Drift Analysis with Fitness Levels for Elitist Evolutionary Algorithms

The fitness level method is a popular tool for analyzing the hitting time of elitist evolutionary algorithms. Its idea is to divide the search space into multiple fitness levels and estimate lower and upper bounds on the hitting time using transition probabilities between fitness levels. However, the lower bound generated by this method is often loose. An open question regarding the fitness level method is what are the tightest lower and upper time bounds that can be constructed based on transition probabilities between fitness levels. To answer this question, {\color{red} we combine drift analysis with fitness levels and define the tightest bound problem as a constrained multi-objective optimization problem subject to fitness levels.} The tightest metric bounds from fitness levels are constructed and proven for the first time. Then linear bounds are derived from metric bounds and a framework is established that can be used to develop different fitness level methods for different types of linear bounds. The framework is generic and promising, as it can be used to draw tight time bounds on both fitness landscapes without and with shortcuts. This is demonstrated in the example of the (1+1) EA maximizing the TwoMax1 function