Multi-Extremal Non-differentiable Optimization Using the Secant Technique

Abstract

This work addresses the global optimization of nonconvex, non-differentiable Hölder continuous functions. It reviews theoretical foundations and established lower-bounding methods, such as the Piyavskii-Shubert and Gourdin-Piyavskii Algorithm (GPA), noting GPA's limitations with analytical parabolic intersections. The primary contribution is the investigation and extension of the secant Piyavskii algorithm (SPA). SPA innovatively approximates these intersections using simpler linear secants, offering enhanced flexibility across various Hölder exponents. Convergence properties of SPA are also discussed. Numerical experiments compared SPA against GPA on test functions, assessing computational time and function evaluations. The results generally show the superior performance of the SPA algorithm in terms of efficiency for solving this difficult class of optimization problems.