A BUNDLE-TYPE QUASI-NEWTON METHOD FOR NONCONVEX NONSMOOTH OPTIMIZATION

In this paper, we propose a bundle-type quasi-Newton method for minimizing a nonconvex nonsmooth function. The method is based on the redistributed bundle method with an on-the-fly convexification technique. At each iteration, the convexification parameter and the prox-parameter are suitably modified to guarantee that the proximal point of a piecewise affine model of a local convexification function approximates well-enough the proximal point of the objective function f at x(k). A quasi-Newton procedure is added at the end of each serious step. Specifically, we construct a suitable search direction d(k) via the BFGS update and monitor the reduction in the norm of the approximate subgradient to recognize whether an Armijo-type line search on f should be executed. Global convergence of the algorithm is established in the sense that there exists an accumulation point of the serious iterations such that it is a stationary point of f. Superlinear convergence is proved under suitable assumptions. Preliminary numerical results are reported to illustrate that the method is efficient and has advantages over the redistributed bundle method.