DC programming and DCA for solving Brugnano-Casulli piecewise linear systems

Piecewise linear optimization is one of the most frequently used optimization models in practice, such as transportation, finance and supply-chain management. In this paper, we investigate a particular piece wise linear optimization that is optimizing the norm of piecewise linear functions (NPLF). Specifically, we are interested in solving a class of Brugnano-Casulli piecewise linear systems (PLS), which can be reformulated as an NPLF problem. Speaking generally, the NPLF is considered as an optimization problem with a nonsmooth, nonconvex objective function. A new and efficient optimization approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithms) is developed. With a suitable DC formulation, we design a DCA scheme, named l(1)-DCA, for the problem of optimizing the l(1)-norm of NPLF. Thanks to particular properties of the problem, we prove that under some conditions, our proposed algorithm converges to an exact solution after a finite number of iterations. In addition, when a nonglobal solution is found, a numerical procedure is introduced to find a feasible point having a smaller objective value and to restart l(1)-DCA at this point. Several numerical experiments illustrate these interesting convergence properties. Moreover, we also present an application to the free-surface hydrodynamic problem, where the correct numerical modeling often requires to have the solution of special PLS, with the aim of showing the efficiency of the proposed method. (C) 2016 Elsevier Ltd. All rights reserved.