A Sparse Quasi-Newton Method Based on Automatic Differentiation for Solving Unconstrained Optimization Problems

Abstract

In our paper, we introduce a sparse and symmetric matrix completion quasi-Newton model using automatic differentiation, for solving unconstrained optimization problems where the sparse structure of the Hessian is available. The proposed method is a kind of matrix completion quasi-Newton method and has some nice properties. Moreover, the presented method keeps the sparsity of the Hessian exactly and satisfies the quasi-Newton equation approximately. Under the usual assumptions, local and superlinear convergence are established. We tested the performance of the method, showing that the new method is effective and superior to matrix completion quasi-Newton updating with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method and the limited-memory BFGS method.

Keywords

• symmetric quasi-Newton method
• unconstrained optimization problems
• matrix completion
• automatic differentiation
• superlinear convergence
• Broyden–Fletcher–Goldfarb–Shanno method