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

A Gradient Sampling method based on Ideal direction for solving  nonsmooth nonconvex optimization problems: convergence analysis and numerical  experiments