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

Least change secant methods, for function minimization, depend on nding a \good" symmetric positive deenite 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 eigen-values of the (scaled) Hessian. A new approach for nding good least change updates is the multicriteria problem of Byrd. This uses the deviation from unity, of the n eigenvalues of the scaled update, as measures of least change. The eecient (multicriteria optimal) class for this problem is the Broyden class on the \good" side of the symmetric rank one (SR1) update. It is called the Broyden EEcient 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 multicriteria notions of eeciency and proper eeciency yield a region of updates that contains the well known updates studied to date. This provides a uniied framework for deriving updates. First, the inverse eecient class is found. It is then shown that the Broyden eecient class and inverse eecient class are in fact also proper eecient classes. Then, allowing sizing and an additional function in the multi-criteria problem, results in a two parameter eecient region of updates that includes many of the updates studied to date, e.g., it includes the Oren-Luenberger self-scaling updates, as well as the Broyden EEcient Class. This eecient region, called the Self-Scaling EEcient Region, is proper eecient and lies between two curves, where the rst curve is determined by the sized SR1 updates while the second curve consists of the optimal conditioned updates. Numerical tests are included that compare updates inside and outside the eecient region.