A smooth proximity measure for optimality in multi-objective optimization using Benson's method

Multi-objective optimization problems give rise to a set of trade-off Pareto-optimal solutions. To evaluate a set-based multi-objective optimization algorithm, such as an evolutionary multi-objective optimization (EMO) algorithm, for its convergence and diversity attainment, more than one performance metrics are required. To measure the convergence aspect, a new Karush-Kuhn-Tucker proximity measure (KKTPM) was recently proposed based on the extent of satisfaction of KKT optimality conditions on the augmented achievement scalarization function (AASF) formulation. However, the Pareto-optimality of a point depends on a parameter needed to be used in the AASF formulation. In this paper, we use Benson's method as a scalarized version of the multi-objective optimization problem, mainly because it is parameter-less and is a popularly used in the multi-criterion decision-making (MCDM) literature. The proposed Benson's method based metric (B-KKTPM) is applied to optimized solutions of popular EMO algorithms on standard two to 10-objective test problems and to a few engineering design problems. B-KKTPM is able to determine relative closeness of a set of trade-off solutions from the strictly efficient solutions without any prior knowledge of them. To reduce the computational cost of solving an optimization problem to compute B-KKTPM, we also propose a direct, but approximate, method. The obtained results from our extensive study indicates that (i) the proposed optimization based and direct B-KKTPMs can be used for a termination check for any optimization algorithm, and (ii) the direct B-KKTPM method can be used as a replacement of the optimization-based version for a good trade-off between computational time and accuracy.