Quasi-Newton methods for minimizing a quadratic function subject to uncertainty

TL;DR

This paper studies the behavior of quasi-Newton methods, especially BFGS and memoryless variants, for quadratic minimization under gradient evaluation errors, highlighting their performance differences and proposing a chance-constrained approach.

Contribution

It analyzes quasi-Newton methods under gradient errors, compares BFGS and memoryless variants, and introduces a chance-constrained model for improved search directions.

Findings

Memoryless quasi-Newton often outperforms BFGS with large errors.

Chance-constrained search directions can improve final accuracy.

BFGS behaves well with small errors, but less so with large errors.

Abstract

We investigate quasi-Newton methods for minimizing a strictly convex quadratic function which is subject to errors in the evaluation of the gradients. The methods all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. A BFGS quasi-Newton method is empirically known to behave very well on a quadratic problem subject to small errors. We also investigate large-error scenarios, in which the expected behavior is not so clear. In particular, we are interested in the behavior of quasi-Newton matrices that differ from the identity by a low-rank matrix, such as a memoryless BFGS method. Our numerical results indicate that for large errors, a memory-less quasi-Newton method often outperforms a BFGS method. We also consider a more advanced model for generating search directions, based on solving a chance-constrained optimization problem. Our results indicate that such a model often gives a slight advantage in final accuracy, although the computational cost is significantly higher.