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

Abstract

In this paper we show how to use drift analysis in the case of two random variables X1, X2, when the drift is approximatively given by A • (X1, X2) T for a matrix A. The non-trivial case is that X1 and X2 impede each other's progress, and we give a full characterization of this case. As application, we develop and analyze a minimal example TwoLin of a dynamic environment that can be hard. The environment consists of two linear function f1 and f2 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 TwoLin, but does not find the shared optimum in polynomial time for large χ.