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1. A NEW HYBRID WYL-AMRI CONJUGATE
GRADIENT METHOD WITH SUFFICIENT
DESCENT CONDITION FOR UNCONSTRAINED
OPTIMIZATION
Ibrahim S. Mohammed*
Mustafa Mamat
Abdelrhaman Abashar
Kamil Uba Kamfa

2. Contents :-
* Introduction
* Objectives of the research
* Conjugate Gradient Version
* New method and Algorithm
* Numerical Results
* Conclusion
* References

3. * Introduction
Conjugate gradient method (CG) are designed to solve large scale unconstrained
optimization problem. In general, the method has the form
𝒎𝒊𝒏 𝒙 ∈𝑹 𝒏 𝒇(𝒙) (1.1)
where 𝒇 ∶ 𝑹 𝒏
→ 𝑹 is continuously differentiable . Conjugate gradient methods are
iterative methods of the form
𝒙 𝒌+𝟏 = 𝒙 𝒌 + 𝜶 𝒌 𝒅 𝒌 (1.2)
where 𝒙 𝒌+𝟏 is the current iterate point , 𝜶 𝒌 > 𝟎 is step length which is computed
by carrying out a line search , 𝒅 𝒌 is the search direction of the conjugate gradient
method define by
𝒅 𝒌
−𝒈 𝒌 , 𝒊𝒇 𝒌 = 𝟎
−𝒈 𝒌 + 𝜷 𝒌 𝒅 𝒌, 𝒊𝒇 𝒌 ≥ 𝟏
(1.3)
Some classical formula’s for 𝜷 𝒌 are given as follows
𝜷 𝒌
𝑯𝑺
=
𝒈 𝒌
𝑻
(𝒈 𝒌−𝒈 𝒌−𝟏)
(𝒈 𝒌−𝒈 𝒌−𝟏) 𝑻 𝒅 𝒌−𝟏
(1.4)

4. Cont : Introduction
𝜷 𝒌
𝑭𝑹
=
𝒈 𝒌
𝑻
𝒈 𝒌
𝒈 𝒌−𝟏
𝑻 𝒈 𝒌−𝟏
(1.5)
𝜷 𝒌
𝑷𝑹𝑷
=
𝒈 𝒌
𝑻
(𝒈 𝒌−𝒈 𝒌−𝟏)
𝒈 𝒌−𝟏
𝑻 𝒈 𝒌−𝟏
(1.6)
𝜷 𝒌
𝑪𝑫
= −
𝒈 𝒌
𝑻
𝒈 𝒌
𝒅 𝒌−𝟏
𝑻 𝒈 𝒌−𝟏
(1.7)
𝜷 𝒌
𝑫𝒀
=
𝒈 𝒌
𝑻
𝒈 𝒌
(𝒈 𝒌−𝒈 𝒌−𝟏) 𝑻 𝒅 𝒌−𝟏
(1.8)
𝜷 𝒌
𝑳𝑺
= −
𝒈 𝒌
𝑻
(𝒈 𝒌−𝒈 𝒌−𝟏)
𝒅 𝒌−𝟏
𝑻 𝒈 𝒌−𝟏
(1.9)

5. Cont : Introduction
• Zoutendijk proved that FR Method with exact line search is
globally convergent , Al – Baali extended this results to the
strong Wolfe - Powell line search.
• Recently, Wei et al. [7], propose a new CG formula
• Abashar et al ,modified RMIL method to suggest

6. • Research Objective
* To proposed a new formula for solving
unconstrained optimization.
* To analyze the performance these new
formulas based on standard optimizations test
problem functions.
* To proof the sufficient descent conditions of
our new method

7. • Conjugate Gradient Version
(i) Hybrid CG methods (ii) Scaled CG methods. (iii) Three terms CG methods.
.
An important class of conjugate gradient algorithm is the hybrid CG method, for
example Hu and Storey [18] propose
Dai and Yuan [19] suggested two hybrid methods,

8. New method and Algorithm
we propose a new hybrid CG method is a combination between CG methods
(WYL, RMIL)
Algorithm
Step 1: Given
6
0 10, 
 n
Rx , set 00 gd  if 0g then stop.
Step 2: Compute k by applying exact line search.
Step 3: kkkk dxx 1 if 1kg then stop.
Step 4: Compute
hRW
k and generated kd by (1.3).
Step 5: Set 1 kk go to Step 2.

9. Numerical Results
• Test problem functions considered by Andrei.
• Stopping criteria as Hillstrom , 𝑔 𝑘 ≤ 𝜀.
• Subroutine programming using Matlab.
• Using exact line search.
• Performance profile introduced by Dolan and More.

