A hybrid FR-DY conjugate gradient algorithm for unconstrained optimization with application in portfolio selection

In this paper, we present a new hybrid conjugate gradient (CG) approach for solving unconstrained optimization problem. The search direction is a hybrid form of the Fletcher-Reeves (FR) and the Dai-Yuan (DY) CG parameters and is close to the direction of the memoryless Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton approach. Independent of the line search, the search direction of the new approach satisfies the descent condition and possess the trust region. We establish the global convergence of the approach for general functions under the Wolfe-type and Armijo-type line search. Using the CUTEr library, numerical results show that the propose approach is more efficient than some existing approaches. Furthermore, we give a practical application of the new approach in optimizing risk in portfolio selection.