Triangle Steepest Descent: A Geometry-Based Gradient Algorithm with Guaranteed R-Linear Convergence
Abstract
Gradient methods are among the simplest yet most widely used algorithms for unconstrained optimization. Motivated by a geometric property of the steepest descent (SD) method that can alleviate the zigzag behavior in quadratic problems, we develop a new gradient variant called the Triangle Steepest Descent (TSD) method. The TSD method introduces a cycle parameter that governs the periodic combination of past search directions, providing a geometry-driven mechanism to enhance convergence. To the best of our knowledge, TSD is the first formally established geometry-based gradient scheme since Akaike (1959). We prove that TSD is at least R-linearly convergent for strongly convex quadratic problems and demonstrate through extensive numerical experiments that it exhibits superlinear behavior, outperforming the Barzilai-Borwein (BB) method and monotone Dai-Yuan gradient method (DY) in quadratic cases. These results suggest that incorporating geometric information into gradient directions offers a promising avenue for developing efficient optimization algorithms.
Cite
@article{arxiv.2501.16731,
title = {Triangle Steepest Descent: A Geometry-Based Gradient Algorithm with Guaranteed R-Linear Convergence},
author = {Ya Shen and Qing-Na Li and Yu-Hong Dai},
journal= {arXiv preprint arXiv:2501.16731},
year = {2025}
}