Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems

Abstract

It is well known that the sufficient descent condition is very important to the global convergence of the nonlinear conjugate gradient method. In this paper, some modified conjugate gradient methods which possess this property are presented. The global convergence of these proposed methods with the weak Wolfe-Powell (WWP) line search rule is established for nonconvex function under suitable conditions. Numerical results are reported.

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“…Email: xrli68@163.com 收稿日期：2011 年 3 月 12 日；修回日期：2011 年 3 月 20 日；录用日期：2011 年 3 月 21 日 摘 要：Yuan [15] 提出了修改的 PRP 共轭梯度方法，该方法能保证参数 k [1,2], , [7], ( ) …”

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Abstract:Yuan [15] proposed a modified PRP conjugate gradient method which can ensure that the scalar 0  k  holds and the search direction possesses the sufficient descent property without any line search. This technique has been extended to other conjugate gradient methods, but the convergence has been not given.

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A Three-Term Conjugate Gradient Algorithm with Quadratic Convergence for Unconstrained Optimization Problems

“…The PRP algorithm [6,7] is one of the most effective CG algorithms and its convergence can be found (see [7,9,10], etc.). Powell [9] suggested that should not be less than zero; then many new CG formulas are proposed (see [11][12][13][14][15][16][17], etc.) to ensure the scalar ≥ 0.…”

Modified nonlinear conjugate gradient method with sufficient descent condition for unconstrained optimization

“…This counter-example also indicates that the method has a drawback that it is impossible to be convergent when the objective function is nonconvex. Therefore, during the past few years, much effort has been investigated to create new formulae for b k , which not only possesses global convergence for general functions but also is superior to original method from the computation point of view (see [8][9][10][11][12][13][14][15][16][17]). …”