Some nonlinear conjugate gradient methods based on spectral scaling secant equations

Abstract

Three conjugate gradient methods based on the spectral equations are proposed. One is a conjugate gradient method based on the spectral scaling secant equation proposed by Cheng and Li (J Optim Thoery Appl 146:305–319, 2010), which gives the most efficient Dai–Kou conjugate gradient method with sufficient descent in Dai and Kou (SIAM J Optim 23:296–320, 2013) family of conjugate gradient methods. The other is a special case in Dai and Liao (Appl Math Optim 43:87–101, 2001) family of conjugate gradient methods, which adopts a special parameter, an approximation to the spectral of the Hessian of the objective function. Another is a sufficient descent conjugate gradient method, which is based on the second one and can be viewed as a three-term conjugate gradient method based on a spectral scaling secant equation. Convergence properties of the three methods are discussed, and numerical results imply that conjugate gradient methods based on the spectral secant equations perform well, especially when the sufficient descent condition is satisfied.