Finding search directions in quasi-Newton methods for minimizing a quadratic function subject to uncertainty

Shen Peng (), Gianpiero Canessa (), David Ek () and Anders Forsgren ()
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Shen Peng: Xidian University
Gianpiero Canessa: KTH Royal Institute of Technology
David Ek: KTH Royal Institute of Technology
Anders Forsgren: KTH Royal Institute of Technology

Computational Optimization and Applications, 2025, vol. 91, issue 1, No 5, 145-171

Abstract: Abstract We investigate quasi-Newton methods for minimizing a strongly convex quadratic function which is subject to errors in the evaluation of the gradients. In particular, we focus on computing search directions for quasi-Newton methods that all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. The BFGS quasi-Newton method may be seen as an ideal method in exact arithmetic and is empirically known to behave very well on a quadratic problem subject to small errors. We investigate large-error scenarios, in which the expected behavior is not so clear. We consider memoryless methods that are less expensive than the BFGS method, in that they generate low-rank quasi-Newton matrices that differ from the identity by a symmetric matrix of rank two. In addition, a more advanced model for generating the search directions is proposed, based on solving a chance-constrained optimization problem. Our numerical results indicate that for large errors, such a low-rank memoryless quasi-Newton method may perform better than a BFGS method. In addition, the results indicate a potential edge by including the chance-constrained model in the memoryless quasi-Newton method.

Keywords: Quadratic programming; Quasi-Newton method; Stochastic quasi-Newton method; Chance constrained model (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-025-00661-4

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