Optimality Conditions for Double-sparsity Constrained Optimization  

双稀疏约束优化的最优性条件

作　　者：WANG Dongrui XIU Naihua ZHOU Shenglong 王东瑞;修乃华;周声龙(北京交通大学数学与统计学院,北京100044)

机构地区：[1]School of Mathematics and Statistics,Beijing Jiaotong University,Beijing,100044,P.R.China

出　　处：《数学进展》2024年第6期1145-1157,共13页Advances in Mathematics（China）

基　　金：Supported by the National Key R&D Program of China(No.2023YFA1011100);NSFC(No.12131004)。

摘　　要：Sparse optimization has witnessed advancements in recent decades,and the step function finds extensive applications across various machine learning and signal processing domains.This paper integrates zero norm and the step function to formulate a doublesparsity constrained optimization problem,wherein a linear equality constraint is also taken into consideration.By defining aτ-Lagrangian stationary point and a KKT point,we establish the first-order and second-order necessary and sufficient optimality conditions for the problem.Furthermore,we thoroughly elucidate their relationships to local and global optimal solutions.Finally,special cases and examples are presented to illustrate the obtained theorems.稀疏优化在近几十年来取得了显著进展,而阶跃函数在各种机器学习和信号处理领域都有广泛的应用.本文整合零范数和阶跃函数,建立一个双稀疏和等式约束优化问题.通过定义τ-拉格朗日稳定点和KKT点,我们建立了优化问题的一阶和二阶必要和充分最优性条件.此外,我们还深入阐明了它们与局部最优解和全局最优解之间的关系.最后,本文提供了特殊情况和示例来说明所得到的定理.

关 键 词：double-sparsity constrained optimization Lagrangian stationary point KKT point optimality condition 

分 类 号：O224[理学—运筹学与控制论]