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Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes

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Does the refined Slater's condition hold also in the infinite-dimensional case?

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 …
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