A trust-region method for box-constrained nonlinear semidefinite programs

We propose a trust-region method for nonlinear semidefinite programs with box-constraints. The penalty barrier method can handle this problem, but the size of variable matrices available in practical time is restricted to be less than 500. We develop a trust-region method based on the approach of Coleman and Li (1996) that utilizes the distance to the boundary of the box-constraints into consideration. To extend this method to the space of positive semidefinite matrices, we device a new search direction by incorporating the eigenvectors of the variable matrix into the distance to the boundary. In this paper, we establish a global convergence of the proposed method, and preliminary numerical experiments show that our method solves the problems with the size being larger than 5000, and it is faster than the feasible direction for functions with nonlinearity higher than quadratic.