Analysis of Sparse Quasi-Newton Updates with Positive Definite Matrix Completion

Abstract

Based on the idea of maximum determinant positive definite matrix completion, Yamashita (Math Prog 115(1):1–30, 2008) proposed a new sparse quasi-Newton update, called MCQN, for unconstrained optimization problems with sparse Hessian structures. In exchange of the relaxation of the secant equation, the MCQN update avoids solving difficult subproblems and overcomes the ill-conditioning of approximate Hessian matrices. However, local and superlinear convergence results were only established for the MCQN update with the DFP method. In this paper, we extend the convergence result to the MCQN update with the whole Broyden’s convex family. Numerical results are also reported, which suggest some efficient ways of choosing the parameter in the MCQN update the Broyden’s family.