<p>An inexact-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{O} (\epsilon _g^{-3/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msubsup> <mi>ϵ</mi> <mi>g</mi> <mrow> <mo>-</mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> iterations to achieve the norm of the gradient smaller than <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\epsilon _g$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϵ</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation> for given <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\epsilon _g>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϵ</mi> <mi>g</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and at most <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal O(\max \{ \epsilon _g^{-3/2 }, \epsilon _H^{-3} \} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mo movablelimits="true">max</mo> <mrow> <mo stretchy="false">{</mo> <msubsup> <mi>ϵ</mi> <mi>g</mi> <mrow> <mo>-</mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>ϵ</mi> <mi>H</mi> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msubsup> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> iterations to reach a second-order stationary point respectively. Notably, the proposed algorithm remains applicable even in cases of the gradient and Hessian of the objective function are unknown. Numerical experiments are performed with gradient and Hessian being approximated by forward finite-differences to illustrate the theoretical results and numerical comparison.</p>

An Adaptive Cubic Regularization Inexact-Newton Method on Riemannian Manifolds

期刊论文
发表日期：

2025年11月3日

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Mauricio S. Louzeiro,
Gilson N. Silva,
Jinyun Yuan,
Daoping Zhang

摘要

An inexact-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most $\mathcal{O} (\epsilon _g^{-3/2})$ $O (ϵ_{g}^{- 3 / 2})$ iterations to achieve the norm of the gradient smaller than $\epsilon _g$ $ϵ_{g}$ for given $\epsilon _g>0$ $ϵ_{g} > 0$ , and at most $\mathcal O(\max \{ \epsilon _g^{-3/2 }, \epsilon _H^{-3} \} )$ $O (max {ϵ_{g}^{- 3 / 2}, ϵ_{H}^{- 3}})$ iterations to reach a second-order stationary point respectively. Notably, the proposed algorithm remains applicable even in cases of the gradient and Hessian of the objective function are unknown. Numerical experiments are performed with gradient and Hessian being approximated by forward finite-differences to illustrate the theoretical results and numerical comparison.