Note on an extension of "Davidon" methods to nondifferentiable functions

1. This note summarizes a paper [4] to appear in full elsewhere. It presents an algorithm for the minimization of a general (not necessarily differentiable) convex function. Its central idea is the construction of descent directions as projections of the origin onto the convex hull of previously calculated subgradients as long as satisfactory progress can be made. Using projection to obtain a direction,of movement for the conjugate gradient method is not new [5,8], but previous methods have projected onto the affine manifold spanned by the gradients, rather than onto the convex hull. Alternatively, Demjanov [2] has employed a projection onto the convex hull of a local set of subgradients to obtain a direction of steepest descent. The present algorithm may be said to combine these ideas. It is also connected with proposals of Bertsekas and Mitter [ 1 ] (by Theorem 2 below), and has many points in common with a method due to Wolfe [ 7].