On the complexity of finding first-order critical points in constrained nonlinear optimization

Coralia Cartis, Nicholas I M Gould, Philippe L. Toint

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Abstract

The complexity of finding e-Approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(e-2) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-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. © 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Original languageEnglish
Pages (from-to)93-106
Number of pages14
JournalMathematical Programming
Volume144
Issue number1-2
DOIs
Publication statusPublished - 2014

Keywords

  • Constrained nonlinear optimization
  • Evaluation complexity
  • Worst-case analysis

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