Open Problems in Nonlinear Conjugate Gradient Algorithms for Unconstrained Optimization

The paper presents some open problems associated to the nonlinear conjugate gradient algorithms for unconstrained optimization. Mainly, these problems refer to the initial direction; the conjugacy condition, the step length computation; new formula for conjugate gradient parameter computation based on function's values, the influence of accuracy of line search procedure, how we can take the problem's structure on conjugate gradient algorithms, how we can consider the second order information in these algorithms, what the most convenient restart procedure is, what the best hybrid conjugate gradient algorithm is, scaled conjugate gradient algorithms, what the most suitable stopping criterion in conjugate gradient algorithms is, etc.