We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented. 1. Introduction As we all know, many problems of considerable practical importance can be related to the solution of nonsmooth optimization of problems (NSOPs) and system of nonsmooth equations. In general, optimization a function is one of the most important problems of real life and plays a fundamental role in mathematics and its applications in the other disciplines such as control theory, optimal control, engineering, and economics. Nonsmooth optimization is one of the research areas in computational mathematics, applied mathematics, and engineering design optimization and also is widely used in many of practical problems. It is necessary to know that several important methods for solving difficult smooth problems lead directly to the need to solve nonsmooth problems, which are either smaller in dimension or simpler in structure. For instance, decomposition methods for solving very large scale smooth problems produce lower-dimensional nonsmooth problems; penalty methods for solving constrained smooth problems result in unconstrained nonsmooth problems; nonsmooth equation methods for solving smooth variational inequalities and smooth nonlinear complementarity problems give arise to systems of nonsmooth equations (see ). The well-known methods for nonsmooth optimization include subgradient method, cutting-planes method, analytic center cutting-planes method, bundle method, trust region method, and bundle trustering method (see ). Note that the most difficult type of optimization problem to solve is a nonsmooth problem. Nonsmooth optimization refers to the more general problem of minimizing functions that are typically not differentiable at their minimizers. The focus of this paper is the numerical solution of NSOPs and system of nonsmooth equations. The techniques for solving the minimization problems and nonsmooth equations are closely related. The outline of the paper
X. Chen, Z. Nashed, and Q. Liqun, “Convergence of Newton's method for singular smooth and nonsmooth equations using adaptive outer inverses,” SIAM Journal on Optimization, vol. 7, no. 2, pp. 445–462, 1997.