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When applying the globalized BFGS algorithm (Quasi-Newton Method, optimization, minimization) to approximate the minimum of a function using the Quasi-Newton-Method, sometimes one can get a negative gradient when trying to figure out the search direction.

What could be possible methods (or algorithms) to calculate an approximation of the hessian matrix, which has to be updated for the next step (iteration)?

Thanks in advance!

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    $\begingroup$ What is a "negative gradient"? Is it that all components are negative? Why do you think this is problematic? $\endgroup$ – LutzL Jun 17 at 4:55
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    $\begingroup$ Presumably the author of the question means that sometimes the search direction is not a descent direction. In that case the typical approach is to throw out the approximate Hessian and restart the algorithm with a steepest descent step. $\endgroup$ – Brian Borchers Jun 18 at 4:11

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