Runtime Analysis of the (1+1) EA on Weighted Sums of Transformed Linear Functions

TL;DR

This paper analyzes the runtime of the (1+1) EA on a class of functions that are weighted sums of two transformed linear functions, extending known results for linear functions to more complex objective functions.

Contribution

It introduces a new class of objective functions and proves that the (1+1) EA finds optimal solutions in expected time O(n log n), generalizing previous linear function results.

Findings

Expected runtime of O(n log n) for the (1+1) EA on these functions.

Optimal solutions are obtained with a mutation rate depending on overlapping bits.

Generalizes linear function analysis to weighted sums of transformed linear functions.

Abstract

Linear functions play a key role in the runtime analysis of evolutionary algorithms and studies have provided a wide range of new insights and techniques for analyzing evolutionary computation methods. Motivated by studies on separable functions and the optimization behaviour of evolutionary algorithms as well as objective functions from the area of chance constrained optimization, we study the class of objective functions that are weighted sums of two transformed linear functions. Our results show that the (1+1) EA, with a mutation rate depending on the number of overlapping bits of the functions, obtains an optimal solution for these functions in expected time O(n log n), thereby generalizing a well-known result for linear functions to a much wider range of problems.