On the Use of Implicit Updates in Minimum Curvature Multi-step Quasi-Newton Methods

Summary Multi-step quasi-Newton methods for optimization employ, at each iteration, an interpolating polynomial in the variable space to construct a multi-step version of the well-known Secant Equation (the relation which constrains the updating of the Hessian approximation). There is some freedom in the choice of the interpolating polynomial and this freedom is exploited, in the case of two-step methods, by the so-called “Minimum Curvature” algorithms, which produce the’ smoothest’ interpolation, in the sense of obtaining the polynomial with the smallest possible second derivative (measured in some suitable norm). Typically, these norms are defined by a positive-definite matrix and, in this paper, we will consider and compare the use of different matrices in defining the norm. In particular, we will describe the construction of implicit methods, in which, as we will demonstrate, there is no requirement to compute the matrix defining the norm explicitly.