On the evaluation complexity of cubic regularization methods for potentially rank-deficient nonlinear least-squares problems and its relevance to constrained nonlinear optimization

Coralia Cartis, Nick Gould, Philippe Toint

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Abstract

We propose a new termination criterion suitable for potentially singular, zero or nonzero residual, least-squares problems, with which cubic regularization variants take at most O(ε-3/2) residual- and Jacobian-evaluations to drive either the Euclidean norm of the residual or its gradient belowε this is the best known bound for potentially rank-deficient nonlinear least-squares problems. We then apply the new optimality measure and cubic regularization steps to a family of least-squares merit functions in the context of a target-following algorithm for nonlinear equality-constrained problems; this approach yields the first evaluation complexity bound of order ε-3/2 for nonconvexly constrained problems when higher accuracy is required for primal feasibility than for dual first-order criticality.

Original languageEnglish
Pages (from-to)1553-1574
Number of pages22
JournalSIAM Journal on Optimization
Volume23
Issue number3
DOIs
Publication statusPublished - 29 Oct 2013

Keywords

  • Constrained nonlinear optimization
  • Cubic regularization methods
  • Evaluation complexity
  • Least-squares problems
  • Worst-case analysis

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