Complexity of a Projected Newton-CG Method for Optimization with Bounds

Yue Xie; Stephen J. Wright

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arXiv · 2103.15989

Complexity of a Projected Newton-CG Method for Optimization with Bounds

Yue Xie·Stephen J. Wright

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Abstract

This paper describes a method for solving smooth nonconvex minimization problems subject to bound constraints with good worst-case complexity guarantees and practical performance. The method contains elements of two existing methods: the classical gradient projection approach for bound-constrained optimization and a recently proposed Newton-conjugate gradient algorithm for unconstrained nonconvex optimization. Using a new definition of approximate second-order optimality parametrized by some tolerance $\epsilon$ (which is compared with related definitions from previous works), we derive complexity bounds in terms of $\epsilon$ for both the number of iterations required and the total amount of computation. The latter is measured by the number of gradient evaluations or Hessian-vector products. We also describe illustrative computational results on several test problems from low-rank matrix optimization.

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Yue Xie, Stephen J. Wright. 2021-03-29. Complexity of a Projected Newton-CG Method for Optimization with Bounds. https://arxiv.org/abs/2103.15989

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