A seven-point algorithm for piecewise smooth univariate minimization

TL;DR

This paper introduces a novel seven-point algorithm for univariate minimization of piecewise smooth functions without derivative information, using quadratic underestimators to efficiently find the minimum.

Contribution

The paper presents a new derivative-free optimization algorithm that constructs quadratic underestimators to locate minima of piecewise smooth functions.

Findings

Effective in minimizing piecewise smooth functions without derivatives

Converges reliably to the global minimum in tested scenarios

Provides a systematic approach for derivative-free optimization

Abstract

In this paper, we construct an algorithm for minimising piecewise smooth functions for which derivative information is not available. The algorithm constructs a pair of quadratic functions, one on each side of the point with smallest known function value, and selects the intersection of these quadratics as the next test point. This algorithm relies on the quadratic function underestimating the true function within a specific range, which is accomplished using a adjustment term that is modified as the algorithm progresses.