PROJECTED HESSIAN UPDATING ALGORITHMS FOR NONLINEARLY CONSTRAINED OPTIMIZATION.

Jorge Nocedal*, Michael L. Overton

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

135 Scopus citations

Abstract

We consider the problem of minimizing a smooth function of n variables subject to m smooth equality constraints. We begin by describing various approaches to Newton's method for this problem, with emphasis on the recent work of Goodman. This leads to the proposal of a Broyden-type method which updates an n multiplied by (n-m) matrix approximating a 'one-sided projected Hessian' of a Lagrangian function. This method is shown to converge Q-superlinearly. We also give a new short proof of the Boggs-Tolle-Wang necessary and sufficient condition for Q-superlinear convergence of a class of quasi-Newton methods for solving this problem. Finally, we describe an algorithm which updates an approximation to a 'two-sided projected Hessian', a symmetric matrix of order n-m which is generally positive definite near a solution. We present several new variants of this algorithm and show that under certain conditions they all have a local two-step Q-superlinear convergence property, even though only one set of gradients is evaluated per iteration. Numerical results are presented, indicating that the methods may be very useful in practice.

Original languageEnglish (US)
Pages (from-to)821-850
Number of pages30
JournalSIAM Journal on Numerical Analysis
Volume22
Issue number2
StatePublished - Oct 1 1985

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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