矩阵0/1复合优化模型与最优性分析  

Matrixed 0/1 composite optimization model and optimality analysis

作　　者：张鹏鹤 修乃华[2] 戚厚铎 Penghe Zhang;Naihua Xiu;Houduo Qi

机构地区：[1]香港理工大学应用数学系,中国香港999077 [2]北京交通大学数学与统计学院,北京100044 [3]香港理工大学数据科学及人工智能学系,中国香港999077

出　　处：《中国科学:数学》2025年第2期397-414,共18页Scientia Sinica:Mathematica

基　　金：香港研究资助局基金(批准号:PolyU15309223);国家重点研发计划(批准号:2023YFA1011100);国家自然科学基金(批准号:12131004)资助项目。

摘　　要：本文从多分类支持向量机、多标签分类和双边排序等实际应用中抽象凝练出一类矩阵0/1复合优化问题,其目标函数涉及离散的0/1损失函数、非光滑的极大值函数以及线性项三重复合结构.本文给出现有相关工作中5种稳定点在这一问题下的具体形式,分析在特定条件下各稳定点之间以及与该问题解之间的关系,并进一步建立矩阵0/1复合优化问题的二阶必要和充分最优性条件.尽管矩阵0/1复合优化属于非凸非连续问题,本文的最优性结论均无需约束规范假设.In this paper,a novel matrixed o/1 composite optimization problem is abstracted from practical applications including multi-class support vector machine,multi-label learning,and bipartite ranking.Its objective function involves three layers of composition:the discrete 0/1 loss function,the nonsmooth maximum function,and a linear term.Inspired by some related works,we define five types of stationary points for this problem and examine the interrelations among these stationary points and their connection to minimizers of the problem.Additionally,second-order necessary and sufficient optimality conditions for matrixed 0/1 composite optimization problems are established.Although the matrixed 0/1 composite optimization essentially belongs to nonconvex and discrete composite problems,the optimality results derived in this paper do not require any constraint qualification.

关 键 词：矩阵0/1复合优化 矩阵0/1损失函数 稳定点 二阶最优性条件 

分 类 号：G63[文化科学—教育学]