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arXiv · 2501.16731

Triangle Steepest Descent: A Geometry-Based Gradient Algorithm with Guaranteed R-Linear Convergence

Ya Shen·Qing-Na Li·Yu-Hong Dai
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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 $j$ 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.

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Ya Shen, Qing-Na Li, Yu-Hong Dai. 2025-01-28. Triangle Steepest Descent: A Geometry-Based Gradient Algorithm with Guaranteed R-Linear Convergence. https://arxiv.org/abs/2501.16731

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