# Edinburgh Research Group in Optimization - Technical Report ERGO 11-005

Technical Report ERGO 11-005

On the complexity of finding first-order critical points in constrained nonlinear programming
Coralia Cartis, Nicholas I. M. Gould and Philippe L. Toint


The complexity of finding ε-approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(ε-2) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order shorts-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply.


Evaluation complexity, worst-case analysis, constrained nonlinear optimization




Written: 13 April 2011


Published in Mathematical Programming.