Two modified conjugate gradient methods for solving unconstrained optimization and application

Conjugate gradient methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems as they do not require the storage of any matrices. In order to obtain a theoretically effective and numerically efficient method, two modified conjugate gradient methods ( called the MCB1 and MCB2 methods ) are proposed. In which the coefficient βk in the two proposed methods is inspired by the structure of the conjugate gradient parameters in some existing conjugate gradient methods. Under the strong Wolfe line search, the sufficient descent property and global convergence of the MCB1 method are proved. Moreover, the MCB2 method generates a descent direction independently of any line search and produces good convergence properties when the strong Wolfe line search is employed. Preliminary numerical results show that the MCB1 and MCB2 methods are effective and robust in minimizing some unconstrained optimization problems and each of these modifications outperforms the four famous conjugate gradient methods. Furthermore, the proposed algorithms were extended to solve the problem of mode function.