# Globalized BFGS (Quasi-Newton method) condition

I didn't find any information on the internet about the globalized $$\textit{BFGS}$$ method. Wikipedia only talks about the normal BFGS, without this globalized condition:

Assume direction of descent $$d = -H_f^{-1}(x^k)\cdot \nabla f(x^k)$$. Instead of calculating the inverse of the Hessian matrix we use an approximation method with two Rank-1 Updates, the result of this method is the Matrix $$B^{-1}$$.

Now, the condition of the globalized BFGS method:

$$\nabla f(x^k)^T\cdot d \leq -\rho||d||$$ with $$\rho$$ being a constant (e.g. $$0.001$$)

If we happen to have this condition unfulfilled (this would probably be the result of a wrong Hessian-Matrix approximation), we choose $$d = -\nabla f(x^k)$$ for this step and also reset the BFGS method by either:

$$\circ$$ (1) Setting $$B^{-1} = I$$ for our next iteration

$$\circ$$ (2) Doesn't do anthing leave $$B^{-1}$$ as is for next iteration

I don't really see a difference in (1) or (2). Assume we would choose (1) for our implementation, due to the formula $$d = -B_f^{-1}(x^k)\cdot \nabla f(x^k)$$ with $$B = I$$ would result in $$d = -\nabla f(x^k)$$.

With (2) - how can we be sure that we are able to fullfill said condition again ever? Assume we never update $$B^{-1}$$ again because our $$d$$ doesn't fulfill the condition, we would always put $$d = -\nabla f(x^k)$$. In a different scenario we might be able to fulfill the condition in step $$k + 10$$, now we are doing the two Rank-1 updates, but with a matrix $$B$$ from step $$k$$ - so $$10$$ steps ago! How could that work ever!?

All in all I don't really get this resetting or globalized BFGS method.

I would really appreciate if someone could explain to me why $$(1)$$ or $$(2)$$ would be a good choice.