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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?

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  • $\begingroup$ Could you elaborate a bit more on which problem you want to apply the conjugate gradient method? As Theorem 2 of the linked document is exactly your statement for the problem "computing the minimum of a quadratic function", I suppose you want to solve a different problem, right? $\endgroup$ Jan 5 '18 at 18:33
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You may have a look into this survey paper on conjugate gradient methods for continuously differentiable functions. There you can see that a lot of assumptions on the objective and the computed directions $d_i$ are required to only guarantee convergence (see for instance their Theorem 2).

So, the answer to your question is no, in general you need more than dimension-many steps to have convergence (if there is convergence at all) for general, non-linear functions.

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