A Semiglobal, Practical, Strict Pseudogradient Property for Iterative Methods

TL;DR

This paper introduces the SPSP property for iterative methods, showing it ensures stability and robustness in optimization algorithms, including those with errors, and models a broad class of methods.

Contribution

It defines the SPSP property for search directions and demonstrates its importance for stability and robustness in iterative optimization algorithms.

Findings

SPSP search directions lead to semiglobal, practical asymptotic stability.

The SPSP property is robust to perturbations.

Applicable to algorithms with absolute and relative errors.

Abstract

We consider a class of iterative numerical methods and introduce the notion of semiglobally, practically, strictly pseudogradient (SPSP) search directions. We demonstrate the relevance of the SPSP property in modelling a variety of optimization algorithms, including those subject to absolute and relative errors. We show that the attractors of iterative methods with SPSP search directions exhibit semiglobal, practical, asymptotic stability. Moreover, the SPSP property is robust in the sense that perturbations of SPSP search directions also have the SPSP property.