Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

• Published in: Mathematical programming Vol. 163; no. 1-2; pp. 359 - 368
• Main Authors: Birgin, E. G., Gardenghi, J. L., Martínez, J. M., Santos, S. A., Toint, Ph. L.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2017; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applied mathematics; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Complexity; Continuity (mathematics); Derivatives; Full Length Paper; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Nonlinear programming; Nonlinearity; Numerical Analysis; Optimization; Regularization methods; Studies; Texts; Theoretical; Evaluation complexity; 68Q25; Unconstrained optimization; 90C30; 65K05; Nonlinear optimization; 49M37; High-order models; 90C60; Regularization
• ISSN: 0025-5610; 1436-4646; 1436-4646
• DOI: 10.1007/s10107-016-1065-8

The worst-case evaluation complexity for smooth (possibly nonconvex) unconstrained optimization is considered. It is shown that, if one is willing to use derivatives of the objective function up to order p (for p ≥ 1 ) and to assume Lipschitz continuity of the p -th derivative, then an ϵ -approximate first-order critical point can be computed in at most O ( ϵ - ( p + 1 ) / p ) evaluations of the problem’s objective function and its derivatives. This generalizes and subsumes results known for p = 1 and p = 2 .