On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming

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 $\mathcal{O}(\epsilon^{-2})$ function evaluations to reduce the size of a first-order criticality measure below $\epsilon$. Specializing this result to the case when the composite objective is an exact penalty function allows us to consider the objective- and constraint-evaluation 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 $\epsilon$ of a KKT point is at most $\mathcal{O}(\epsilon^{-2})$ problem evaluations, which is the same in order as the function-evaluation complexity of steepest-descent methods applied to unconstrained, nonconvex smooth optimization.