Practical methods of optimization; (2nd ed.)
Practical methods of optimization; (2nd ed.)
Trust-region methods
On the Convergence of Successive Linear-Quadratic Programming Algorithms
SIAM Journal on Optimization
Modified Gauss-Newton scheme with worst case guarantees for global performance
Optimization Methods & Software
Recursive Trust-Region Methods for Multiscale Nonlinear Optimization
SIAM Journal on Optimization
Nonlinear programming without a penalty function or a filter
Mathematical Programming: Series A and B
Mathematical Programming: Series A and B
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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.