Analysis of the $(\mu/\mu_I,\lambda)$-$\sigma$-Self-Adaptation Evolution Strategy with Repair by Projection Applied to a Conically Constrained Problem

TL;DR

This paper provides a theoretical analysis of a self-adaptive evolution strategy with projection repair applied to conically constrained problems, predicting its dynamics and steady-state behavior.

Contribution

It introduces closed-form approximations and evolution equations to analyze the strategy's performance and steady-state behavior on conically constrained optimization problems.

Findings

Strategy behaves like optimizing a sphere model in steady state.

Parents stay away from the cone boundary rather than on it.

Theoretical predictions closely match experimental results.

Abstract

A theoretical performance analysis of the $(\mu/\mu_I,\lambda)$-$\sigma$-Self-Adaptation Evolution Strategy ($\sigma$SA-ES) is presented considering a conically constrained problem. Infeasible offspring are repaired using projection onto the boundary of the feasibility region. Closed-form approximations are used for the one-generation progress of the evolution strategy. Approximate deterministic evolution equations are formulated for analyzing the strategy's dynamics. By iterating the evolution equations with the approximate one-generation expressions, the evolution strategy's dynamics can be predicted. The derived theoretical results are compared to experiments for assessing the approximation quality. It is shown that in the steady state the $(\mu/\mu_I,\lambda)$-$\sigma$SA-ES exhibits a performance as if the ES were optimizing a sphere model. Unlike the non-recombinative $(1,\lambda)$-ES, the parental steady state behavior does not evolve on the cone boundary but stays away from the boundary to a certain extent.