From a Scalar to a Matrix Setting for the Dai--Liao Parameter
As is well known, both the numerical performance and the theoretical properties of the Dai--Liao conjugate gradient algorithm are highly dependent on the adjustment of its key parameter. Here, we first employ the well-known Harmonic--Geometric--Arithmetic--Quadratic mean inequality to reinforce the optimality of two previously proposed scalar adaptive settings of the Dai--Liao parameter. In other words, we show how these parameter choices are capable of enhancing the well-conditioning of the Dai--Liao search direction matrix by shrinking the intervals containing its singular values. A similar analysis is also carried out for scaled memoryless quasi--Newton updating formulas in order to further justify the optimality of two classical scaling parameters associated with these updates. Then, as the main contribution of this work, we move from the classical scalar setting of the Dai--Liao parameter to a matrix setting aimed at enhancing flexibility and diversity within optimization methods. In particular, we show that, under such a matrix formulation of the parameter, a well-known three-term conjugate gradient algorithm emerges as a member of the proposed extended Dai--Liao class of algorithms while enjoying several computationally attractive properties. Among these, the well-conditioning of the associated search direction matrix is especially noteworthy, as well as the ability to make more explicit use of the second-order information of the model. Finally, to provide practical evidence supporting the proposed matrix setting of the Dai--Liao parameter, we conduct a series of numerical experiments on standard benchmark test problems and report the results in detail. Generally speaking, the proposed framework is shown to retain both theoretical soundness and computational reliability.