In this paper, we make a modification to the LS conjugate gradient method and propose a descent LS method. The method can generates sufficient descent direction for the objective function. We prove that the method is globally convergent with an Armijo-type line search. Moreover, under mild conditions, we show that the method is globally convergent if the Armijo line search or the Wolfe line sea...

In [Z.Wei, S. Yao, L. Liu, The convergence properties of some new conjugate gradient methods, Applied Mathematics and Computation 183 (2006) 1341–1350], Wei et al. proposed a new conjugate gradient method called WYL method which has good numerical experiments and some excellent properties such as b k P 0. In this paper, we prove that while tk 6 1 c 2L kgkk kdkk , the sufficient descent conditio...

A modified Polak-Ribière-Polyak conjugate gradient algorithm which satisfies both the sufficient descent condition and the conjugacy condition is presented. These properties are independent of the line search. The algorithms use the standard Wolfe line search. Under standard assumptions we show the global convergence of the algorithm. Numerical comparisons with conjugate gradient algorithms usi...

Conjugate gradient methods are probably the most famous iterative methods for solving large scale optimization problems in scientific and engineering computation, characterized by the simplicity of their iteration and their low memory requirements. In this paper, we propose a new conjugate gradient method which is based on the MBFGS secant condition by modifying Perry’s method. Our proposed met...

It is well-known that the search direction generated by the standard (unmodified) PRP nonlinear conjugate gradient method is not necessarily a descent direction of the objective function, which brings difficulty for its global convergence for general functions. However, to our surprise, it is easily proved in this short note that the unmodified PRP method still converges globally even for nonco...

In this paper, a new nonlinear conjugate gradient method is proposed for large-scale unconstrained optimization. The sufficient descent property holds without any line searches. We use some steplength technique which ensures the Zoutendijk condition to be held, this method is proved to be globally convergent. Finally, we improve it, and do further analysis.

We analyze the conjugate gradient (CG) method with variable preconditioning for solving a linear system with a real symmetric positive definite (SPD) matrix of coefficients A. We assume that the preconditioner is SPD on each step, and that the condition number of the preconditioned system matrix is bounded above by a constant independent of the step number. We show that the CG method with varia...