Two-Dimensional Drift Analysis: Optimizing Two Functions Simultaneously Can Be Hard

03/28/2022

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by   Duri Janett, et al.

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In this paper we show how to use drift analysis in the case of two random variables X_1, X_2, when the drift is approximatively given by A· (X_1,X_2)^T for a matrix A. The non-trivial case is that X_1 and X_2 impede each other's progress, and we give a full characterization of this case. As application, we develop and analyze a minimal example TwoLinear of a dynamic environment that can be hard. The environment consists of two linear function f_1 and f_2 with positive weights 1 and n, and in each generation selection is based on one of them at random. They only differ in the set of positions that have weight 1 and n. We show that the (1+1)-EA with mutation rate χ/n is efficient for small χ on TwoLinear, but does not find the shared optimum in polynomial time for large χ.