability, i.e. the conflict between each pair of objectives and the proportion of non-dominated solutions as the number of objectives is varied, and finally (iii) the change of PDP's difficulty when the number of objectives is increased. Results show that the regarded objectives are actually in conflict and that the problem is more difficult to solve while more objectives are considered.