In real-world applications, many optimization problems have the time-linkage property, that is, the objective function value relies on the current solution as well as the historical solutions. Although the rigorous theoretical analysis on evolutionary algorithms (EAs) has rapidly developed in recent two decades, it remains an open problem to theoretically understand the behaviors of EAs on time-linkage problems. This article takes the first step to rigorously analyze EAs for time-linkage functions. Based on the basic OneMax function, we propose a time-linkage function where the first bit value of the last time step is integrated but has a different preference from the current first bit. We prove that with probability $1-o(1)$ , randomized local search and (1 + 1) EA cannot find the optimum, and with probability $1-o(1)$ , $(\mu +1)$ EA is able to reach the optimum.

中文翻译：

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具有时间连锁性的适应度函数的进化算法分析

在实际应用中，很多优化问题都具有时间关联性，即目标函数值依赖于当前解以及历史解。尽管对进化算法 (EA) 的严格理论分析在最近二十年迅速发展，但从理论上理解 EA 在时间连锁问题上的行为仍然是一个悬而未决的问题。本文迈出了严格分析 EA 的时间链接函数的第一步。基于基本的 OneMax 函数，我们提出了一个时间链接函数，其中集成了最后一个时间步的第一个比特值，但与当前第一个比特有不同的偏好。我们用概率证明 $1-o(1)$ , 随机局部搜索和 (1 + 1) EA 无法找到最优值，并且有概率 $1-o(1)$ , $(\mu +1)$ EA 能够达到最优。