Polak Ribiere Polyak Conjugate Research Articles

A modified PRP conjugate gradient method for unconstrained optimization and nonlinear equations

A modified Polak Ribiere Polyak(PRP) conjugate gradient(CG) method is proposed for solving unconstrained optimization problems. The search direction generated by this method satisfies sufficient descent condition at each iteration and this method inherits one remarkable property of the standard PRP method. Under the standard Armijo line search, the global convergence and the linearly convergent rate of the presented method is established. Some numerical results are given to show the effectiveness of the proposed method by comparing with some existing CG methods.

The modified PRP conjugate gradient algorithm under a non-descent line search and its application in the Muskingum model and image restoration problems

In this paper, a modified Polak–Ribiere–Polyak (PRP) method, which possesses the following desired properties for unconstrained optimization problems, is presented. (i) The search direction of the given method has the gradient value and the function value. (ii) A non-descent backtracking-type line search technique is proposed to obtain the step size $$\alpha _k$$ and construct a point. (iii) The method inherits an important property of the classical PRP method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. (iv) The strongly global convergence and R-linear convergence of the modified PRP method for nonconvex optimization are established under some suitable assumptions. (v) The numerical results show that the modified PRP method not only is interesting in practical computation but also has better performance than the normal PRP method in estimating the parameters of the nonlinear Muskingum model and performing image restoration.