### Technical Report ERGO 11-002

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

*Coralia Cartis, Nicholas I. M. Gould and Philippe L. Toint*

##### 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 objectiveand 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 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.

##### Download:

ERGO-11-002.pdf

##### History:

Written: 8 February 2011

Revised: 1 May 2011

##### Status:

Published in *SIAM Journal on Optimization*.