The conjugate gradient method plays a very important role in several fields, to solve problems of large sizes. To improve the efficiency of this method, a lot of work has been done; in this paper, we propose a new modification of PRP method to solve a large scale unconstrained optimization problems in relation with Strong Wolf Powell line search property, when the latter was used under some conditions, a global convergence result was proved. In comparison with other known methods the efficiency of this method proved that it is better in the number of iterations and in time on $90$ proposed problems by use of Matlab.

This paper proposes a novel hybrid conjugate gradient method for nonparametric statistical inference.The proposed method is a convex combination of the modified linear search (MLS) and Fletcher-Reeves (FR) methods, and it inherits the advantages of both methods. The FR method is known for its fast convergence, while the MLS method is known for its robustness to noise. The proposed method combines these advantages to achieve both fast convergence and robustness to noise. Our method is evaluated on a variety of nonparametric statistical problems, including kernel density estimation, regression, and classification. The results show that the new method outperforms the MLS and FR methods in terms of both accuracy and efficiency.

This paper presents a new hybrid conjugate gradient method for solving  nonlinear unconstrained optimization problems; it is based on a combination of $RMIL$  (Rivaie-Mustafa-Ismail-Leong)  and $hSM$  (hybrid Sulaiman- Mohammed) methods. The proposed algorithm enjoys the sufficient descent condition without depending on any line search; moreover, it is globally convergent under the usual and Strong Wolfe line search assumptions.  The performance of the algorithm is demonstrated through numerical experiments on a set of 100 test functions from [1] and four image restoration problems with two noise levels. The numerical comparisons with four existing methods show that the proposed method is promising and effective.

The conjugate gradient method is one of the most important ideas in sci-entific computing. It is applied to solving linear systems of equations and nonlinear optimization problems. In this paper, based on a variant of the Hestenes–Stiefel (HS) method and Polak–Ribière–Polyak (PRP) method, two modified conjugate gradient methods (named MHS∗ and MPRP∗) are presented and analyzed. The search direction of the presented methods fulfills the sufficient descent condition at each iteration. We establish the global convergence of the proposed algorithms under normal assumptions and Strong Wolfe line search. Preliminary elementary numerical experi-ment results are presented, demonstrating the promise and the effective-ness of the proposed methods. Finally, the proposed methods are further extended to solve the problem of conditional model regression function.

In this paper, we formulate and study the duality problems in Wolfe type for optimization problems with vanishing constraints in nonsmooth case, whereas Mishra et al. (Ann Oper Res 243(1): 249–, 272, 2016) investigated them in smooth case. Also, we derive the weak, Strong and strict converse duality results for the problems with Lipschitzian functions utilizing the Clarke subdifferential.