It is known that Pareto dominance has its own weaknesses as the selection criterion in evolutionary multi-objective optimization. Algorithms based on Pareto dominance can suffer from slow convergence to the optimal front, inferior performance on problems with many objectives, etc. Non-Pareto criterion, such as decomposition-based criterion and indicator-based criterion, has already shown promising results in this regard, but its high selection pressure may lead the algorithm to prefer some specific areas of the problem’s true Pareto front, especially when the front is highly irregular. In this paper, we propose a bi-criterion evolution framework of Pareto criterion and non-Pareto criterion, which attempts to make use of their strengths and compensates for each other’s weaknesses. The proposed framework consists of two parts, Pareto criterion evolution and non-Pareto criterion evolution. The two parts work collaboratively, with an abundant exchange of information to facilitate each other’s evolution. Specifically, the non-Pareto criterion evolution leads the Pareto criterion evolution forward and the Pareto criterion evolution compensates the possible diversity loss of the non-Pareto criterion evolution. The proposed framework keeps the freedom on the implementation of the non-Pareto criterion evolution part, thus making it applicable for any non-Pareto-based algorithm. In the Pareto criterion evolution, two operations, population mainte- nance and individual exploration, are presented. The former is to maintain a set of representative nondominated individuals, and the latter is to explore some promising areas which are undeveloped (or not well-developed) in the non-Pareto criterion evolution. Experimental results have shown the effectiveness of the proposed framework. The bi-criterion evolution works well on seven groups of 42 test problems with various characteristics, including those where Pareto-based algorithms or non-Pareto- based algorithms struggle.
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