On the Convergence of Interior-Point Methods for Bound-Constrained Nonlinear Optimization Problems with Noise

TL;DR

This paper studies the convergence of a modified barrier method for bound-constrained nonlinear optimization problems affected by bounded noise, proving convergence properties, proposing a practical stopping test, and analyzing active-set identification.

Contribution

It introduces a noise-tolerant convergence analysis for barrier methods, including a new stopping criterion and active-set identification under noisy evaluations.

Findings

The method converges with noise levels diminishing to zero.

The proposed stopping test effectively detects convergence without parameter estimates.

Active constraints can be identified despite noisy second derivatives.

Abstract

We analyze the convergence properties of a modified barrier method for solving bound-constrained optimization problems where evaluations of the objective function and its derivatives are affected by bounded and non-diminishing noise. The only modification compared to a standard barrier method is a relaxation of the Armijo line-search condition. We prove that the algorithm generates iterates at which the size of the barrier function gradient eventually falls below a threshold that converges to zero if the noise level converges to zero. Based on this result, we propose a practical stopping test that does not require estimates of unknown problem parameters and identifies iterations in which the theoretical threshold is reached. We also analyze the local convergence properties of the method when noisy second derivatives are used. Under a strict-complementarity assumption, we show that iterates stay in a neighborhood around the optimal solution once it is entered. The neighborhood is defined in a scaled norm that becomes narrower for variables with active bound constraints as the barrier parameter is decreased. As a consequence, we show that active bound constraints can be identified despite noise. Numerical results demonstrate the effectiveness of the stopping test and illustrate the active-set identification properties of the method.