An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould & Toint (Part I, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter even second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most O(ǫ −3/2 ) iterations to drive the objective's gradient below the desired accuracy ǫ, and O(ǫ −3 ), to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved by Nesterov & Polyak (Math. Programming 108(1), 2006, pp 177-205) for their Algorithm 3.3 which minimizes the cubic model globally on each iteration. Our approach is more general, and relevant to practical (large-scale) calculations, as ARC allows the cubic model to be solved only approximately and may employ approximate Hessians.

An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould & Toint (Part I, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter even second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most O(ǫ −3/2) iterations to drive the objective's gradient below the desired accuracy ǫ, and O(ǫ −3), to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved by Nesterov & Polyak (Math. Programming 108(1), 2006, pp 177-205) for their Algorithm 3.3 which minimizes the cubic model globally on each iteration. Our approach is more general, and relevant to practical (large-scale) calculations , as ARC allows the cubic model to be solved only approximately and may employ approximate Hessians.

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 ε. 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 ε in O(1/ε 2 ) iterations. Furthermore, RTR returns a point where also the Riemannian Hessian's least eigenvalue is larger than −ε in O(1/ε 3 ) iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy ε (up to constants) and hence are sharp in that sense.These are the first deterministic results for global rates of convergence to approximate first-and second-order Karush-Kuhn-Tucker points on manifolds. They apply in particular for optimization constrained to compact submanifolds of R n , under simpler assumptions.