Bundle Methods, Cutting-Plane Algorithms and σ-Newton Directions

Abstract

Recently Lemaréchal and Zowe [7] have introduced a theoretical second-order model for minimizing a real, not necessarily differentiable, convex function defined on]Rn. This model approximates the convex function f along any fixed direction d and is based on the variation with respect to a of the perturbed directional derivative f ^′_σ (x,d) (all definitions in convex analysis used in this paper can be found in the classical book by Rockafallar [9]). With this help, a second-order expansion of f(x+d) — f(x), depending on σ ≥ 0, is obtained at the current iterate x and a σ-Newton direction is naturally defined as a direction which minimizes this expansion (when f is twice continuously differentiable on a neighborhodd of x and σ = 0, then this direction coincides with the classical Newton direction).