Worst-case evaluation complexity of a quadratic penalty method for nonconvex optimization

This paper addresses the worst-case evaluation complexity of a version of the standard quadratic penalty method for smooth non convex optimization problems with constraints. The method analysed allows inexact solution of the subproblems and do not require prior knowledge of the Lipschitz constants related with the problem. When an approximate feasible point is used as starting point, ( ) it is shown that the referred method takes at most O log(s(-1) 0 e(-2)) outer iterations to generate an e-approximate KKT point, where s(0) is the first penalty parameter. For equality constrained problems, this bound yields to an evaluation complexity bound of O (e(-4)), when s0 = e(-2) and suitable first-order methods are used as inner solvers. complexity bound of O (e(-(P+1)/P)) is established when appropriFor problems having only linear equality constraints, an evaluation ate P-order methods (P = 2) are used as inner solvers. Illustrative numerical results are also presented and corroborate the theoretical predictions.