An analysis of reduced Hessian methods for constrained optimization

Richard H. Byrd*, Jorge Nocedal

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


We study the convergence properties of reduced Hessian successive quadratic programming for equality constrained optimization. The method uses a backtracking line search, and updates an approximation to the reduced Hessian of the Lagrangian by means of the BFGS formula. Two merit functions are considered for the line search: the ℓ1 function and the Fletcher exact penalty function. We give conditions under which local and superlinear convergence is obtained, and also prove a global convergence result. The analysis allows the initial reduced Hessian approximation to be any positive definite matrix, and does not assume that the iterates converge, or that the matrices are bounded. The effects of a second order correction step, a watchdog procedure and of the choice of null space basis are considered. This work can be seen as an extension to reduced Hessian methods of the well known results of Powell (1976) for unconstrained optimization.

Original languageEnglish (US)
Pages (from-to)285-323
Number of pages39
JournalMathematical Programming
Issue number1-3
StatePublished - Nov 1990


  • Constrained optimization
  • nonlinear programming
  • quasi-Newton methods
  • reduced Hessian methods
  • successive quadratic programming

ASJC Scopus subject areas

  • Software
  • General Mathematics


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