Nonsmooth Analysis

• J. R. Birge and Liqun Qi, "Semiregularity and generalized subdifferentials with applications to stochastic programmming," Mathematics of Operations Research 4 (1993) 982-1005.

  Abstract

  The Michel-Penot subdifferential of a locally Lipschitzian function is the principal part of the Clarke subdifferential. It coincides with the G-derivative at differentiable points. A locally Lipschitzian function can be determined by its Michel-Penot subdifferential uniquely up to an additive constant though this cannot be done by its Clarke subdifferential if the set of abnormal points is not negligible. A set-valued operator is the Michel-Penot subdifferential of a locally Lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition. Various calculus rules hold for the Michel-Penot subdifferential. Equalities hold for these rules at a point under semiregularity, which is weaker than regularity. For a locally Lipschitzian function in a separable Banach space, semiregularity holds everywhere except for a Haar zero set. Applications in optimization are discussed.

• J.R. Birge, L. Qi, and Z. Wei, " Two methods for nonsmooth optimization," Technical Report 95-3, Deparment of Industrial and Operations Engineering, University of Michigan, 1995.

  Abstract

  Two methods for solving nonsmooth minimization problems are proposed and their convergence properties are studied. The first method is a locally superlinear convergent method for solving nonsmooth convex optimization problems. An application of this method relaxes the convergence conditions of Rockafellar's results for the proximal point algorithm. The second method extends the proximal point algorithm to solve nonsmooth nonconvex optimization problems. It applies the Moreau-Yosida regularization as well as the bundle method to a sequence of convex functions. In particular, it extends Gueler's proximal point algorithm with better convergence properties.