Assessing the performance of Pareto front (PF) approximations is a key issue in the field of evolutionary multi/many-objective optimization. Inverted Generational Distance (IGD) has been widely accepted as a performance indicator for evaluating the comprehensive quality for a PF approximation. However, IGD usually becomes infeasible when facing a realworld optimization problem as it needs to know the true PF a priori. In addition, the time complexity of IGD grows quadratically with the size of the solution/reference set. To address the aforementioned issues, a grid-based IGD (Grid-IGD) is proposed to estimate both convergence and diversity of PF approximations for multi/many-objective optimization. In Grid-IGD, a set of reference points is generated by estimating PFs of the problem in question, based on the representative nondominated solutions of all the approximations in a grid environment. To reduce the time complexity, Grid-IGD only considers the closest solution within the grid neighborhood in the approximation for every reference point. Grid-IGD also possesses other desirable properties such as Pareto compliance, immunity to dominated/duplicate solutions and no need of normalization. In the experimental studies, Grid-IGD is verified on both the artificial and real PF approximations obtained by five many-objective optimizers. Effects of the grid specification on the behavior of Grid-IGD are also discussed in detail theoretically and experimentally.

Feature selection (FS) for classification is crucial for large-scale images and bio-microarray data using machine learning. It is challenging to select informative features from high-dimensional data which generally contains many irrelevant and redundant features. These features often impede classifier performance and misdirect classification tasks. In this article, we present an efficient FS algorithm to improve classification accuracy by taking into account both the relevance of the features and the pairwise features correlation in regard to class labels. Based on conditional mutual information and entropy, a new supervised similarity measure is proposed. The supervised similarity measure is connected with feature redundancy minimization evaluation and then combined with feature relevance maximization evaluation. A new criterion max-relevance and min-supervised-redundancy (MRMSR) is introduced and theoretically proved for FS. The proposed MRMSR-based method is compared to seven existing FS approaches on several frequently studied public benchmark datasets. Experimental results demonstrate that the proposal is more effective at selecting informative features and results in better competitive classification performance.