Mathematical Programming

, Volume 63, Issue 1, pp 129–156

Representations of quasi-Newton matrices and their use in limited memory methods


  • Richard H. Byrd
    • Computer Science DepartmentUniversity of Colorado
  • Jorge Nocedal
    • Department of Electrical Engineering and Computer ScienceNorthwestern University
  • Robert B. Schnabel
    • Computer Science DepartmentUniversity of Colorado

DOI: 10.1007/BF01582063

Cite this article as:
Byrd, R.H., Nocedal, J. & Schnabel, R.B. Mathematical Programming (1994) 63: 129. doi:10.1007/BF01582063


We derive compact representations of BFGS and symmetric rank-one matrices for optimization. These representations allow us to efficiently implement limited memory methods for large constrained optimization problems. In particular, we discuss how to compute projections of limited memory matrices onto subspaces. We also present a compact representation of the matrices generated by Broyden's update for solving systems of nonlinear equations.

Key words

Quasi-Newton methodconstrained optimizationlimited memory methodlarge-scale optimization

Copyright information

© The Mathematical Programming Society, Inc. 1994