$${\text {B}}$$ B -subdifferentials of the projection onto the matrix simplex

An important tool in matrix optimization problems is the strong semismoothness of the projection mapping onto the cone of real symmetric positive semidefinite matrices, and the explicit formula for its $${\text {B}}$$ B (ouligand)-subdifferentials. In this paper, we examine the corresponding results for the so-called matrix simplex, that is, the set of real symmetric positive semidefinite matrices whose traces are equal to one. This result complements the current literature and enlarges the toolbox of matrix spectral operators whose $${\text {B}}$$ B -subdifferentials are explicitly formulated. Since the matrix simplex frequently arises in subproblems for solving matrix optimization problems, the derived results can potentially serve as a useful tool for efficiently solving these problems. As an illustration, we present a numerical example to demonstrate that the proposed approach can outperform the existing approaches which used projection mapping onto positive semidefinite matrix cone directly.