A general convergence analysis method for evolutionary multi-objective optimization algorithm

Convergence analysis of multi-objective optimization algorithm has been an area of vital interest to the research community. With this regard, a number of approaches have been proposed and studied. However, these studies and developed proposals cannot cope with more than 3-dimensional optimization problems. Generally speaking, interpolation planes are formed by 3-dimension data. So, when the dimensionality of the Pareto front is more than 3, the dimensionality of Pareto front will be reduced to 3 by involving principal component analysis. This may lead to some important data being missed. Due to missing data, the formed interpolation plane is usually inaccurate and uneven. This will give rise to difficulties to evaluate the distance between the Pareto front and the optimal Pareto front. Subsequently, it is not easy to evaluate exact convergence time and with this regard the existing solutions lack general. Having this in mind, this paper develops a general convergence analysis (GCAM) for evolutionary multi-objective optimization algorithm (EMOA). In this approach, two originality aspects come to existence: one associates with the interpolation plane convergence analysis while the second concerns the improved drift analysis of evolutionary algorithm. Firstly, for more than 3-dimensional space, the dimensionality of the Pareto front set becomes reduced to 3 through a locally linear embedding. This overcomes the irregular interpolation plane problem and produce a high-quality interpolation. Secondly, this study originally analyzes the convergence of EMOA by engaging an improved drift analysis. Finally, we determine the first stopping time of EMOA by analyzing the convergence metric. The experimental results demonstrate that the proposed method exhibits better performance in comparison with CAD, CAL, and CAC. Specifically, the error proportion of SMS-EMOA, AR-MOEA, SPEA2+SDE, GFM-MOEA has been decreased by 12%, 15%, 21%, 19% and 17%, respectively.