# Running Time Analysis of the (1+1)-EA for Robust Linear Optimization

## TL;DR

This paper analyzes the expected running time of the (1+1)-EA algorithm for solving robust linear optimization problems under uncertainty, providing conditions for polynomial-time solutions in different robust scenarios.

## Contribution

It offers the first theoretical analysis of the (1+1)-EA's performance on robust linear optimization with cardinality constraints, focusing on deletion-robust and worst-case scenarios.

## Key findings

- Derived tight bounds for robust parameters ensuring polynomial running time.
- Identified conditions under which the (1+1)-EA efficiently solves robust linear problems.
- Disclosed the potential of EAs for practical robust optimization applications.

## Abstract

Evolutionary algorithms (EAs) have found many successful real-world applications, where the optimization problems are often subject to a wide range of uncertainties. To understand the practical behaviors of EAs theoretically, there are a series of efforts devoted to analyzing the running time of EAs for optimization under uncertainties. Existing studies mainly focus on noisy and dynamic optimization, while another common type of uncertain optimization, i.e., robust optimization, has been rarely touched. In this paper, we analyze the expected running time of the (1+1)-EA solving robust linear optimization problems (i.e., linear problems under robust scenarios) with a cardinality constraint $k$. Two common robust scenarios, i.e., deletion-robust and worst-case, are considered. Particularly, we derive tight ranges of the robust parameter $d$ or budget $k$ allowing the (1+1)-EA to find an optimal solution in polynomial running time, which disclose the potential of EAs for robust optimization.