I am interested in the "most" simple form of the semi-infinite programming (SIP): \begin{align} \min_{x\in \mathbb R^n} \hspace{0.5cm} & f(x)\\ \text{s.t.} \hspace{0.5cm} & \langle a(t), x\rangle \leq \lambda, \forall t\in T. \end{align} where
- $f: \mathbb R^n\rightarrow \mathbb R^n$ is smooth convex, e.g. $f(x)=||y-x||_2^2$,
- $T$ is infinite, e.g. $T=[0,1]$
- $a: T\rightarrow \mathbb R^n$ is continuous
- $\lambda>0$
Where I can find the KKT conditions for such problem?
I tried to google:
- Wiki: don't have KKT conditions
- A book: but this only talks about linear SIP, i.e. $f(x)=\langle c, x\rangle$.
- An article: but this talks about non-smooth and multi-objective problems, which is very general, and of course too complicated for me.