The quasi-newton equation (QN) plays a key role in contemporary nonlinear optimization. In this paper, we present a new symmetric rank-one (SR1) method by using preconditioning conjugate gradient (PCG) method for solving unconstrained optimization problems. The suggested method has an algorithm in which the usual SR1 Hessian is updated. We show that the new quasi-newton (SR1) method maintains the Quasi-Newton condition and the positive definite property. Numerical experiments are reported which produces by the new algorithm better numerical results than those of the normal (SR1) method by using PCG algorithm based on the number of iterations (NOI) and the number of functions evaluation (NOF).

In this paper, we s uggest a new conjugate gradient method for solving nonlinear uncons trained optimization problems by using three term conjugate gradient method , We give a descent condi tion and the sufficient descent condition of the suggested method.

Conjugate gradient (CG) methods which are usually generate descent search directions, are beneficial for large-scale unconstrained optimization models, because of its low memory requirement and simplicity. This paper studies the three-term CG method for unconstrained optimization. The modified a three-term CG method based on the formal * which is suggested by Kafaki and Ghanbari [11], and using some well-known CG formulas for unconstrained optimization. Our proposed method satisfies both (the descent and the sufficient descent) conditions. Furthermore, if we use the exact line search the new proposed is reduce to the classical CG method. The numerical results show that the suggested method is promising and exhibits a better numerical performance in comparison with the three-term (ZHS-CG) method from an implementation of the suggested method on some normal unconstrained optimization test functions.

This paper, proposes a new conjugate gradient method for unconstrained optimization based on Dai-Liao (DL) formula; descent condition and sufficient descent condition for our method are provided. The numerical results and comparison show that the proposed algorithm is potentially efficient when we compare with (PR) depending on number of iterations (NOI) and the number of functions evaluation (NOF).

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