Data-compatibility of algorithms

TL;DR

This paper introduces a data-compatibility approach to constrained optimization, focusing on solutions that are sufficiently close to the optimal set and value, offering an alternative to traditional convergence analysis.

Contribution

It proposes a data-compatibility framework for constrained optimization and demonstrates its application to the Hybrid Subgradient Method (HSM), including a string-averaging variant.

Findings

Demonstrates data-compatibility of HSM in convex optimization

Introduces a string-averaging HSM method

Discusses relevance to disjoint constraint sets

Abstract

The data-compatibility approach to constrained optimization, proposed here, strives to a point that is "close enough" to the solution set and whose target function value is "close enough" to the constrained minimum value. These notions can replace analysis of asymptotic convergence to a solution point of infinite sequences generated by specific algorithms. We consider a problem of minimizing a convex function over the intersection of the fixed point sets of nonexpansive mappings and demonstrate the data-compatibility of the Hybrid Subgradient Method (HSM). A string-averaging HSM is obtained as a by-product and relevance to the minimization over disjoint hard and soft constraints sets is discussed.