期刊
MATHEMATICAL PROGRAMMING
卷 130, 期 2, 页码 295-319出版社
SPRINGER
DOI: 10.1007/s10107-009-0337-y
关键词
Nonlinear optimization; Unconstrained optimization; Cubic regularization; Newton's method; Trust-region methods; Global complexity bounds; Global rate of convergence
类别
资金
- EPSRC [GR/S42170]
- Engineering and Physical Sciences Research Council [EP/F005369/1] Funding Source: researchfish
- EPSRC [EP/F005369/1] Funding Source: UKRI
An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program, doi:10.1007/s10107-009-0286-5, 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177-205, 2006) and a proposal by Weiser, Deuflhard and Erdmann (Optim Methods Softw 22(3):413-431, 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 second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most O(epsilon(-3/2)) iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below the desired accuracy epsilon and O(epsilon(-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 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.
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