  In this paper, a modification to the Gradient Sampling (GS) method for
minimizing nonsmooth nonconvex functions is presented. One drawback in GS
method is the need of solving a Quadratic optimization Problem (QP) at each
iteration, which is time-consuming especially for large scale objectives. To
resolve this difficulty, we propose a new descent direction, namely Ideal
direction, for which there is no need to consider any quadratic or linear
optimization subproblem. It is shown that, this direction satisfies Armijo step
size condition and can be used to make a substantial reduction in the objective
function. Furthermore, we prove that using Ideal directions preserves the
global convergence of the GS method. Moreover, under some moderate assumptions,
we present an upper bound for the number of serious iterations. Using this
upper bound, we develop a different strategy to study the convergence of the
method. We also demonstrate the efficiency of the proposed method using small,
medium and large scale problems in our numerical experiments.
