Is the example of convex programming problem (convex program) for which $\exists \lambda = (\lambda_0, ... ,\lambda_m)$ s.t. Karush-Kuhn-Tucker conditions are met but $\dot{x}$ isn't minimum?
X - linear space. $f_i: X \rightarrow \mathbb{R}\cup \{+\infty\}$ - convex functions, $i \in \{0,m\}$
Convex programming problem: $$ \cases{f_0(x) \rightarrow \inf \\ f_i(x)\leq 0, i \in \{1,m\}}$$
For $\lambda = (\lambda_0, ... ,\lambda_m)$ let $L(x, \lambda) = \sum f_i(x)\lambda_i$
$\dot{x} - \text{solution}$
Karush-Kuhn-Tucker conditions:
- $ \displaystyle\min_{x \in X} L(x, \lambda) = L(\dot{x}, \lambda)$
- $ \lambda_i \geq 0, \quad i \in \{0,m\}$
- $ \lambda_i f_i(\dot{x}) = 0, \quad i \in \{1,m\}$