Optimal breakpoint selection method for piecewise linear approximation

TL;DR

This paper introduces RAM, an optimal method for selecting breakpoints in piecewise linear approximation that minimizes error for both convex and concave functions, improving approximation accuracy.

Contribution

The paper presents RAM, a novel iterative rotational adjusting method for optimal breakpoint placement in piecewise linearization, with proven optimality and broad applicability.

Findings

RAM effectively minimizes approximation error.

The method converges to optimal breakpoint positions.

Numerical experiments validate the method's efficiency.

Abstract

Piecewise linearization is a key technique for solving nonlinear problems in transportation network design and other optimization fields, in which generating breakpoints is a fundamental task. This paper proposes an optimal breakpoint selection method, rotational adjusting method (RAM), to minimize the approximation error between the original function and the piecewise linear function with limited number of pieces, applicable to both convex or concave function. RAM rotationally adjusts the location of breakpoints based on its adjacent breakpoints, and the optimal positions would be reached after several iterations. The optimality of the method is proved. Numerical experiments are conducted on the logarithmic function.