凸随机锥优化的增广Lagrange随机近似方法  

An augmented Lagrangian stochastic approximation method for convex stochastic conic optimization

作　　者：刘昊洋 张继宏 张立卫 Haoyang Liu;Jihong Zhang;Liwei Zhang

机构地区：[1]东北财经大学金融学院,大连116025 [2]沈阳理工大学理学院,沈阳110168 [3]东北大学工业智能与系统优化国家级前沿科学中心,沈阳110819

出　　处：《中国科学:数学》2025年第2期283-300,共18页Scientia Sinica:Mathematica

基　　金：国家重点研发计划(批准号:2022YFA1004000);国家自然科学基金(批准号:12371298)资助项目。

摘　　要：本文考虑由期望定义的凸随机锥优化,提出了一种增广Lagrange随机近似(augmented Lagrangian stochastic approximation,ALSAcp)方法求解这一问题,分析了该方法的期望后悔值.在合适条件下,证明了如果算法中的参数选择得当,该方法对目标下降和约束违背的后悔值都是O(T^(−1/2)),其中T表示迭代次数.此外还证明了,该方法目标下降和约束违背的后悔值至少以1−1/T的概率都不超过O(log(T)/√T).In this paper,we consider the convex stochastic conic optimization defined by expectations.An augmented Lagrangian stochastic approximation(ALSAcp)method is proposed to solve this convex stochastic conic optimization problem and the regrets of this method are analyzed.Under mild conditions,we show that this method exhibits O(T^(−1/2))regret for both objective reduction and constraint violation if parameters in the algorithm are properly chosen,where T denotes the number of iterations.Moreover,we show that,with at least 1−1/T probability,the method has no more than O(log(T)/√T)for both objective descent regret and constraint violation regret.

关 键 词：随机近似 增广LAGRANGE函数 目标下降后悔值 约束违背后悔值 高概率后悔界 凸随机锥优化 

分 类 号：O224[理学—运筹学与控制论]