On how to solve large-scale log-determinant optimization problems

We propose a proximal augmented Lagrangian method and a hybrid method, i.e., employing the proximal augmented Lagrangian method to generate a good initial point and then employing the Newton‐CG augmented Lagrangian method to get a highly accurate solution, to solve large‐scale nonlinear semidefinite programming problems whose objective functions are a sum of a convex quadratic function and a log‐determinant term. We demonstrate that the algorithms can supply a high quality solution efficiently even for some ill‐conditioned problems.