A new conjugate gradient algorithm with cubic Barzilai-Borwein stepsize for unconstrained optimization.

In this paper, a new conjugate gradient (CG) algorithm in Dai-Liao (DL) family is presented for solving unconstrained optimization problems. The proposed algorithm tries to adjust positive values for the so-called DL parameter by using quadratic and/or cubic models of the objective function. More precisely, the cubic regularization model of the objective function is properly employed when the non-positive curvature is detected. Besides, the CG parameter is introduced so that the generated CG directions are descent. Under some standard assumptions, we establish the convergence property of the new proposed algorithm. Numerical results on some test problems are reported. The results show that the new algorithm performs well and is competitive with CG_DESCENT method.