Prove the equivalence of the following expressions
- $\left\|\left(\nabla^2 (_)−_\right)(_{+1}−_)\right\|=o\left(\left\|_{+1}−_\right\|\right)$.
2.$\left\|\nabla(_{+1})−\nabla (_)−_(_{+1}−_)\right\|=o\left(\left\|_{+1}−_\right\|\right).$
I never work with small or if someone can help me with a hint i appreciate.
Where $x_{k+1}=x_k+\alpha p_k$ where $\alpha \in \mathbb{R}$ and $p$ is a descent direction and $H_k$ is a matrix how approaches the Hessian, I traid to use the limit definition on Hessian matrix size i have $\lim_{\alpha \longrightarrow 0 } \dfrac{\nabla(_{+1})−\nabla (_)}{\beta}$ if i remplace on 1
- $\left\|\left(\lim_{\alpha \longrightarrow 0 } \dfrac{\nabla(_{+1})−\nabla (_)}{\alpha}−_\right)(_{+1}−_)\right\|=o\left(\left\|_{+1}−_\right\|\right)$.
Here is close of what i need but i dont see how to continue.