I am revisiting conjugate gradient here and have this question: Can we say the algorithm will converge with exact $k$ steps ($k$ is number of variables) in conjugate gradient method using exact line search? (Assuming a general non-linear function but we can do exact line search efficiently.)
Because the key idea of conjugate gradient method is "tuning next variable will not mess up with previously tuned variable". So, if exactly line search is used, we should be about to figure out the optimal value for each variable one at a time. With $k$ steps, we can reach the optimal point.
Am I correct?