用于非线性优化的新型分数阶加速梯度下降方法及其在姿态识别中的应用
A Novel Fractional-Order Accelerated Gradient Descent Method for Nonlinear Optimization with Application to Posture Recognition
浏览论文内容
中文总结 AI 辅助
本文提出适用于光滑与非光滑优化的Caputo分数阶加速梯度下降（CFAGD）方法，结合自适应参数与Armijo线搜索，理论上线性收敛，数值验证显示其性能优于竞争方法，可应用于姿态识别。
中文摘要 AI 辅助
本文针对无约束优化问题提出了一种Caputo分数阶加速梯度下降（CFAGD）方法，该方法既适用于光滑目标函数，也适用于一类非光滑目标函数。所提方法引入了自适应（β_k）参数，该参数在迭代过程中通过启发式方式更新，以优化搜索方向；同时结合Caputo分数阶导数与自适应（β^k）参数，保留非整数阶导数的记忆特性。通过基于Armijo条件的不精确线搜索技术选择合适的步长，核心思路是用正参数缩放步长，改善迭代点趋近最优解时的表现，生成下降序列。在强凸性与有界Hessian假设下，证明了该方法的线性收敛性。含神经网络示例的数值验证进一步表明，CFAGD方法相比现有竞争方法，能实现更快、更稳定的性能。
英文摘要
This article proposes a Caputo fractional accelerated gradient descent (CFAGD) method for unconstrained optimization problems that is applicable to both smooth and a class of non-smooth objective functions. The proposed approach incorporates an adaptive (\b{eta}k)-parameter, which is heuristically updated throughout the iterative process to improve the search direction. Furthermore, the method employs the Caputo fractional derivative together with the adaptive (\b{eta}^k)-parameter, thereby preserving the memory characteristics associated with non-integer-order derivatives. A suitable step-size is selected via an inexact line-search technique based on the Armijo condition. The central idea is to scale the step-size by a positive parameter to improve the behavior of the iterates as they approach an optimal point, thereby generating a descent sequence. Under strong convexity and bounded Hessian assumptions, linear convergence of the proposed method is established. Numerical validations, including neural-network-based examples, further indicate that the CFAGD method can achieve faster and more stable performance than competing approaches.
