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Conjugate Gradient Methods with Non-monotone Armijo Line Search Technique

Author: LiuJiXia
Tutor: YinHongYou
School: Nanjing University of Aeronautics and Astronautics
Course: Operational Research and Cybernetics
Keywords: Non-monotone line search Conjugate gradient methods Global convergence Modified FR method Modified PRP method Hybrid conjugate gradient method
CLC: O224
Type: Master's thesis
Year: 2009
Downloads: 42
Quote: 1
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Abstract


Conjugate gradient method is a class of very important numerical algorithms for unconstrained optimization problems. A good property of the conjugate gradient method is its lower storage and good convergence property. Therefore, they are particularly welcome in solving large-scale optimization problems. However, conjugate gradient methods do not necessarily guarantee to get descent directions for the objective function. Although some of them can produce descent directions, they depend on the line search technique. On the other hand, the technique of non-monotone line search has received many successful applications and extensions in unconstrained optimization and the speed of convergence of conjugate gradient methods under non-monotone line search, which doesn’t request for descent direction at each step. Non-monotone line search technique plays an important role in solving unconstrained optimization problems. Ever since the technique was presented, optimization researchers become more and more interested in it.This thesis mainly analyzes the modified conjugate gradient methods, which are very effective in producing the sufficient descent directions. The global convergences of the modified FR method, the modified PRP method as well as the hybrid FR-PRP method, are discussed respectively. Firstly, we propose a modified FR method and consider the use of the new non-monotone Armijo line search technique in the modified FR method. We also obtain the global convergence of the non-monotone modified FR method under appropriate conditions. Secondly, we combine the modified PRP method proposed in reference[1] with non-monotone Armijo line search technique and prove the global convergence of the new method. We also test the two new methods with the classical test functions. Preliminary numerical results show the new methods are effective comparing with non-monotone standard FR algorithm and non-monotone standard PRP algorithm. At last, we propose a hybrid conjugate gradient method on FR method and PRP method, which has taken the advantages of the two methods. We prove it can ensure the global convergence under non-monotone Wolfe line search technique.

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CLC: > Mathematical sciences and chemical > Mathematics > Operations Research > Optimization of the mathematical theory
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