A simple Newton method for local nonsmooth optimization

Abstract

Superlinear convergence has been an elusive goal for black-box nonsmooth optimization. Even in the convex case, the subgradient method is very slow, and while some cutting plane algorithms, including traditional bundle methods, are popular in practice, local convergence is still sluggish. Faster variants depend either on problem structure or on analyses that elide sequences of "null" steps. Motivated by a semi-structured approach to optimization and the sequential quadratic programming philosophy, we describe a new bundle Newton method that incorporates second-order objective information with the usual linear approximation oracle. One representative problem class consists of maxima of several smooth functions, individually inaccessible to the oracle. Given as additional input just the cardinality of the optimal active set, we prove local quadratic convergence. A simple implementation shows promise on more general functions, both convex and nonconvex, and suggests first-order analogues.