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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Gould, Nicholas I. M. Cartis, Coralia Toint, Philippe L.
Copyright Year	2011
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 $\mathcal{O}(\epsilon^{-2})$ 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.
Starting Page	1721
Ending Page	1739
Page Count	19
File Format	PDF
ISSN	10526234
DOI	10.1137/11082381X
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	4
Volume Number	21
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-12-22
Access Restriction	Subscribed
Subject Keyword	global rate of convergence nonlinear programming trust region methods Methods based on necessary conditions exact penalty methods nonsmooth optimization quadratic regularization methods Analysis of algorithms and problem complexity steepest descent methods Mathematical programming methods Nonconvex programming, global optimization global complexity bounds Nonlinear programming Implicit function theorems; global Newton methods Abstract computational complexity for mathematical programming problems Methods of nonlinear programming type
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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