Analysis of Multiobjective Evolutionary Algorithms on Fitness Function With Time-Linkage Property

The time-linkage property, which means that the optimization problem not only relies on the current solution but also on historical solutions, is common in real-world applications. Although the theoretical studies on multiobjective evolutionary algorithms (MOEAs) have been rapidly developed in decades, there exists no theoretical analyses for MOEAs on time-linkage problems. This letter aims to take the first step toward rigorously analyzing MOEAs on time-linkage problems. To be specific, we constructed a multiobjective optimization problem with the time-linkage property based on the benchmark COCZ problem, called
$COCZ_{TL}$
. For this problem, we proved that GSEMO cannot find the Pareto front, but nondominated sorting genetic algorithm II can find it with a probability of at least
$\Omega {}({1}/{n})$
. Furthermore, we proposed an algorithm with auxiliary populations called GSEMO/Ps that is based on GSEMO. The results show that GSEMO/Ps can find the Pareto front of
$COCZ_{TL}$
and the expected runtime is at most
$O(n^{3})$
. This letter provides insight into understanding the optimization behaviors of MOEAs in the multiobjective optimization problems with the time-linkage property.