Optimizing Monotone Functions Can Be Difficult

TL;DR

This paper investigates how the (1+1) evolutionary algorithm's efficiency in optimizing monotone pseudo-Boolean functions depends critically on the mutation probability constant, revealing cases of both polynomial and exponential run-times.

Contribution

It demonstrates that the mutation probability constant c significantly influences optimization difficulty, with new bounds and a counterexample showing exponential run-time for large c.

Findings

For c<1, the algorithm finds the optimum in Θ(n log n) iterations.

At c=1, an upper bound of O(n^{3/2}) is established.

For c>33, the algorithm likely does not find the optimum within exponential time.

Abstract

Extending previous analyses on function classes like linear functions, we analyze how the simple (1+1) evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotone. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant $c$ in the mutation probability $p(n) = c/n$ can make a decisive difference. We show that if $c < 1$, then the (1+1) evolutionary algorithm finds the optimum of every such function in $\Theta(n \log n)$ iterations. For $c=1$, we can still prove an upper bound of $O(n^{3/2})$. However, for $c > 33$, we present a strictly monotone function such that the (1+1) evolutionary algorithm with overwhelming probability does not find the optimum within $2^{\Omega(n)}$ iterations. This is the first time that we observe that a constant factor change of the mutation probability changes the run-time by more than constant factors.