# A Tight Runtime Analysis for the $(\mu + \lambda)$ EA

## TL;DR

This paper provides a precise asymptotic runtime analysis of the A with arbitrary nd mbda populations on the OneMax problem, extending and tightening previous bounds for these evolutionary algorithms.

## Contribution

It derives the first tight asymptotic runtime bounds for the A with general nd mbda on OneMax, resolving a long-standing open problem.

## Key findings

- Established the asymptotically tight runtime formula for A on OneMax.
- Improved previous runtime bounds for the A with nd mbda populations.
- Provided methods applicable to analyze other population-based evolutionary algorithms.

## Abstract

Despite significant progress in the theory of evolutionary algorithms, the theoretical understanding of evolutionary algorithms which use non-trivial populations remains challenging and only few rigorous results exist. Already for the most basic problem, the determination of the asymptotic runtime of the $(\mu+\lambda)$ evolutionary algorithm on the simple OneMax benchmark function, only the special cases $\mu=1$ and $\lambda=1$ have been solved. In this work, we analyze this long-standing problem and show the asymptotically tight result that the runtime $T$, the number of iterations until the optimum is found, satisfies \[E[T] = \Theta\bigg(\frac{n\log n}{\lambda}+\frac{n}{\lambda / \mu} + \frac{n\log^+\log^+ \lambda/ \mu}{\log^+ \lambda / \mu}\bigg),\] where $\log^+ x := \max\{1, \log x\}$ for all $x > 0$. The same methods allow to improve the previous-best $O(\frac{n \log n}{\lambda} + n \log \lambda)$ runtime guarantee for the $(\lambda+\lambda)$~EA with fair parent selection to a tight $\Theta(\frac{n \log n}{\lambda} + n)$ runtime result.