A Note on Minimization Algorithms which make Use of Non-quardratic Properties of the Objective Function
Max Biggs · IMA Journal of Applied Mathematics · 1973
* The scalar n* in (1) is intended to reflect the non-quadratic properties of/(x). A value off/* is chosen at each iteration so that r}*s£yk is, in some sense, a better estimate than fifvfc of the true directional second derivative 8jV/(xt)5fc. In the original paper it was proposed that n* be calculated by expressing f(xk+adk) in the form c£(a) = A\«—a\+b, A > 0, p > 1 (3) and determining the dominant degree, p, and hence a suitable n*, using the value of the function and gradient at xk and xk+8t. However, while computational experience has shown that efficient algorithms for unconstrained minimization can be based on (1), the calculation of//* from (3) has proved a little cumbersome in that it involves the solution of a pair of non-linear equations. In order to avoid this we now propose calculating n* by representing f(xk+tx8k) by the cubic model '(a) = d/da q>(a) and if //* is defined by