Quasi-Newton Methods Using Multiple Secant Equations.

Abstract

Abstract : The author investigated quasi-Newton methods for unconstrained optimization and systems of nonlinear equations where each approximation to the Hessian or Jacobian matrix obeys several secant equations. For systems of nonlinear equations, this work is just a simplification and generalization of previous work by Barnes and Gay and Schnabel. For unconstrained optimization, the desire that the Hessian approximation obey more than one secant equation may be inconsistent with the requirements that it be symmetric. Presented are very simple necessary and sufficient conditions for there to exist symmetric, or symmetric and positive definite, updates that obey multiple secant equations. If these conditions are satisfied, one can derive generalizations of all the standard symmetric updates, including the PSB, DFP, and BFGS, that satisfy multiple secant equations. The author shows how to successfully specify multiple secant equations for unconstrained optimization, and that algorithms using these secant equations and the generalized PSB, DFP or BFGS updates are locally and q-superlinearly convergent under standard assumptions.