×

A Newton-CG augmented Lagrangian method for semidefinite programming. (English) Zbl 1213.90175

A new augmented Lagrangian method for solving semidefinite programming (SDP) problems is proposed and studied. The objective function of the inner problems in this method is only once and not twice continuously differentiable so that these problems are solved by a semismooth Newton conjugate gradient (CG) method. The local superlinear or quadratic rate of convergence of the latter method is proven under natural assumptions. In particular, it is shown that the positive definiteness of the generalized Hessian of the objective function of each inner problem, which is a key property for ensuring the efficiency of an inexact semismooth Newton CG method, is equivalent to the constraint nondegeneracy of the corresponding dual problem. Furthermore, the local linear rate of convergence of the overall Newton-CG augmented Lagrangian method is verified. Numerical results for a variety of large-scale SDP problems demonstrate that the proposed method is very efficient and can solve such problems with high accuracy.

MSC:

90C06 Large-scale problems in mathematical programming
90C22 Semidefinite programming
90C25 Convex programming
65F10 Iterative numerical methods for linear systems
PDFBibTeX XMLCite
Full Text: DOI Link