Nonsmoothness is a typical characteristic of numerous optimization problems originating from both real world and scientific applications. Well known examples from practical optimization are minimax problems used in robust optimization and the reformulation of a constrained optimization problem by adding nonsmooth penalty terms of constraint violations to the original function. Although there are plenty of publications dealing with nonsmooth analysis and optimization, there are only a few software tools available for nonsmooth optimization problems. Therefore, the purpose of this thesis is to develop, implement, and examine an algorithm for unconstrained, nonconvex, and nonsmooth optimization problems. It will be assumed that all nondifferentiabilities occurring in the objective function are caused by the absolute value and those functions that can be expressed in terms of the absolute value as the maximum and minimum function. The idea of the developed optimization method LiPsMin is the minimization of composite piecewise differentiable objective functions via successive piecewise linearization overestimated by a quadratic term. The minimization of the resulting local quadratic subproblem benefits from additional information obtained by exploiting the structure of the underlying piecewise linearization. Convergence results of LiPsMin towards first order optimal points are developed and the numerical performance of the algorithm is investigated.