Nonsmooth DC optimization support vector machines method for piecewise linear regression

Abstract

A new regression method called the adaptive piecewise linear support vector regression (A-PWLSVR) is introduced. We use the L1-risk function to define regression errors and apply the support vector machine approach in combination with the piecewise linear regression to develop a model for regression problems. We formulate the model as an unconstrained nonconvex nonsmooth optimization problem, where the objective function is represented as a difference of two convex (DC) functions. To address the nonconvexity of the problem a novel incremental approach is proposed. This approach builds the piecewise linear estimates by applying an adaptive selection procedure for the model parameters. The approach enables us to select starting points being rough approximations of the solution. The double bundle method for nonsmooth DC optimization is applied to solve the optimization problems. The proposed A-PWLSVR method is evaluated on several synthetic and real-world data sets for regression and compared with some mainstream regression methods.