Journal title
IMA Journal of Numerical Analysis
DOI
10.1093/imanum/drx080
Issue
1
Volume
39
Last updated
2024-04-15T17:57:13.027+01:00
Page
1-33
Abstract
We consider the minimization of a cost function $f$ on a manifold $M$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance $\varepsilon$. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of $f$ to the tangent spaces of $M$, both of these algorithms produce points with Riemannian gradient smaller than $\varepsilon$ in $O(1/\varepsilon^2)$ iterations. Furthermore, RTR returns a point where also the Riemannian Hessian's least eigenvalue is larger than -$\varepsilon$ in $O(1/\varepsilon^3)$ iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy $\varepsilon$ (up to constants) and hence are sharp in that sense. These are the first general results for global rates of convergence to approximate first- and second-order KKT points on manifolds. They apply in particular for optimization constrained to compact submanifolds of $\mathbb{R}^n$, under simpler assumptions.
Symplectic ID
808769
Submitted to ORA
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Publication type
Journal Article
Publication date
07 Feb 2018