A hybrid method combining the FR conjugate gradient method and the WYL conjugate gradient method is proposed for unconstrained optimization problems. The presented method possesses the sufficient descent property under the strong Wolfe-Powell (SWP) line search rule relaxing the parameter

Consider the following

The following formula for

The sufficient descent condition

In the following section, the algorithm is stated. The properties and the global convergence of the new method are proved in Section

Now we describe our algorithm as follows.

Choose an initial point

If

Compute step size

Let

Calculate the search direction

Set

In the following, we assume that

(i) The function

(ii) In an open convex, set

The strong Wolfe-Powell (SWP) search rule is to find a step length

The following theorem shows that the hybrid algorithm with the SWP line search possesses the sufficient condition (

Let the sequences

By the definition

By (

Denote

Suppose that Assumption

By the same way, if Assumption

Suppose that Assumption

By (

Suppose that Assumption

By (

Secondly, using (

The weak Wolfe-Powell line search is to find a step length

From the computation point of view, one of the well-known formulas for

Let Assumption

The following Property

Suppose that

Let Assumption

By (

Let the sequences

Finally, by Lemmas

Let the sequences

In this section, we report some results of the numerical experiments. It is well known that there exist many new conjugate gradient methods (see [

If

If

Dolan and Moré [

Requiring a baseline for comparisons, they compared the performance on problem

The performance of solver

According to the above rules, we know that one solver whose performance profile plot is on top right will win over the rest of the solvers.

Figures

Performance profiles of these three methods (NI).

Performance profiles of these three methods (NFN).

From Figures

This paper gives a hybrid conjugate gradient method for solving unconstrained optimization problems. Under the SWP line search, this method possesses the sufficient descent condition only with the parameter

For further research, we should study the new method with the nonmonotone line search technique. Moreover, more numerical experiments for large practical problems (such as the problems [

The authors are very grateful to the anonymous referees and the editors for their valuable suggestions and comments, which improved our paper greatly. This work is supported by China NSF grants 10761001 and the Scientific Research Foundation of Guangxi University (Grant no. X081082).