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.