A Generalized Markov-Chain Modelling Approach to $(1,\lambda)$-ES Linear Optimization: Technical Report

TL;DR

This paper extends Markov-chain models of the $(1,\lambda)$-ES linear optimization algorithm by exploring non-normal step distributions, especially those with Archimedean copulas, to better utilize problem-specific information.

Contribution

It provides sufficient conditions on step distribution types, beyond normality, for the success of a constant step-size $(1,\lambda)$-ES on linear problems with constraints.

Findings

Identifies conditions for non-normal distributions to ensure algorithm success.

Analyzes the impact of copula-based distributions on optimization performance.

Extends previous models to broader distribution classes.

Abstract

Several recent publications investigated Markov-chain modelling of linear optimization by a $(1,\lambda)$-ES, considering both unconstrained and linearly constrained optimization, and both constant and varying step size. All of them assume normality of the involved random steps, and while this is consistent with a black-box scenario, information on the function to be optimized (e.g. separability) may be exploited by the use of another distribution. The objective of our contribution is to complement previous studies realized with normal steps, and to give sufficient conditions on the distribution of the random steps for the success of a constant step-size $(1,\lambda)$-ES on the simple problem of a linear function with a linear constraint. The decomposition of a multidimensional distribution into its marginals and the copula combining them is applied to the new distributional assumptions, particular attention being paid to distributions with Archimedean copulas.