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Adaptive cubic regularisation methods for unconstrained optimization. II: Worst-case function- and derivative-evaluation complexity. (English) Zbl 1229.90193
Summary: An adaptive regularisation framework using cubics (ARC) was proposed for unconstrained optimization and analysed in Part I [Math. Program. 127, No. 2 (A), 245–295 (2011; Zbl 1229.90192)], generalizing at the same time an unpublished method due to A. Griewank [“The modification of Newton’s method for unconstrained optimization by bounding cubic terms”, Technical Report NA/12, 1981, DAMTP, University of Cambridge (1981)], an algorithm by Y. Nesterov and B. T. Polyak [Math. Program. 108, No. 1 (A), 177–205 (2006; Zbl 1142.90500)] and a proposal by M. Weiser, P. Deuflhard and B. Erdmann [Optim. Methods Softw. 22, No. 3, 413–431 (2007; Zbl 1128.74007)]. 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 second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most \(O(\varepsilon^{-3/2})\) iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below the desired accuracy \({\varepsilon}\), and \(O(\varepsilon^{-3})\) iterations, to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved for Algorithm 3.3 of Nesterov and Polyak [loc. cit.] which minimizes the cubic model globally on each iteration. Our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate Hessians.

MSC:
90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
49M37 Numerical methods based on nonlinear programming
49M15 Newton-type methods
58C15 Implicit function theorems; global Newton methods on manifolds
90C60 Abstract computational complexity for mathematical programming problems
68Q25 Analysis of algorithms and problem complexity
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