Sharp bounds in perturbed smooth optimization

TL;DR

This paper derives precise bounds on how solutions to convex smooth optimization problems change under various perturbations, with applications in statistics, machine learning, and control.

Contribution

It provides explicit error bounds and solution difference expansions for perturbed convex optimization problems, enhancing robustness analysis.

Findings

Explicit solution difference bounds for perturbed problems

Error terms quantified in solution expansions

Applicable to linear, quadratic, and smooth perturbations

Abstract

This paper studies the problem of perturbed convex and smooth optimization. The main results describe how the solution and the value of the problem change if the objective function is perturbed. Examples include linear, quadratic, and smooth additive perturbations. Such problems naturally arise in statistics and machine learning, stochastic optimization, stability and robustness analysis, inverse problems, optimal control, etc. The results provide accurate expansions for the difference between the solution of the original problem and its perturbed counterpart with an explicit error term.