Another hybrid conjugate gradient method as a convex combination of WYL and CD methods

Conjugate gradient (CG) methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems. In this paper, a new hybrid conjugate gradient (CG) method is presented and analyzed for solving unconstrained optimization problems, where the parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>β</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> βₖ is a convex combination of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>β</m:mi> <m:mi>k</m:mi> <m:mi>WYL</m:mi> </m:msubsup> </m:math> βₖWYL and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>β</m:mi> <m:mi>k</m:mi> <m:mi>CD</m:mi> </m:msubsup> </m:math> βₖCD . Under the strong Wolfe line search, the new method possesses the sufficient descent condition and the global convergence properties. The preliminary numerical results show the efficiency of our method in comparison with other CG methods. Furthermore, the proposed algorithm HWYLCD was extended to solve the problem of a mode function.