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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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Has the following generalization of monotropic programming been studied in the literature?
I am interested in problems of the form
$$\min_{x \in C} \sum_{i=1}^n\sum_{j=1}^n f(x_i,x_j)$$
where $C$ is a convex subset of $\mathbb{R}^{n}$, and $f \colon \mathbb{R}^{2} \to \mathbb{R}$ is convex. …
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When is a convex function continuous on its domain?
Consider a lower-semicontinuous convex function $f\colon \mathbb{R}^n \to \mathbb{R}$ with domain $C = \{x \in \mathbb{R}^d: f(x) < \infty\}$. I am interested in understanding under what conditions $f …