Consider the problem $$\text{minimize}\,\,\, -8x_{1}+x_{2} \\ \text{subject to}\,\,\,\,\,x_{2} \leq 8, \\ (x_{1}-4)^{2} - x_{2} \leq 8 $$
By using the method of Lagrange multipliers, I found the optimal point to occur at $\hat{x} = (x_{1},x_{2}) = (8,8)$ with multiplier vector $\hat{\lambda}=(0,1)$.
I was able to show after the fact that the problem satisfies the Linear Independence Constraint Qualification (LICQ), since the gradients of the constraint functions evaluated at $\hat{x}$ are linearly independent. And so, $\hat{x}$ does satisfy the Karush-Kuhn-Tucker (KKT) conditions - i.e., these first-order necessary conditions for optimality are satisfied.
Also, since the problem is convex, these conditions are also sufficient to determine optimality.
The next thing that I need to do, and what I need help with is finding the tangent cone to the feasible set at the optimal solution $\hat{x} = (8,8)$.
My notes/text defines the tangent cone as the set $T_{X}(x)$ of all tangent directions for $X \subset \mathbb{R}^{n}$ at $x\in X$. And, a direction $d$ is called tangent to a set $X \subset \mathbb{R}^{n}$ at the point $x \in X$ and scalars $\tau_{k}>0$, $k = 1,2, \cdots $ such that $\tau_{k} \downarrow 0$ and $$d = \lim_{k \to \infty}\frac{x^{k}-x}{\tau_{k}}.$$
It also defines the tangent cone as the closure of the cone of feasible directions at $x \in X$: $$ T_{X}(x) = \overline{K_{X}(x)}= \overline{cone(X-x)}, $$ where $cone(X-x)$ is the cone generated by the convex set $X$.
The feasible set in question here IS convex, and below is a sketch of it:
The arrows indicate the direction in which I think the tangent cone is supposed to go, but I'm not sure. Could somebody please help me with this? Unfortunately, I'm still not confident when it comes to figuring out tangent cones...
I was thinking maybe I should first find the tangent line at that point, but then where do I go from there? And how will the constraint qualification conditions come into play?
Thank you ahead of time.
At the optimal point, the slope of the parabola is $8$. Thus, from your pictures, it is clear that $T_X(8,8)$ is the convex cone generated by the vectors $(-1,0)$ and $(-1,-8)$, i.e., $$T_X(8,8) = \big\{ \alpha(-1,0)+\beta(-1,-8) \mid \alpha\geq 0,\beta\geq 0\big\}.$$