Somewhat based on: Reformulation of optimization problem using kkt and lagrange conditions
Say I have the following optimization problem: $$ \begin{aligned} \min_{z}\min_{y} \, &\frac{1}{2} y^T \bar{H}y,\\ &\begin{aligned} \text{s.t. } &\bar{E}y = 0,\\ &\bar{A}y \leq \bar{b} + \bar{d}z, \end{aligned} \end{aligned} $$ (with $y \in \mathbb{R}^{n}, z\in \{0,1\}^{2r},\bar{H}\in \mathbb{R}^{n\times n},\bar{E} \in \mathbb{R}^{m_1\times n},\bar{A}\in \mathbb{R}^{m_2\times n},\bar{b} \in \mathbb{R}^{m_2},\bar{d} \in \mathbb{R}^{m_2\times 2r}).$
Also, importantly, $\bar{H}$ is indefinite.
My goal is to reformulate the inner minimization problem over $y$ using the KKT conditions.
I am wondering under which circumstances the KKT conditions are actually first order necessary conditions. From my understanding and from what I gathered from my previous question (see link above), the minimum has to exist in order for the KKT conditions to be necessary. Thus, I would say that in the following cases they are indeed necessary:
- $\bar{H}$ is semidefinite on the feasible set $\mathcal{F} = \left\{y \, \middle| \, \bar{E}y = 0 \, \land \, \bar{A}y \leq \bar{b} + \bar{d}z\right\}$
- the feasible set $\mathcal{F}$ is bounded
Are there any other situations which I did not mention (assuming my findings are correct) under which I can apply the KKT conditions to the optimization problem at hand?