10. Cont: Numerical Results
TABLE 1: A LIST OF PROBLEM FUNCTIONS
No Function Dim Initial Points
1 Six hump 2 (1, 1), (2, 2), (5,5), (10, -10)
2 Three hump 2 (24, 24), (29, 29), (33, 33), (50, 50)
3 Booth 2 (10, -10), (20, 20), (50, 50), (100, 100)
4 Treccani 2 (5, 5), (10,10), (-20, 20), (-50, -50)
5 Matyas 2 (1, 1), (5, 5), (10, 10), (50, 50)
6 Extended Maratos 2, 4 (0,0,0,0), (0.5,5, 0.5, 5), (10, 0.5, 10, 0.5), (70, 70, 70, 70)
7 Ext FREUD & ROTH 2, 4 (13, 13, 13, 13), (21, 21, 21, 21), (25, 25, 25,25), (23, 23, 23, 23)
8 Generalized Trig 2, 4, 10 (0.5, 5, …, 5),(5, 10, …, 10),(7,7, …, 7),(50, 50, …, 50)
9 Fletcher 2, 4, 10 (23, 23, …, 23), (45, 45, …, 45), (50, 5,…, 5), (70, 70, …,70)
10 Extended Penalty 2, 4, 10, 100 (0.5,5, …,5),(10,-0.5…,-0.5), (105,105, …,105), (130,130, …,130)
11 Raydan 1 2, 4, 10, 100 (1, 1, …,1), (3, 3, …, 3), (5, 5, …, 5), (-10, -10, …, -10)
12 Hager 2, 4, 10, 100 (3, -3, …, -3),(21, 21, …, 21), (-23, 23, …, 23), (23, 23, …, 23)
13 Rosenbrock 2, 4, 10, 100, 500, 1000, 10000 (7, 7, …, 7), (13, 13, …, 13), (23, 23, …, 23), (35, 35, …, 35)
14 Shallow 2, 4, 10, 100, 500, 1000, 10000 (21, -21, …, -21), (21, 21, …, 21), (50,50, …, 50),(130, 130, …, 130)
15 Tridiagonal 1 2, 4, 10, 100, 500, 1000, 10000 (0, 0,…, 0), (1, -1, …, -1), (17, -17, …, -17), (30, 30, …, 30)
16 Ext White & Holst 2, 4, 10, 100, 500, 1000, 10000 (-5, -5, …, -5), (2, -2, …, -2), (3, -3, …, -3), (7, -7, …, -7)
17 Ext Denschnb 2, 4, 10, 100, 500, 1000, 10000 (8, 8, …, 8), (11, 11, …,11), (12, 12, …, 12),(13, 13, …, 13)
18 Diagonal 4 2, 4, 10, 100, 500, 1000, 10000 (2, 2, …, 2), (5, 5, …,5), (10, 10, …, 10), (15, 15, …, 15)

12. Cont: Numerical Results
• Performance Profile based on Number
iterations

13. Cont: Numerical Results
• Performance Profile based on CPU time

14. • Conclusion
* AMRI was able to solve 95% of test problems.
* WYL solve 97% of test problems.
* SW-A solve all test problems.

15. • References
* M.R. Hestenes, E.L. Stiefel, Methods of conjugate gradients for solving linear
systems, J. Res. Natl. Bur. Stand. Sec. B 49 1952, pp. 409–432
*Z. Wei, S. Yao, L. Liu, The convergence properties of some new conjugate
gradient methods, Appl. Math. Comput. 183 2006, pp. 1341–1350.
*G. Zoutendijk, Nonlinear programming computational methods, in:J.Abadie
(Ed.), Integer and Nonlinear Programming, North-Holland, Amsterdam, 1970,
pp. 37–86.
*M.Rivaie,M.Mamat,L. June, M.Ismail, a new class of nonlinear conjugate
gradient coefficient with global convergence properties, Appl.Math.comput.218
,2012, pp.11323-11332.
. * Y.H. Dai, Y. Yuan. An efficient hybrid conjugate gradient method for
unconstrained optimization. Ann. Oper. Res. 103, 2001, pp. 33–47.
* E. Dolan, J.J.More, Benchmarking optimization software with performance
profile, Math. Prog. 91, 2002, pp. 201–213.
*Y. F. Hu, C. Storey . Global convergence result for conjugate gradient
methods. J.Optim.Theory.Appl., 71,1991, pp. 399-405.
* N. Andrei, An unconstrained optimization test functions collection, Adv.
Modell. Optim. 10, 2008, pp. 147–161.

16. •
Thank You

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