@article{5ac8cbc632ea4bd299695730256141c3,
title = "On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming",
abstract = "We estimate the worst-case complexity of minimizing an unconstrained, nonconvex composite objective with a structured nonsmooth term by means of some first-order methods. We find that it is unaffected by the nonsmoothness of the objective in that a first-order trust-region or quadratic regularization method applied to it takes at most O(ε-2) function evaluations to reduce the size of a first-order criticality measure below ε. Specializing this result to the case when the composite objective is an exact penalty function allows us to consider the objective- and constraintevaluation worst-case complexity of nonconvex equality-constrained optimization when the solution is computed using a first-order exact penalty method. We obtain that in the reasonable case when the penalty parameters are bounded, the complexity of reaching within ε of a KKT point is at most O(ε-2) problem evaluations, which is the same in order as the function-evaluation complexity of steepest-descent methods applied to unconstrained, nonconvex smooth optimization. {\textcopyright} 2011 Society for Industrial and Applied Mathematics.",
keywords = "nonlinear optimization, composite minimization, complexity, nonconvex problems.",
author = "Coralia Cartis and Nick Gould and Philippe Toint",
note = "Publication code : FP SB092/2011/06 ; SB04977/2011/06",
year = "2011",
month = jan,
day = "1",
doi = "10.1137/11082381X",
language = "English",
volume = "21",
pages = "1721--1739",
journal = "SIAM Journal on Optimization",
issn = "1052-6234",
publisher = "Society for Industrial and Applied Mathematics",
number = "4",
}