• 论文 •

### 基于二次多项式的积极集光滑化max函数及其在无约束minimax问题中的应用

1. 山西师范大学, 临汾 041004
• 收稿日期:2019-07-20 发布日期:2021-05-13
• 基金资助:
国家自然科学基金（11701350）资助.

Zhou Zhengyong, Yang Qi. ACTIVE SET SMOOTHING MAX FUNCTION BASED ON QUADRATIC POLYNOMIAL AND ITS APPLICATION TO UNCONSTRAINED MINIMAX PROBLEMS[J]. Mathematica Numerica Sinica, 2021, 43(2): 192-209.

### ACTIVE SET SMOOTHING MAX FUNCTION BASED ON QUADRATIC POLYNOMIAL AND ITS APPLICATION TO UNCONSTRAINED MINIMAX PROBLEMS

Zhou Zhengyong, Yang Qi

1. Shanxi Normal University, Linfen 041004, China
• Received:2019-07-20 Published:2021-05-13

In this paper, based on the piecewise quadratic polynomial equation, a smoothing max function with the active set strategy is proposed. The piecewise quadratic polynomial equation is transformed to a general quadratic polynomial equation by calculating the index set related to the smoothing max function. Then, by the properties of roots for the quadratic polynomial equation, a stable calculation strategy for the smoothing max function is given. The smoothing max function is continuously differentiable, its gradient function is locally Lipschitz continuous and strongly semi-smooth. It is only related to the component functions with large function value, hence, it is suitable for problems containing max function defined by a large number of complex component functions. In order to show the efficiency of the smoothing max function, a smoothing method based on the smoothing max function is proposed for unconstrained minimax problems with multiple complex component functions. Preliminary numerical experiments show the feasibility and efficiency of the smoothing max function.

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