一类非凸Bregman梯度法的线性收敛研究  

Linear Convergence of a Class of Non-convex Bregman Gradient Algorithm

作　　者：李蝶 郭科[1] LI Die;GUO Ke(School of Mathematics&Information,China West Normal University,Nanchong Sichuan 637009,China)

机构地区：[1]西华师范大学数学与信息学院,四川南充637009

出　　处：《西华师范大学学报（自然科学版）》2025年第1期30-35,共6页Journal of China West Normal University(Natural Sciences)

基　　金：国家自然科学基金项目(11801455,11971238);四川省自然科学基金杰出青年科学基金项目(2023NSFSC1922)。

摘　　要：梯度下降算法是一类求解无约束优化问题的重要方法,其研究中光滑性的假设具有重要作用。Bregman梯度下降算法是对梯度下降算法的一种推广,本质上可以看作将经典的光滑性削弱成相对光滑性时自然产生的。文章研究了Bregman梯度下降算法求解相对强quasar-凸和相对光滑问题的线性收敛性,证明了当目标函数为相对强quasar-凸且相对光滑时,Bregman梯度下降算法产生的函数值序列具有线性收敛速度,同时,给出了迭代序列的收敛性。Gradient descent algorithm is an important method for solving unconstrained optimization problems,and the assumption of smoothness plays a crucial role in its research.Bregman gradient descent algorithm is an extension of the gradient descent algorithm,and it can be essentially seen as a natural outgrowth when the classical smoothness is reduced to relative smoothness.This paper studies the linear convergence of Bregman gradient descent algorithm for solving relatively strongly quasar-convex and relatively smooth problems.It is proved that when the objective function is relatively strongly quasar-convex and relatively smooth,the sequence of function value produced by Bregman gradient descent algorithm has a linear convergence rate.Meanwhile,the convergence of iterative sequence is also given.

关 键 词：相对光滑 强quasar-凸 相对强quasar-凸 Bregman梯度下降算法 线性收敛率 

分 类 号：O224[理学—运筹学与控制论]