In this paper we introduce a new approach to solve constrained nonlinear non-smooth prograing probles ith any desirable accuracy even hen the objective function is a non-smooth one. In this approach for any given desirable accuracy, all the nonlinear functions of original problem (in objective function and in constraints) are approximated by a piecewise linear functions. We then represent an efficient algorithm to find the global solution of the later problem. The obtained solution has desirable accuracy and the error is completely controllable. One of the main advantages of our approach is that the approach can be extended to problems with non-smooth structure by introducing a novel definition of Global Weak Differentiation in the sense of L1 norm. Finally some numerical examples are given to show the efficiency of the proposed approach to solve approximately constraints nonlinear non-smooth programming problems.