Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions.
Assume we have the following convex optimization problem:
$$ \begin{align} \text{minimize } \; &H(x) \; \text{ s.t.} \\ &G_i(x) \leq 0 , i \in \{1,\dotsc,m\} \\ &x \in K \end{align} $$
Now assume we have estimators of $H$ and $G_i$, namely $\mathbb H_n, \mathbb G_{n,i}$ which converge uniformly almost surely, i.e.
$$ \begin{align} &\sup_{x\in K} \left\vert H(x) - \mathbb H_n(x)\right\vert \to 0 \; \text{almost surely}\\ &\sup_{x\in K} \left\vert G_i(x) - \mathbb G_{n,i}(x)\right\vert \to 0 \; \text{almost surely } \; , i \in \{1,\dotsc,m\} \end{align} $$
Now we can consider the same (possibly non-convex) optimization problem with $H$ and $G_i$ replaced by their estimators. Can we say anything about the convergence of the sequence of minimizers? In particular, are any consistency results available?
Of course, if $G_i$ was known, then the theory of M-estimators would apply, but here we are also estimating the convex set which constrains the problem!
