A Diagonal-Sparse Quasi-Newton Method for Unconstrained Optimization Problem
无约束优化问题的对角稀疏拟牛顿法

In this paper, we present a diagonal-sparse quasi-Newton method for unconstrained optimization problems. The method is similar to quasi-Newton method, but restricts the quasi-Newton matrix to a sparse matrix, and uses approximate quasi-Newton condition to determine a search direction and uses Armijo's line search rule to define a step-size at each iteration. It avoids the storage and computation of some matrices in its iteration, so that it is suitable for solving large scale optimization problems. Under some mild assumptions, we prove the global convergence and linear convergence rate, and futher analyze the superlinear convergence property of this method. Numerical experiments show that the diagonal-sparse quasi-Newton method is suitable to solve large scale problems, especially the problems in which the Hesse matrix of objective functions is sparse. Numerical results also show that the new method is more efficient than other similar methods, such as Cauchy method, conjugate gradient method, etc.