Matrixed 0/1 composite optimization model and optimality analysis

In this paper, a novel matrixed 0/1 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.