Journal
IMA JOURNAL OF NUMERICAL ANALYSIS
Volume 32, Issue 4, Pages 1662-1695Publisher
OXFORD UNIV PRESS
DOI: 10.1093/imanum/drr035
Keywords
nonlinear optimization; convex constraints; cubic regularization; regularization; numerical algorithms; global convergence; worst-case complexity
Categories
Funding
- Royal Society [14265]
- EPSRC [EP/E053351/1, EP/F005369/1, EP/G038643/1]
- European Science Foundation through the OPTPDE
- Engineering and Physical Sciences Research Council [EP/E053351/1, EP/I013067/1] Funding Source: researchfish
- EPSRC [EP/E053351/1, EP/I013067/1] Funding Source: UKRI
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The adaptive cubic regularization algorithm described in Cartis et al. (2009, Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results. Math. Program., 127, 245-295; 2010, Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity [online]. Math. Program., DOI: 10.1007/s10107-009-0337-y) is adapted to the problem of minimizing a nonlinear, possibly nonconvex, smooth objective function over a convex domain. Convergence to first-order critical points is shown under standard assumptions, without any Lipschitz continuity requirement on the objective's Hessian. A worst-case complexity analysis in terms of evaluations of the problem's function and derivatives is also presented for the Lipschitz continuous case and for a variant of the resulting algorithm. This analysis extends the best-known bound for general unconstrained problems to nonlinear problems with convex constraints.
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