A Computationally Fast Convergence Measure and Implementation for Single-, Multiple- and Many-Objective Optimization

A previous study suggested a Karush-Kuhn-Tucker proximity measure (KKTPM) that is able to identify relative closeness of any point from the theoretical optimum point without actually knowing the exact location of the optimum point. However, the drawback of the KKTPM metric is that it requires a new optimization problem to be solved for each solution to find its convergence measure. In this paper, we propose a number of approximate formulations of the KKTPM measure by studying its calculation procedure so that the KKTPM measure can be computed in a computationally fast manner and without solving the inherent optimization problem. The approximate KKTPM values are evaluated in comparison with the original exact optimization-based KKTPM value on standard single-objective, multiobjective, and many-objective optimization problems. In all cases, our proposed "estimated" approximate method is found to achieve a strong correlation of KKTPM values with the exact values, and achieve such results in two or three orders of magnitude smaller computational time. We also present a parser-based KKTPM computational procedure, which can be used independently or in conjunction with an evolutionary multiobjective optimization procedure.