Uncrowded Hypervolume-based Multi-objective Optimization with Gene-pool Optimal Mixing

Domination-based multi-objective (MO) evolutionary algorithms (EAs) are today arguably the most frequently used type of MOEA. These methods however stagnate when the majority of the population becomes non-dominated, preventing further convergence to the Pareto set. Hypervolume-based MO optimization has shown promising results to overcome this. Direct use of the hypervolume however results in no selection pressure for dominated solutions. The recently introduced Sofomore framework overcomes this by solving multiple interleaved single-objective dynamic problems that iteratively improve a single approximation set, based on the uncrowded hypervolume improvement (UHVI). It thereby however loses many advantages of population-based MO optimization, such as handling multimodality. Here, we reformulate the UHVI as a quality measure for approximation sets, called the uncrowded hypervolume (UHV), which can be used to directly solve MO optimization problems with a single-objective optimizer. We use the state-of-the-art gene-pool optimal mixing evolutionary algorithm (GOMEA) that is capable of efficiently exploiting the intrinsically available greybox properties of this problem. The resulting algorithm, UHV-GOMEA, is compared to Sofomore equipped with GOMEA, and the domination-based MO-GOMEA. In doing so, we investigate in which scenarios either domination-based or hypervolume-based methods are preferred. Finally, we construct a simple hybrid approach that combines MO-GOMEA with UHV-GOMEA and outperforms both.

A Population Size Reduction Approach for Nondominated Sorting-Based Optimization Algorithms

The solution set of any multi-objective optimization problem can be expressed as an approximation set of Pareto front. The number of solution candidates in this set could be large enough to cover the entire Pareto front as a general belief. However, among the sufficiently close points on the objective space, almost same accurate solutions can obtain. Hence, in this set, it is possible to eliminate some of the solutions without detriment to the overall performance. Therefore, in this research, the authors propose a population size reduction method which systematically reduced the population size based on the distance and angle relations between any consecutive solutions. The results are evaluated based on two-objective benchmark problems and compared with the results of NSGA-II algorithm with respect to three different performance evaluation metrics.

Approximation Set of the Interval Set in Pawlak's Space

The interval set is a special set, which describes uncertainty of an uncertain concept or setZwith its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two approximations (i.e., upper approximation setR¯(Z) and lower approximation setR_(Z)) are presented, respectively. The disadvantages of using upper-approximation setR¯(Z) or lower-approximation setR_(Z) as approximation sets of the uncertain set (uncertain concept)Zare analyzed, and a new method for looking for a better approximation set of the interval setZis proposed. The conclusion that the approximation setR0.5(Z) is an optimal approximation set of interval setZis drawn and proved successfully. The change rules ofR0.5(Z) with different binary relations are analyzed in detail. Finally, a kind of crisp approximation set of the interval setZis constructed. We hope this research work will promote the development of both the interval set model and granular computing theory.