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. © 2011 Society for Industrial and Applied Mathematics.
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Cartis, C., Gould, N., & Toint, P. (2011). On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming. SIAM Journal on Optimization, 21(4), 1721-1739. https://doi.org/10.1137/11082381X