A PR Exercise – a.k.

a.k. from thus spake a.k.

In the last few posts we've been looking at the BFGS quasi-Newton algorithm for minimising multivariate functions. This uses iteratively updated approximations of the Hessian matrix of second partial derivatives in order to choose directions in which to search for univariate minima, saving the expense of calculating it explicitly. A particularly useful property of the algorithm is that if the line search satisfies the Wolfe conditions then the positive definiteness of the Hessian is preserved, meaning that the implied locally quadratic approximation of the function must have a minimum.
Unfortunately for large numbers of dimension the calculation of the approximation will still be relatively expensive and will require a significant amount of memory to store and so in this post we shall take a look at an algorithm that only uses the vector of first partial derivatives.