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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)
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.
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.
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