# Analysis of the $(\mu/\mu_I,\lambda)$-CSA-ES with Repair by Projection Applied to a Conically Constrained Problem

## TL;DR

This paper provides a theoretical analysis of a specific evolution strategy with step size adaptation applied to a conically constrained linear optimization problem, predicting its dynamics and steady states.

## Contribution

It introduces a stochastic iterative model for the strategy and derives equations predicting its behavior, validated by comparison with actual algorithm runs.

## Key findings

- Theoretical predictions match real algorithm dynamics.
- Steady state values are accurately predicted.
- Model simplifies fluctuations to analyze mean behavior.

## Abstract

Theoretical analyses of evolution strategies are indispensable for gaining a deep understanding of their inner workings. For constrained problems, rather simple problems are of interest in the current research. This work presents a theoretical analysis of a multi-recombinative evolution strategy with cumulative step size adaptation applied to a conically constrained linear optimization problem. The state of the strategy is modeled by random variables and a stochastic iterative mapping is introduced. For the analytical treatment, fluctuations are neglected and the mean value iterative system is considered. Non-linear difference equations are derived based on one-generation progress rates. Based on that, expressions for the steady state of the mean value iterative system are derived. By comparison with real algorithm runs, it is shown that for the considered assumptions, the theoretical derivations are able to predict the dynamics and the steady state values of the real runs.