Worst case approach in convex minimization problems with uncertain data

TL;DR

This paper introduces a worst case scenario approach to quantify errors in convex minimization problems with uncertain data, decomposing errors into approximation and data uncertainty components, demonstrated through nonlinear reaction-diffusion examples.

Contribution

It presents a novel method to analyze and compare errors from data uncertainty and approximation in convex problems using duality gap and worst case analysis.

Findings

Decomposition of worst case error into two computable parts.

Application to nonlinear reaction-diffusion problems.

Numerical examples illustrating the approach.

Abstract

This paper concerns quantitative analysis of errors generated by incompletely known data in convex minimization problems. The problems are discussed in the mixed setting and the duality gap is used as the fundamental error measure. The influence of the indeterminate data is measured using the worst case scenario approach. The worst case error is decomposed into two computable quantities, which allows the quantitative comparison between errors resulting from the inaccuracy of the approximation and the data uncertainty. The proposed approach is demonstrated on a paradigm of a nonlinear reaction-diffusion problem together with numerical examples.