Some combined techniques of spectral conjugate gradient methods with applications to robotic and image restoration models

The spectral conjugate gradient (SCG) methods ensure the descent property of an iterative scheme by scaling the first term of the search direction in a conjugate gradient
(CG) method. The formulations of SCG algorithms in Li et al. (J. Comput. Appl. Math.
350, 372–379 2019) and Amini and Faramarzi (J. Comput. Appl. Math. 417 2023)
rely on the double-bounded property for their convergence properties to be achieved.
However, these modifications ignored some terms in their SCG parameters to ensure
sufficient descent. Consequently, this study aims to propose a sufficient descent Polak,
Ribière and Polyak (PRP) SCG method for approximating solutions to large-scale optimization problems without ignoring any term in the SCG parameter and relaxing the
double-bounded property. Firstly, the spectral parameter is motivated by some intriguing theoretical features of the extended conjugacy condition, as well as the quadratic
convergence property of the quasi-Newton method. Secondly, based on various standard test problems, the numerical results reveal the method’s advantages compared to
some popular CG methods. Lastly, the method demonstrates positive outcomes when
applied to solve time-varying and inverse optimization problems involving robotic
control and image restoration models.