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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (6): 851-869.doi: 10.3969/j.issn.1005-3085.2023.06.001

• •    下一篇

带有CVaR罚的分布鲁棒指数跟踪模型：易求解的转化

王茹钰1,  胡耀忠2,  张 超1   

  1. 1. 北京交通大学数学与统计学院，北京 100044
    2. 加拿大阿尔伯塔大学数学与统计科学系，埃德蒙顿 T6G 2G1
  • 收稿日期:2023-09-15 接受日期:2023-10-16 出版日期:2023-12-15 发布日期:2024-02-15
  • 通讯作者: 张 超 E-mail: zc.njtu@163.com
  • 基金资助:
    国家自然科学基金 (12171027). 

A Distributionally Robust Index Tracking Model with the CVaR Penalty: Tractable Reformulation

WANG Ruyu1,  HU Yaozhong2,  ZHANG Chao1   

  1. 1. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044
    2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1
  • Received:2023-09-15 Accepted:2023-10-16 Online:2023-12-15 Published:2024-02-15
  • Contact: C. Zhang. E-mail address: zc.njtu@163.com
  • Supported by:
    The National Natural Science Foundation of China (12171027).

摘要：

提出了一种带有条件在险价值（CVaR）惩罚的分布鲁棒指数跟踪模型，该模型将分布鲁棒优化的思想与CVaR惩罚相结合。模型中概率的不确定性通过随机向量的一阶和二阶矩的置信区域来描述。将该模型由一个“min-max-min”形式的优化问题，等价转化为一个非光滑最小化问题。同时提供了一个近似求解带有连续型随机向量的非光滑极小化问题的离散化方案，通过该方案离散化之后的目标函数包含众多但有限个的非光滑函数的最大化。在较弱的条件下证明了离散化之后的模型收敛到原问题等价转化后的非光滑连续分布模型。采用了光滑投影梯度（Smoothing Projected Gradient, SPG）方法求解离散化后的模型，并证明了由SPG方法产生的迭代点序列的任何聚点都是离散化之后模型的全局最小值点。在2008年1月至2023年7月的纳斯达克日度指数数据集上与先进模型进行比较，数值结果验证了所提模型以及SPG方法的有效性。

关键词: 指数跟踪, 分布鲁棒优化, 条件在险价值, 非光滑, 光滑投影梯度方法

Abstract:

We propose a distributionally robust index tracking model with the conditional value-at-risk (CVaR) penalty. The model combines the idea of distributionally robust optimization for data uncertainty and the CVaR penalty to avoid large tracking errors. The distribution ambiguity is described through a confidence region based on the first-order and second-order moments of the random vector involved. We reformulate the model in the form of a min-max-min optimization into an equivalent nonsmooth minimization problem. We further give an app-roximate discretization scheme for the possible continuous random vector of the nonsmooth minimization problem, whose objective function involves the maximum of numerous but finite nonsmooth functions. The convergence of the discretization scheme to the equivalent nonsmooth reformulation is shown under mild conditions. A smoothing projected gradient (SPG) method is employed to solve the discretization scheme. Any accumulation point is shown to be a global minimizer of the discretization scheme. Numerical results on the NASDAQ index dataset from January 2008 to July 2023 demonstrate the effectiveness of our proposed model and the efficiency of the SPG method, compared with several state-of-the-art models and corresponding methods for solving them.

Key words: index tracking, distributionally robust optimization, conditional value-at-risk, nonsmooth, smoothing projected gradient method

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