On the Evaluation Complexity of Composite Function Minimization with Applications to Nonconvex Nonlinear Programming.

Journal article

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 objective- and constraintevaluation 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. © 2011 Society for Industrial and Applied Mathematics.

Authors: Cartis, C; Gould, N; Toint, P

• Journal: SIAM Journal on Optimization
• Volume: 21
• Issue: 4
• Pages: 1721-1739
• Publication date: 2011-01-01
• DOI: 10.1137/11082381X
• EISSN: 1095-7189
• ISSN: 1052-6234
• Language: English
• Keywords: Trust region methods; Nonsmooth optimization; Exact penalty methods; Quadratic regularization methods; Nonlinear programming; Global rate of convergence; Global complexity bounds; Steepest descent methods