  In this paper, we combine the positive aspects of the Gradient Sampling (GS)
and bundle methods, as the most efficient methods in nonsmooth optimization, to
develop a robust method for solving unconstrained nonsmooth convex optimization
problems. The main aim of the proposed method is to take advantage of both GS
and bundle methods, meanwhile avoiding their drawbacks. At each iteration of
this method, to find an efficient descent direction, the GS technique is
utilized for constructing a local polyhedral model for the objective function.
If necessary, via an iterative improvement process, this initial polyhedral
model is improved by some techniques inspired by the bundle and GS methods. The
convergence of the method is studied, which reveals the following positive
features (i) The convergence of our method is independent of the number of
gradient evaluations required to establish and improve the initial polyhedral
models. Thus, the presented method needs much fewer gradient evaluations in
comparison to the original GS method. (ii) As opposed to GS type methods, the
objective function need not be continuously differentiable on a full measure
open set in $\mathbb R^n$ to ensure the convergence for the class of convex
problems. Apart from the mentioned advantages, by means of numerical
simulations, we show that the presented method provides promising results in
comparison with GS methods, especially for large scale problems. Moreover, in
contrast with bundle methods, our method is not very sensitive to the accuracy
of supplied gradients.
