Non-smooth singular Newton's method for positive semidefinite solution of nonlinear matrix equations

In this paper, we propose a new method for solving conic constrained nonlinear matrix equations. With the use of the orthogonal projection onto the positive semidefinite cone of matrices, the conic constrained equation is transformed to a non‐smooth unconstrained equation which is solved by the non‐smooth Newton's method. Here, we use an explicit expression of the Clarke generalised Jacobian of the projection onto the cone of positive semidefinite matrices as developed by several authors. We prove under natural assumptions that the method converges locally and superlinearly.