Hindawi Publishing Corporation Journal of Mathematics Volume 2013, Article ID 750834, 10 pages http://dx.doi.org/10.1155/2013/750834 Research Article A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations Hamid Reza Erfanian, M. H. Noori Skandari, and A. V. Kamyad Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad 9177948953, Iran Correspondence should be addressed to Hamid Reza Erfanian; erfanian@usc.ac.ir Received 29 August 2012; Accepted 19 November 2012 Academic Editor: Ellina Grigorieva Copyright © 2013 Hamid Reza Erfanian et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 frst 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. Ten, we apply the results for solving system of nonsmooth equations. Finally, for efciency 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 disci- plines such as control theory, optimal control, engineering, and economics. Nonsmooth optimization is one of the research areas in computational mathematics, applied mathematics, and engi- neering design optimization and also is widely used in many of practical problems. It is necessary to know that several important methods for solving difcult 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; non- smooth equation methods for solving smooth variational inequalities and smooth nonlinear complementarity prob- lems give arise to systems of nonsmooth equations (see [1]). Te 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 [2]). Note that the most difcult 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 diferentiable at their minimizers. Te focus of this paper is the numerical solution of NSOPs and system of nonsmooth equations. Te techniques for solving the minimization problems and nonsmooth equations are closely related. Te outline of the paper is as follows. In Section 2, we introduce the reader to a new generalized derivative (GD) for one variable and multivariable functions (see Kamyad et al. [3]). A new approach for NSOP based on GD is studied in Section 3. Also, using the last section, an approach for solving system of nonsmooth equations is considered in Section 4. Some conclusive remarks are given in Section 5. Finally, we