Global convergence properties of the two new dependent Fletcher–Reeves conjugate gradient methods

Abstract

In this paper, we propose two new dependent Fletcher–Reeves conjugate gradient methods arising from different choice for the scalar β_k. We make two different kinds of estimations of upper bounds of ∣β_k∣ with respect to ${\beta }_k^{\mathrm{FR}}$, which are based on Abel Theorem of non-convergent series of positive items. With several different line searches, global convergence results are established for the two new methods which extend the previous dependent Fletcher–Reeves conjugate gradient methods.

Introduction

Consider the unconstrained optimization problem

$$\underset{x\in R^n}{\min }f(x)\text{,}$$

where f : Rⁿ → R is continuously differentiable. Conjugate gradient methods for solving (1.1) are iterative methods of the form

$$x_{k+1}=x_k+{\alpha }_k{d}_k\text{,}$$

$$d_k=\left\{\begin{array}{cc}-g_k\text{,}\hfill & k=1\text{;}\hfill \\ -g_k+{\beta }_k{d}_{k-1}\text{,}\hfill & k⩾2\text{,}\hfill \end{array}\right.$$

where g_k = ∇f(x_k), α_k is a step-length obtained by a one-dimensional line search and β_k is a scalar. The well-known formulas for β_k are the Fletcher–Reeves (FR) [18] formula

$${\beta }_k^{\mathrm{FR}}=\frac{\Vert g_k{\Vert }^2}{\Vert g_{k-1}{\Vert }^2}$$

and the Polak–Ribière–Polyak (PRP) [19], [20] formula

$${\beta }_k^{\mathrm{PRP}}=\frac{g_k^{\mathrm{T}}(g_k-g_{k-1})}{\Vert g_{k-1}{\Vert }^2}\text{,}$$

where ∥ · ∥ means the Euclidean norm. The conjugate gradient method is available for large-scale unconstrained optimization because its storage is relatively small. Number results in [17] shows that if f is easy to be computed and if its dimension n is very large, the conjugate gradient method is still the best choice for solving (1.1). Convergence properties of the FR conjugate gradient method have been studied by many authors, including [1], [3], [4], [5], [6], [7], [8], [9], [10], [13], [15], [16], [18], [19], [20].

Many efforts have been devoted to investigate the global convergence properties of the FR conjugate gradient method. Powell [8] and Zoutendijk [13] proved that the FR conjugate gradient method with exact line searches is globally convergent on general functions. Al-Baali [1] extended this result to inexact line searches. However, the numerical performance of the FR conjugate gradient method is often much slower than that of the PRP conjugate gradient method. The PRP conjugate gradient method has a good numerical performance [9], but does not have such good convergence property [8]. Due to this, some dependent FR conjugate gradient methods are proposed [5], [6], [7], [15], [16]. In this kind of methods, the region of β_k with respect to ${\beta }_k^{\mathrm{FR}}$ is introduced and if β_k is chosen suitably in this region, then the method not only has the global convergence property but also has the nice numerical experience.

Gilbert and Nocedal [5] investigated global convergence properties of the dependent FR conjugate gradient method with β_k satisfying $|{\beta }_k|⩽{\beta }_k^{\mathrm{FR}}$, provided that the line search satisfies the strong Wolfe conditions

$$f(x_k)-f(x_k+{\alpha }_k{d}_k)⩾-\delta {\alpha }_k{g}_k^{\mathrm{T}}{d}_k\text{,}$$

$$|g(x_k+{\alpha }_k{d}_k{)}^{\mathrm{T}}{d}_k|⩽\sigma |g_k^{\mathrm{T}}{d}_k|\text{,}$$

where 0 < δ < σ < 1. If β_k satisfies $|{\beta }_k|⩽c{\beta }_k^{\mathrm{FR}}$ (where c > 1), they have given an example to indicate that even exact line search cannot guarantee the global convergence property of the dependent FR conjugate gradient method.

