Let $X$ be an infinite-dimensional Banach space. I have the following optimization problem.
$$\begin{array}{ll} \underset{x \in X}{\text{minimize}} & f(x)\\ \text{subject to} & g_1(x) \leq 0\\ & g_2(x) \leq 0\\ & g_3(x) = 0 \end{array}$$
where functions $f: X \to \mathbb R$ and $g_1: X \to \mathbb R$ are convex and functions $g_2, g_3: X \to \mathbb R$ are affine. I am interested in understanding under which conditions strong duality holds for this problem.
I know that if $X$ were finite-dimensional then I could use the "refined" Slater's condition that says that strong duality is attained if there exists an $\overline x \in \mbox{ri}(X)$ such that $g_1(\overline x) < 0$ and $g_2(\overline x) \leq 0$ and $g_3(\overline x) = 0$. In other words, strict feasibility for the convex constraint and standard feasibility for the affine constraint and of course for the equality constraint.
Does something similar exist when $X$ is an infinite-dimensional Banach space, e.g., $L_p$?
