A new subspace minimization conjugate gradient method based on conic model for large-scale unconstrained optimization

Abstract

Conjugate gradient method is one of the most efficient methods for large-scale unconstrained optimization and has attracted focused attention of researchers. With the emergence of more and more large-scale problems, subspace technique becomes particularly important. Recently, many scholars have studied subspace minimization conjugate gradient methods, where the iteration direction is generally obtained by minimizing a quadratic approximate model in a specific subspace. Considering that the conic model contains more information than the quadratic model, and may perform better when the objective function is not close to quadratic, in this paper, a improved conjugate gradient method is presented. Specially, in each iteration, a quadratic or conic model is dynamically selected. The search direction is obtained by minimizing the selected model in a three-dimensional subspace spanned by the current gradient and latest two search directions. It is proved that the search direction satisfies the sufficient descent condition under given conditions. With the modified nonmonotone Wolfe line search, we establish the global convergence and the R-linear convergence of the proposed method under mild assumptions. Numerical experiments indicate that the proposed method is efficient.