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### Common subdifferentials of convex function

Let $f: \mathbb R^n \rightarrow \mathbb R$ be a convex function. By a subdifference of $f$ in $x\in \mathbb R^n$ we mean an $h\in \mathbb R^n$ such that $f(x) \geq f(p)+<x-p,h>$ for all ...
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I am currently trying to understand a certain proof. The author uses the term dual convex pair for a pair $(\phi,\psi)$ of convex functions defined on subsets $X,Y$ of $\mathbb R^n$ satisfying: $$... 0answers 119 views ### Convex subsets, Normed spaces, Separating hyperplane I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If S_1,S_2 are disjoint non-empty convex subsets of a real vector space X (which may be infinite dimensional) then ... 0answers 29 views ### What Projections preserve Pseudocontractiveness? Let f: \mathbb{R}^n \rightarrow \mathbb{R}^n be a pseudocontraction, i.e.,$$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$for all x,y \in ... 1answer 29 views ### Equality constrained Quadratic Program Consider the QP$$ x^* = \arg \min_{\displaystyle x \in \mathbb{R}^n{\geq 0}} \ \frac{1}{2} x^\top P x + q^\top x \ \text{ sub. to: } A x = b, $$where P \succ 0. Without the non-negativity ... 1answer 59 views ### Existence of global minimum Could someone help me with this problem? Let C, D convex and closed sets such that the intersection is empty. I want to show that the function f: \mathbb{R^n} \to \mathbb{R} defined by f(x) = ... 1answer 117 views ### A bounded subset in \mathbb R^2 which is “nowhere convex”? Let F : \mathbb S^1 \to \mathbb R^2 represents a simple closed curve C in \mathbb R^2. The Jordan curve theorem says that the curves bounds a interior domain \Omega and \partial \Omega= C. ... 1answer 29 views ### A characterization of differentiability of a convex function Let \phi : \mathbb R^n \to \mathbb R be a convex function. For all point x\in \mathbb R^n, define the subdifferential as$$\partial \phi(x) = \{ y\in \mathbb R^n | \ \phi(z) \geq \phi(x) + ...
I first give two definitions. Def1: A set $S$ is a cone if $x \in S, \lambda \geq 0 \implies \lambda x \in S$. Def2: Let $S$ be any set (we may assume $\mathbb{R}^n$ with the usual Euclidean ...