A Newton-CG Augmented Lagrangian Method For The Convex Quadratically Constrained Quadratic Semidefinite Programs

We propose a Newton-CG augmented Lagrangian method for solving the convex quadrat-ically constrained quadratic semidefinite programs (CQCQSDP). In order to analyze the localconvergence of our proposed method, we explicitly express the Robinson’s CQ, the strong sec-ond order sufficient condition, constrain nondegenercy, especially characterize the Lipschiz con-tinuity of the corresponding solution mapping at the origin. For the inner problem, we provethat the positive definiteness of the generalized Hessian matrix of the augmented Lagrangianobjective functions is actually equivalent to the constrain nondegenercy of the Wolfe duality ofthe primal problem, which ensures the local superlinear convergence of using an inexact semis-mooth Newton-CG method to solve the inner problem. Numerical experiments show that theproposed method works quite well, and is effective for the large-scale CQCQSDP problems.