Hu and Storey [6] proved the dependent FR conjugate gradient method is globally convergent with the strong Wolfe line search if β_k satisfies

$$\sigma \left|\frac{{\beta }_k}{{\beta }_k^{\mathrm{FR}}}\right|⩽\overline{\sigma }\text{,}\Vert g_k{\Vert }^4\sum _{j=1}^k\prod _{i=j}^{k-1}{l}_i⩽\mathit{ck}\text{,}$$

where 0 < σ < 1, $0<\overline{\sigma }<\frac{1}{2}$, $l_i=|\frac{{\beta }_i}{{\beta }_i^{\mathrm{FR}}}{|}^2$ and c > 0. Liu et al. [7], [16] had the same result if $0<\overline{\sigma }⩽\frac{1}{2}$. It should be pointed out that the global convergence of the previous dependent FR conjugate gradient method can be obtained only under the strong Wolfe line search.

Dai and Yuan [4] investigated global convergence properties of the FR conjugate gradient method with the generalized Wolfe conditions, i.e. (1.6) and

$${\sigma }_1{g}_k^{\mathrm{T}}{d}_k⩽g(x_k+{\alpha }_k{d}_k{)}^{\mathrm{T}}{d}_k⩽-{\sigma }_2{g}_k^{\mathrm{T}}{d}_k\text{,}$$

where 0 < δ < σ₁ < 1 and σ₂ > 0. They also proved that σ₁ + σ₂ ⩽ 1 is a sufficient condition for ensuring $g_k^{\mathrm{T}}{d}_k⩽0$ for all k. It is easy to see that (1.7) can be viewed as a special case of (1.9) with σ₁ = σ₂.

Dai [3] also investigated global convergence properties of the FR conjugate gradient method with the Armijo line search, which is to determine the smallest integer m ⩾ 0 such that, if one defines α_k = λ^m, then

$$f(x_k+{\alpha }_k{d}_k)-f(x_k)⩽\delta {\alpha }_k{g}_k^{\mathrm{T}}{d}_k\text{,}$$

where λ ∈ (0, 1) and δ ∈ (0, 1).

Wang and Zhang [15] proposed an s-dependent GFR conjugate gradient method and made two different kinds of estimations of upper bounds of β_k with respect to ${\beta }_k^{\mathrm{FR}}$

$${\beta }_k^2⩽{\hat{\sigma }}_k(s)({\beta }_k^{\mathrm{FR}}{)}^2\text{,}$$

where ${\hat{\sigma }}_2(s)=1$, ${\hat{\sigma }}_k(s)={\prod }_{j\in {\Omega }_1}\frac{{\Gamma }_{k-1}^{(j)}(s)}{{\Gamma }_{k-2}^{(j)}(s)}$, s ∈ {1, 2}; k ⩾ 3, Ω₁ is a partition of set {1, 2, … , t}, t ⩾ 1 is an integer and the definition of ${\Gamma }_k^{(j)}(s)$ is as follows:

$${\gamma }_i=\frac{|g_i^{\mathrm{T}}{d}_i|}{\Vert g_i{\Vert }^2}\text{,}{\Gamma }_k^{(0)}(s)=\sum _{i=1}^k{\gamma }_i^s\text{,}$$

$${\Gamma }_k^{(t)}(s)=\sum _{i=1}^k\frac{{\gamma }_i^s}{{\prod }_{j=0}^{t-1}{\Gamma }_i^{(j)}(s)}\text{.}$$

They proved global convergence of the s-dependent GFR conjugate gradient method with some various line searches.

In this paper, we give new restriction to β_k which is wider than the previous restriction, i.e. β_k satisfies

$$\sum _{j=1}^k\prod _{i=j+1}^k{\left(\frac{{\beta }_i}{{\beta }_i^{\mathrm{FR}}}\right)}^2{\gamma }_j^s⩽c\prod _{j=0}^t{\Gamma }_k^{(j)}(s)\text{,}$$

where c > 0 is a constant, t ⩾ 1 is an integer and s ∈ {1, 2}. Further, we also define ${\prod }_{i\in \Omega }{\left(\frac{{\beta }_i}{{\beta }_i^{\mathrm{FR}}}\right)}^2=1$ when Ω = ϕ.

Generally, the two regions of β_k defined by (1.14) do not conclude each other.

In Section 2, we will investigate global convergence properties of the two new dependent FR conjugate gradient methods with the generalized Wolfe line search (1.6), (1.9), the Armijo line search (1.10) and the Wolfe line search respectively. In Section 3, we will indicate that some of the global convergent results in [3], [4], [5], [6], [7], [15], [16] can be viewed as a special case of the global convergent results in Section 2.