AN ADAPTIVE CONJUGACY CONDITION AND RELATED NONLINEAR CONJUGATE GRADIENT METHODS

In an attempt to find a reasonable solution for an open problem propounded by Andrei in nonlinear conjugate gradient methods, an adaptive conjugacy condition is proposed. The suggested condition is designed based on an implicit switch from a conjugacy condition to the standard secant equation, using an extended conjugacy condition proposed by Dai and Liao. Following the approach of Dai and Liao, two adaptive nonlinear conjugate gradient methods are proposed based on the suggested adaptive conjugacy condition. An interesting feature of one of the proposed methods is the adaptive switch between the nonlinear conjugate gradient methods proposed by Hestenes and Stiefel, and Perry. Under proper conditions, it is shown that one of the proposed methods is globally convergent for uniformly convex functions and the other is globally convergent for general functions. Numerical results demonstrating the effectiveness of the proposed adaptive approach in the sense of the performance profile introduced by Dolan and Moré are reported.