There are many variants of non-linear conjugate gradient method available ( Flatcher-Reeves, Polak-Rebiere, Dai-Yuan). In case of minimization of quadratic function when first search direction is chosen to be negative of gradient then all of these algorithms are equivalent and possess finite termination property.
However, in case where first search direction is arbitrary (not equal to steepest descent direction), all these algorithms are not equivalent anymore. Moreover, their convergence properties varies as well. I am interested to know what happens to finite termination propety in this case? In particular which non linear CG still possess finite termination property under this condition.