A feasible second order bundle algorithm for nonsmooth, nonconvex optimization problems with inequality constraints: I. Derivation and convergence

TL;DR

This paper introduces a second order bundle algorithm for nonsmooth, nonconvex optimization with inequality constraints, extending the bundle-Newton method to handle constraints without penalty functions, and proves its global convergence.

Contribution

It develops a new second order bundle algorithm for constrained nonsmooth optimization, avoiding penalty functions and establishing convergence.

Findings

Algorithm successfully handles nonsmooth, nonconvex problems with constraints.

Global convergence is proven under mild assumptions.

The method improves iteration points by solving convex quadratically constrained quadratic programs.

Abstract

This paper extends the SQP-approach of the well-known bundle-Newton method for nonsmooth unconstrained minimization to the nonlinearly constrained case. Instead of using a penalty function or a filter or an improvement function to deal with the presence of constraints, the search direction is determined by solving a convex quadratically constrained quadratic program to obtain good iteration points. Issues arising with this procedure concerning the bundling process as well as in the line search are discussed. Furthermore, global convergence of the method is shown under certain mild assumptions.