Scalable subspace methods for derivative-free nonlinear least-squares optimization

We introduce a general framework for large-scale model-based derivative-free optimization based on iterative minimization within random subspaces. We present a probabilistic worst-case complexity analysis for our method, where in particular we prove high-probability bounds on the number of iteration...

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Bibliographic Details
Published in:Mathematical programming Vol. 199; no. 1-2; pp. 461 - 524
Main Authors: Cartis, Coralia, Roberts, Lindon
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2023
Springer
Subjects:
Algebra
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Numerical Analysis
Theoretical
65K05
90C30
Worst case complexity
90C56
Large-scale optimization
Nonlinear least-squares
Derivative-free optimization
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We introduce a general framework for large-scale model-based derivative-free optimization based on iterative minimization within random subspaces. We present a probabilistic worst-case complexity analysis for our method, where in particular we prove high-probability bounds on the number of iterations before a given optimality is achieved. This framework is specialized to nonlinear least-squares problems, with a model-based framework based on the Gauss–Newton method. This method achieves scalability by constructing local linear interpolation models to approximate the Jacobian, and computes new steps at each iteration in a subspace with user-determined dimension. We then describe a practical implementation of this framework, which we call DFBGN. We outline efficient techniques for selecting the interpolation points and search subspace, yielding an implementation that has a low per-iteration linear algebra cost (linear in the problem dimension) while also achieving fast objective decrease as measured by evaluations. Extensive numerical results demonstrate that DFBGN has improved scalability, yielding strong performance on large-scale nonlinear least-squares problems.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01836-1