A modified three-term PRP conjugate gradient algorithm for optimization models

The nonlinear conjugate gradient (CG) algorithm is a very effective method for optimization, especially for large-scale problems, because of its low memory requirement and simplicity. Zhang et al. (IMA J. Numer. Anal. 26:629-649, 2006) firstly propose a three-term CG algorithm based on the well known Polak-Ribière-Polyak (PRP) formula for unconstrained optimization, where their method has the sufficient descent property without any line search technique. They proved the global convergence of the Armijo line search but this fails for the Wolfe line search technique. Inspired by their method, we will make a further study and give a modified three-term PRP CG algorithm. The presented method possesses the following features: (1) The sufficient descent property also holds without any line search technique; (2) the trust region property of the search direction is automatically satisfied; (3) the steplengh is bounded from below; (4) the global convergence will be established under the Wolfe line search. Numerical results show that the new algorithm is more effective than that of the normal method.


Introduction
We consider the optimization models defined by where the function f : n → is continuously differentiable. There exist many similar professional fields of science that can revert to the above optimization models (see, e.g., [-]). The CG method has the following iterative formula for (.): where x k is the kth iterate point, the steplength is α k > , and the search direction d k is designed by -g k+ , i fk = , where g k = ∇f (x k ) is the gradient and β k ∈ is a scalar. At present, there are many wellknown CG formulas (see [-]) and their applications (see, e.g., [-]), where one of the most efficient formulas is the PRP [, ] defined by where g k+ = ∇f (x k+ ) is the gradient, δ k = g k+g k , and . is the Euclidian norm.
It is not difficult to deduce that d T k+ g k+ =g k+  holds for all k, which implies that the sufficient descent property is satisfied. Zhang et al. proved that the threeterm PRP method has global convergence under Armijo line search technique for general functions but this fails for the Wolfe line search. The reason may be the trust region feature of the search direction that cannot be satisfied for this method. In order to overcome this drawback, we will propose a modified three-term PRP formula that will have not only the sufficient descent property but also the trust region feature. In the next section, a modified three-term PRP formula is given and the new algorithm is stated. The sufficient descent property, the trust region feature, and the global convergence of the new method are established in Section . Numerical results are reported in the last section.

The modified PRP formula and algorithm
Motivated by the above observation, the modified three-term PRP formula is where γ  > , γ  > , and γ  >  are constants. It is easy to see that the difference between (.) and (.) is the denominator of the second and the third terms. This is a little change that will guarantee another good property for (.) and impel the global convergence for Wolfe conditions.
Step : Get stepsize α k by the following Wolfe line search rules: and Step : Let x k+ = x k + α k d k . If the condition g k+ ≤ ε holds, stop the program.
Step : Calculate the search direction d k+ by (.).
Step : Set k := k +  and go to Step .

The sufficient descent property, the trust region feature, and the global convergence
It has been proved that, even for the function f (x) = λ x  (λ >  is a constant) and the strong Wolfe conditions, the PRP conjugate gradient method may not yield a descent direction for an unsuitable choice (see [] for details). An interesting feature of the new three-term CG method is that the given search direction is sufficiently descent.
Lemma . The search direction d k is defined by (.) and it satisfies d T k+ g k+ =g k+ is true and (.) holds with γ = . If k ≥ , by (.), we have Then (.) is satisfied. By (.) again, we obtain where the last inequality follows from . Thus (.) holds for all k ≥  with γ = max{,  + /γ  }. The proof is complete.

Remark
To establish the global convergence of Algorithm , the normal conditions are needed.
The function f has a lower bound and it is differentiable. The gradient g is Lipschitz continuous where L >  a constant.

Lemma . Suppose that Assumption A holds and NTT-PRP-CG-A generates the se-
Then there exists a constant β >  such that where the last equality follows from (.). By (.), we get Setting β ∈ (, -σ Lγ ) completes the proof.

Remark
The above lemma shows that the steplengh α k has a lower bound, which is helpful for the global convergence of Algorithm .
Theorem . Let the conditions of Lemma . hold and {x k , d k , α k , g k } be generated by NTT-PRP-CG-A. Thus we get Proof By (.), (.), and (.), we have Summing the above inequality from k =  to ∞, we have which means that The proof is complete.  Table . The parameters are γ  = , γ  = , γ  = , δ = ., σ = .. The program uses the Himmelblau rule:  the CPU time of operating system in seconds.      A new tool was given by Dolan and Moré [] to analyze the performance of the algorithms. Figures - show that the efficiency of the NTT-PRP-CG-A and the Norm-PRP-A relate to Ni, Nfg, and CPU time, respectively. It is easy to see that these two algorithms are effective for those problems and the given three-term PRP conjugate gradient method is more effective than that of the normal three-term PRP conjugate gradient method. Moreover, the NTT-PRP-CG-A has good robustness. Overall, the presented algorithm has some potential property both in theory and numerical experiment, which is noticeable.

Conclusions
In this paper, based on the PRP formula for unconstrained optimization, a modified threeterm PRP CG algorithm was presented. The proposed method possesses sufficient descent property also holds without any line search technique, and we have automatically the trust region property of the search direction. Under the Wolfe line search, the global convergence was proven. Numerical results showed that the new algorithm is more effective compared with the normal method.