Suppose that $H\succ 0$ and we have a sequence $\{x_j\}_{j\ge 1}$ such that they are the unique solution of the following problems:
$$x_{j} \in \text{argmin}_{x\in \mathbb R^n}(x-x_{j-1})^T\nabla f(x_{j-1})+\frac{1}{2}(x-x_{j-1})^TH(x-x_{j-1})+\frac{1}{2}\rho_j\|x-y_j\|_2^2.$$
Note that $\rho_j \to \infty$ as $j \to \infty$. Further, $x_0$ can be arbitrary and we know the sequence $\{y_j\}_j$ is bounded.
I want to show the sequence $$(x_j-x_{j-1})^T\nabla f(x_{j-1})+\frac{1}{2}(x_j-x_{j-1})^TH(x_j-x_{j-1}), \, j\in \mathbb N$$ is bounded below.
It turns out that if $f$ is convex and Lipshitz continues on the whole space, there is a lower bound (Existence of a lower-bound for an interesting function! and Lipschitz implies bounded gradient).
Unfortunately, I am not finding practical functions $f$ that satisfy these conditions (convexity and Lipshitz continuity on the entire space) together. So, I want to see if finding a lower bound is doable with less restrictive conditions or at least practical ones.
Thank you for your time!