Nonconvex multivariate piecewise-linear fitting using the difference-of-convex representation

Abstract

We address the problem of finding a continuous piecewise-linear (CPWL) approximation of a nonlinear function that satisfies predefined error-tolerances, and keeps the number of polytopes low. We introduce the difference-of-convex (DC) CPWL representation that represents any CPWL function as the difference of two convex CPWL functions. Any convex CPWL function can be represented as the maximum of a set of affine functions, so the polytopes defining a DC CPWL function can be implicitly defined by the affine functions. By simply searching the parameters of affine functions, CPWL approximations can be produced which contain polytopes of any type. We use the DC CPWL representation to develop a CPWL approximation algorithm and prove its finite convergence. We compare the quality of CPWL approximations produced from the proposed algorithm to the best-known existing method. We find that the DC CPWL approximation consistently requires fewer polytopes to meet the same error-tolerance.