 Research
 Open Access
A scaled threeterm conjugate gradient method for unconstrained optimization
 Ibrahim Arzuka^{1}Email author,
 Mohd R Abu Bakar^{1} and
 Wah June Leong^{1}
https://doi.org/10.1186/s1366001612391
© Arzuka et al. 2016
 Received: 30 May 2016
 Accepted: 9 November 2016
 Published: 13 December 2016
Abstract
Conjugate gradient methods play an important role in many fields of application due to their simplicity, low memory requirements, and global convergence properties. In this paper, we propose an efficient threeterm conjugate gradient method by utilizing the DFP update for the inverse Hessian approximation which satisfies both the sufficient descent and the conjugacy conditions. The basic philosophy is that the DFP update is restarted with a multiple of the identity matrix in every iteration. An acceleration scheme is incorporated in the proposed method to enhance the reduction in function value. Numerical results from an implementation of the proposed method on some standard unconstrained optimization problem show that the proposed method is promising and exhibits a superior numerical performance in comparison with other wellknown conjugate gradient methods.
Keywords
 unconstrained optimization
 nonlinear conjugate gradient method
 quasiNewton methods
1 Introduction
2 Conjugate gradient method via memoryless quasiNewton method
2.1 Algorithm (STCG)
Algorithm 1
 Step 1.:

Select an initial point \(x_{o}\) and determine \(f (x_{o} )\) and \(g (x_{o} )\). Set \(d_{o}=g_{o}\) and \(k=0\).
 Step 2.:

Test the stopping criterion \(\Vert g_{k}\Vert \) ≤ ϵ, if satisfied stop. Else go to Step 3.
 Step 3.:

Determine the steplength \(\alpha_{k}\) as follows:
Given \(\delta\in ( 0,1 ) \) and \(p_{1},p_{2}\), with \(0< p_{1}< p_{2}<1\). (i)
Set \(\alpha=1\).
 (ii)Test the relation$$ f (x+\alpha d_{k} )f (x_{k} )\leq\alpha \delta g^{T}_{k} d_{k}. $$(16)
 (iii)
If (16) is satisfied, then \(\alpha_{k}=\alpha\) and go to Step 4 else choose a new \(\alpha\in [p_{1}\alpha,p_{2}\alpha ]\) and go to (ii).
 (i)
 Step 4.:

Determine \(z=x_{k}+\alpha_{k} d_{k}\), compute \(g_{z}=\nabla f (z )\) and \(y_{k}=g_{k}g_{z}\).
 Step 5.:

Determine \(r_{k}=\alpha_{k} g^{T}_{k} d_{k}\) and \(q_{k}=\alpha_{k} y^{T}_{k} d_{k}\).
 Step 6.:

If \(q_{k} \neq0\), then \(\vartheta_{k}=\frac{r_{k}}{q_{k}}\), \(x_{k+1}=x_{k}+\vartheta_{k}\alpha_{k} d_{k}\) else \(x_{k+1}=x_{k}+\alpha_{k} d_{k}\).
 Step 7.:

Determine the search direction \(d_{k+1}\) by (12) where \(\mu_{k}\), \(\varphi_{1}\), and \(\varphi_{2}\) are computed by (11), (13), and (14), respectively.
 Step 8.:

Set \(k:=k+1\) and go to Step 2.
3 Convergence analysis
In this section, we analyze the global convergence of the propose method, where we assume that \(g_{k}\neq0\) for all \(k\geq0\) else a stationary point is obtained. First of all, we show that the search direction satisfies the sufficient descent and the conjugacy conditions. In order to present the results, the following assumptions are needed.
Assumption 1
Proof
Now, we shall state the sufficient descent property of the proposed search direction in the following lemma.
Lemma 3.2
Suppose that Assumption 1 holds on the objective function f then the search direction (12) satisfies the sufficient descent condition \(g_{k+1}^{T} d_{k+1}\leqc\Vert g_{k+1}\Vert ^{2}\).
Proof
Lemma 3.3
Suppose that Assumption 1 holds, then the search direction (12) satisfies the conjugacy condition (27).
Proof
Lemma 3.4
Suppose that Assumption 1 holds then there exists a constant \(p>0\) such that \(\Vert d_{k+1}\Vert \leq P\Vert g_{k+1}\Vert \), where \(d_{k+1}\) is defined by (12).
Proof
In order to establish the convergence result, we give the following lemma.
Lemma 3.5
Proof
Theorem 3.6
Proof
4 Numerical results
5 Conclusion
We have presented a new threeterm conjugate gradient method for solving nonlinear large scale unconstrained optimization problems by considering a modification of the quasiNewton memoryless DFP update of the inverse Hessian approximation. A remarkable property of the proposed method is that both the sufficient and the conjugacy conditions are satisfied and the global convergence is established under some mild assumption. The numerical results show that the proposed method is promising and more efficient than any of the other methods considered.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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