Finding trade-off solutions close to KKT points using evolutionary multi-objective optimisation

Abstract

Despite having a wide-spread applicability of evolutionary optimization procedures over the past few decades, EA researchers still face criticism about the theoretical optimality of obtained solutions. In this paper, we address this issue for problems for which gradients of objectives and constraints can be computed either exactly, or numerically or through subdifferentials. We suggest a systematic procedure of analyzing a representative set of Pareto-optimal solutions for their closeness to satisfying Karush-Kuhn-Tucker (KKT) points, which every Pareto-optimal solution must also satisfy. The procedure involves either a least-square solution or an optimum solution to a set of linear system of equations involving Lagrange multipliers. The procedure is applied to a number of differentiable and non-differentiable test problems and to a highly nonlinear engineering design problem. The results clearly show that EAs are capable of finding solutions close to theoretically optimal solutions in various problems. As a by-product, the error metric suggested in this paper can also be used as a termination condition for an EA application. Hopefully, this study will bring EAs and its research closer to classical optimization studies.