Nonsmooth Newton's method for positive semi definite 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 semi-definite cone of matrices, the conic constrained equation is transformed to a nonsmooth unconstrained equation which is solved by the nonsmooth Newton's method. Here, we use an explicit expression of the Clarke generalized 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 superlinearly (respectively, quadratically).