Optimality Conditions For Sparse Optimization Problems With 0/1 Regular

Optimization model of sparse optimization problems with 0/1 regular is a clas-sical optimization model with a wide range of applications,mainly including support vector machines,1-bit compression sensing,maximum rank correlation and curve problems.However,due to the properties of the 0/1-loss function,such as non-convex,discontinuous,non-differentiable and gradient at differentiable points is 0,it is not easy to optimize directly and then the method based on convex relaxation or smooth approximation dominates the existing research.In 2021,Zhou Sheng-long proposed to directly optimize the 0/1-loss function,established the first-order optimality conditions for the original optimization model of sparse optimization problems with 0/1 regular,and proposed to directly optimize the 0/1-loss function by Newton’s method,which proved that the method has locally quadratic conver-gence under certain conditions.However,the second-order optimality condition for the 0/1-loss optimization is a topic that other scholars have not yet studied.In this paper,first of all,we use the definition to calculate the subdifferential,first-order subderivative and second-order subderivative of the 0/1-loss function.Then,according to the relevant knowledge in variational analysis,we calculate the subdif-ferential,first-order subderivative and second-order subderivative of ‖（Ax+b）₊‖₀,which is the composite function of 0/1 loss function and a ne function.Finally,based on the above expression,different forms of the first-order and the second-order optimization conditions of 0/1-loss optimization are given.