I have the following prbolem
$$ min \space (x_1-1)^2+x_2^2\\ s.t. \space x_1-\lambda^{-1}x_2^2\leq0 $$
Furthermore, it holds $$\lambda\in\mathbb{R}\backslash \{0\}$$
How do I show under which conditions on $\lambda$ do the second-order sufficient optimality conditions for constrained optimization problems hold at $x^*=(0,0)^T$? I already know that LICQ holds in $x^*$ and that it is a KKT point.
Thanks for showing your work. For KKT conditions to be sufficient, we just need both the objective and constraints to be convex. Well the objective is clearly convex.
Constraint is convex only if $\lambda<0$. Can you argue why?
Hint 1: For $\lambda<0$, $\ g(x)=x_1-\lambda^{-1} x_2^2$ is a convex function and $g(x)\leq 0$ is just a sub-level set. For $\lambda>0$, let $S=\{x: g(x) \leq 0\}$. Does $(1, \sqrt{\lambda}) \in S$, $\ (1, -\sqrt{\lambda}) \in S$? What about $(1, 0)$?
Hint 2: $f(x) = (x_1-1)^2 + x_2^2$ Lets look at the Hessian: $$\ H= \begin{bmatrix}2 & 0\\0 & 2\end{bmatrix} $$ Clearly, Hessian is Positive Definite.