counterexample for a DC critical point that is not a limiting-stationary point?

Question

Let $f$ be a DC function defined by $f = g - h$ where $g$ and $h$ are proper, lower semicontinuous and convex functions from $\mathbb{R}^n$ to $\mathbb{R}\cup\{+\infty\}$. A point $x^*$ is called a DC (Difference-of-convex) critical point if $0\in \partial g(x^*)- \partial h(x^*)$, where $\partial$ stands for the classical subdifferential of convex functions defined by $$\partial g(x^*) := \left\{ y\in \mathbb{R}^n: g(z) \geq g(x^*) + \langle y, z-x^* \rangle, \forall z\in \mathbb{R}^n \right\}.$$ A point $x^*$ is called a limiting-stationary point if $0\in \partial^{\text{lim}} f(x^*)$ where the limiting-subdifferential is defined by $$\partial^{\text{lim}} f (x^*) : = \left\{ y\in \mathbb{R}^n: \liminf_{z\to x^*, z\neq x^*} \frac{f(z) - f(x^*) - \langle y, z-x^* \rangle}{\|z-x^*\|}\geq 0 \right\}.$$ I wondered is there any counterexample such that a DC critical point is not a limiting-stationary point?

Answer 1

I think there is a simple but convincing example:

Consider $$g(x)=\max\{0,x\} + \chi_{\{x\geq -1\}}(x), \quad h(x)=\max\{0,-x\} $$ where $\chi_{\{x\geq -1\}}$ is the indicator function of the convex set $\{x\in \mathbb{R}: x\geq -1\}$. Hence, both $g$ and $h$ are proper, lower semicontinuous and convex functions from $\mathbb{R}$ to $\mathbb{R}\cup \{+\infty\}$.

Now, consider the point $x^*=0$. We can show that $x^*$ is a DC critical point but not a limiting-stationary point.

$$\partial g(0)=[0,1], \quad \partial h(0) = [-1,0], \quad \partial^{\text{lim}}f(0) = \{1\}.$$ Hence, $$0\in \partial g(0) - \partial h(0) = [0,2]$$ but $$0\notin \partial^{\text{lim}}f(0).$$