Mathematical Programming

, Volume 63, Issue 1, pp 129-156

First online:

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

  • Richard H. ByrdAffiliated withComputer Science Department, University of Colorado
  • , Jorge NocedalAffiliated withDepartment of Electrical Engineering and Computer Science, Northwestern University
  • , Robert B. SchnabelAffiliated withComputer Science Department, University of Colorado

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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 method constrained optimization limited memory method large-scale optimization