Suppose $f(x):R^n\rightarrow R$ satisfies the next 3 conditions:
1:$\nabla f(x)$ is Lipschitz continuous.
2:$\nabla^2 f(x)$ is Lipschitz continuous.
3:$\| \nabla f(x)\|\rightarrow\infty$ as $\|x\|\rightarrow\infty$.
Is there any simple example of nonlinear and non-quadratic function $f(x)$ which satisfies the last 3 conditions?
These conditions are given in some papers in optimization. But when I try to find such a example, I found it hard to construct a nonlinear and non-quadratic function satisfying all these conditions.
$\textbf{Motivation:}$ I am doing some researches in optimization. These conditions are useful in proving the convergence or error estimation. However, I need to do some numerical experiments to show the effectiveness of my algorithm, which requires some real examples.
Since no one answer this question, I post my own thought here. To the best of my knowledge, there are two examples:
1:$\log(1+\exp(Ax+b))$. Here we use the notation $\exp(x):=\exp(x_1)+..\exp(x_n)$ for vector $x\in R^n$.
2:$x^TAx+cos(Bx)$, Here $A$ is a positive definite matrix, and the notation $cos(x):=cos(x_1)+...+cos(x_n)$ for $x\in R^n$.