An Optimality Theory-Based Proximity Measure for Set-Based Multiobjective Optimization

Set-based multiobjective optimization methods, such as evolutionary multiobjective optimization (EMO) methods, attempt to find a set of Pareto-optimal solutions, instead of a single optimal solution. To evaluate these algorithms for their convergence to the efficient set in multiobjective optimization problems, the current performance metrics require the knowledge of the true Pareto-optimal solutions. In this paper, we develop a theoretically motivated Karush-Kuhn-Tucker proximity measure (KKTPM) that can provide an estimate of the proximity of a set of tradeoff solutions from the true Pareto-optimal solutions without any prior knowledge. Besides theoretical development of the proposed metric, the proposed KKTPM is computed for iteration-wise tradeoff solutions obtained from specific EMO algorithms on two-, three-, five-, and ten-objective optimization problems. Results amply indicate the usefulness of the proposed KKTPM as a metric for evaluating different sets of tradeoff solutions and also as a possible termination criterion for an EMO algorithm. Other possible uses of the proposed metric are also highlighted.