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Let $B_k$ be the approximated Hessian computed with the L-BFGS method.

I know it is possible to compute $(B_k)^{-1}d$ with the two loop recursion algorithm. I would like to know is there is such an algorithm that compute $B_kd$.

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The authors of the two-loop recursion write the following in the reference paper: enter image description here

So no, chances are that such a form doesn't exist for the non-inverse Hessian approximation. Nevertheless, products calculated through the "compact form" aren't too much slower than the two-loop recursion.

Source: Byrd, Richard H., Jorge Nocedal, and Robert B. Schnabel. "Representations of quasi-Newton matrices and their use in limited memory methods." Mathematical Programming 63.1 (1994): 129-156.

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