On averaging and representation properties of the BFGS and related secant updates

In this talk we present several representation theorems and averaging theorems for members of the difference class of secant updates introduced by Broadlie, Gourlay , and Greenstadt in 1973. The BFGS update is a well-known member of this difference class. We begin with the notion of an update kernel and use it to construct a new parametrization of the difference class of secant updates. Next we derive several remarkable formulae for the derivatives of the update kernels. The functional form of the parametrization and the differentiation formulae lead to immediate conjecture and proof of representations of members of the difference class as members of the 1972 Dennis class via the mean-value theorem and representations as members of the 1975 Davidon class via Taylor's theorem. This extends 1976 work of Gay and 1977 work of Schnabel. A major contribution is that the integral form of the mean-value theorem leads to a proof that the BFGS update is point wise the infinite average of all the updates on the one dimension manifold in the Dennis class that connects the DFP secant update to the Greenstadt update. Finally we show that the BFGS update can be expressed as the point wise average of these latter two updates. Analogous results hold for all secant updates that belong to the difference class. The amazing manner in which the component pieces of this theory reinforce each other is yet one more testimony for the true beauty and elegance of mathematics.