AN ALL-INCLUSIVE EFFICIENT REGION OF UPDATES FOR LEAST CHANGE SECANT METHODS

Least change secant methods, for function minimization, depend on find ing a ''good'' symmetric positive definite update to approximate the H essian. This update contains new curvature information while simultane ously preserving, as much as possible, the built-up information from t he previous update. Updates are generally derived using measures of le ast change based on some function of the eigenvalues of the (scaled) H essian. A new approach for finding good least change updates is the mu lticriteria problem of Byrd, which uses the deviation from unity, of t he n eigenvalues of the scaled update, as measures of least change. Th e efficient (multicriteria optimal) class for this problem is the Broy den class on the ''good'' side of the symmetric rank one (SR1) update called the Broyden efficient class. This paper uses the framework of m ulticriteria optimization and the eigenvalues of the scaled (sized) an d inverse scaled updates to study the question of what is a good updat e. In particular, it is shown that the basic multicriteria notions of efficiency and proper efficiency yield a region of updates that contai ns the well-known updates studied to date. This provides a unified fra mework for deriving updates. First, the inverse efficient class is fou nd. It is then shown that the Broyden efficient class and inverse effi cient class are in fact also proper efficient classes. Then, allowing sizing and an additional function in the multicriteria problem, result s in a two parameter efficient region of updates that includes many of the updates studied to date, e.g., it includes the Oren-Luenberger se lf-scaling updates, as well as the Broyden efficient class. This effic ient region, called the self-scaling efficient region, is proper effic ient and lies between two curves, where the first curve is determined by the sized SR1 updates while the second curve consists of the optima l conditioned updates. Numerical tests are included that compare updat es inside and 'outside the efficient region.