无约束优化问题的对角稀疏拟牛顿法

A DIAGONAL-SPARSE QUASI-NEWTON METHOD FOR UNCONSTRAINED OPTIMIZATION PROBLEM

对无约束优化问题提出了对角稀疏拟牛顿法,该算法采用了Armijo非精确线性搜索,并在每次迭代中利用对角矩阵近似拟牛顿法中的校正矩阵,使计算搜索方向的存贮量和工作量明显减少,为大型无约束优化问题的求解提供了新的思路．在通常的假设条件下,证明了算法的全局收敛性,线性收敛速度并分析了超线性收敛特征。数值实验表明算法比共轭梯度法有效,适于求解大型无约束优化问题．

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.