An All-Inclusive Efficient Region of Updates for Least Change Secant Methods

Least change secant methods, for function minimization, depend on finding a “good” symmetric positive definite update to approximate the Hessian. This update contains new curvature information while simultaneously preserving, as much as possible, the built-up information from the previous update. Updates are generally derived using measures of least change based on some function of the eigenvalues of the (scaled) Hessian. A new approach for finding good least change updates is the multicriteria problem of Byrd, which uses the deviation from unity, of the n eigenvalues of the scaled update, as measures of least change. The efficient (multicriteria optimal) class for this problem is the Broyden class on the “good” side of the symmetric rank one (SR1) update called the Broyden efficient class. This paper uses the framework of multicriteria optimization and the eigenvalues of the scaled (sized) and inverse scaled updates to study the question of what is a good update. In particular, it is shown that the basic...