A new direction search of hybrid quasi-Newton

A new hybrid quasi-Newton search direction ( 𝐻𝑄𝑁𝐸𝐼 ) is proposed. It uses the update formula of Broyden–Fletcher–Goldfarb–Shanno (BFGS) with a certain conjugate gradient (CG) parameter by a nested direction. The global convergence analysis and superlinear rate, addtionaly with sufficient descent are proved using exact line search. Finally, the computation comparisons are made with original hybrid parents; BFGS and CG, through the efficiency in terms of iteration numbers and CPU-running time showing the superior of 𝐻𝑄𝑁𝐸𝐼. Therefore, the results marked preference of 𝐻𝑄𝑁𝐸𝐼 from other two producer algorithms.