I have an optimization problem of the form:
\begin{align*}\text{minimize } f(x) \text{ s.t. } x \in X \text{ and } g(x) \le G.\end{align*}I have $f$ as a continuous function, $X$ is both compact and convex, and $G \in \mathbb{R}^n$ where $n > 1$. If there is no feasible $x$, I know that\begin{align*}\inf_{x\in X} \sup_{\lambda \in \mathbb{R}^n : \lambda \ge 0} \left[ f(x) + \langle\lambda, g(x) - G\rangle \right]= \infty.\end{align*}I am wondering if it is true that the supinf will be infinity as well, i.e.,\begin{align*}\sup_{\lambda \in \mathbb{R}^n : \lambda \ge 0}\inf_{x\in X} \left[ f(x) + \langle\lambda, g(x)-G\rangle \right]= \infty.\end{align*}When $n=1$, it is easy to show as we can let $\lambda$ grow unbounded on the real line but in the case when $n>1$, I am not sure how to construct a sequence of $\lambda$s that can take the infimum value go to infinity.
