# Tagged Questions

Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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### general framework for proof of convexity in least square problem

Let $f(x_i,\theta) : X \times \Theta \rightarrow Z$ be a parametric function with parameter $\theta$ that we wish to fit to set of samples $S =\{(x_i,z_i)\}_{i=1}^N$, where $(x_i, z_i) \in X \times Z$....
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### Convex set: extreme points and distance to the origin

I'm fairly sure the following is true, although I wouldn't mind being proven wrong. If true, I would like to see an elegant proof, as my attempts are kind of messy. Let $K\subset\mathbb R^2$ be a ...
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### convexity of composite function

There is a composite function F(x)=f(g(x)): R -> R where f(v): R^2 -> R and g(x): R -> R^2. It is given that f(v) is convex on v and v = g(x) gives a map R -> R^2. If x is in a convex set, is F(x) ...
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Let $\emptyset \neq A \subset \mathbb{R}^n$ be a convex set with $0 \notin A$. Then there exist a $v \in \mathbb{R}^n$ such that $v \cdot a \geq 0$ for all $a \in A$ and there exists $a_0 \in A$ with $... 0answers 18 views ### An inequality of a multivariate function:$f(x_1,…,x_n) \geq \frac{1}{n}\sum_i^n f(x_i,x_i,…,x_i) $Let us assume we have a non linear function$f : \Bbb R^{n+} \to \Bbb R ^+$, and let$x = \{x_1, x_2 , ..., x_n\}$,$x_i \in \Bbb{R}^+$, further define$\bar{x} = \frac{1}{n}\sum_i^n x_i$. My ... 0answers 27 views ### A necessary and sufficient condition for$f(x,y) = \phi(x²+y²)$be a convex function Let$f(x,y) = \phi(x²+y²) , \phi \in C^2$and$\phi$non-decreasing. Proof that$f$is convex in the disk$x²+y² \leq a² \iff 2u \phi''(u) + \phi'(u) \geq 0 \forall u \in [0,a]$Here is my ... 1answer 36 views ### Show non-convexity of a function with vector input How does one go about proving non-convexity of the function d? $$d(v) = 1/2*||F(v)- p||^2$$ $$F(v)=\sum_{i=1}^n l_i*\begin{pmatrix} cos(\sum_{j=1}^i v_j) \\ sin(\sum_{j=1}^i v_j \end{pmatrix}$$ ... 0answers 25 views ### Subdifferential optimality conditions I need help with subdifferential optimality. Let$f(x_1, x_2)=x_1^2 + x_2^2 + |x_1 -x_2 - y|. Find: \begin{align} \min_{x_1, x_2} f(x_1, x_2) \end{align} This is convex, so must have unique ... 0answers 25 views ### minimal representation of convex hull Here is a question about the convex hull. LetS$be a set and$\bar{S}$be the closed convex hull of$S$, i.e.,$\bar{S}$is the smallest convex set that contains$S$. Then is the following claim ... 1answer 109 views ### Solution for$\min_{x^Tx=1} x^TAx-c^Tx\min_{x^Tx=1} x^TAx-c^Tx$looks like a simple QPQC problem. If$A$is positive semi-definite, can I get the solution by first getting$x=A^{-1}c$and then projecting$x:=\dfrac{x}{||x||}$to make ... 0answers 23 views ### Sum of convex and concave functions when one is greater than the other Given two$C^\infty$functions$f,g:\mathbb{R}\to\mathbb{R}$such that$f(x)>g(x)\text{ }\forall x\in\mathbb{R}$. Moreover, we know that$f(x)$is convex while$g(x)$is concave. Now, let's define$...
Imagine that we have a function $f:\mathbb{R}\to\mathbb{R}$ which is convex, that is $f''>0$. We also know that $f'''<0$, that is its first derivative function is concave. Now, we can define its ...
Consider the function $f(x,y)=\frac{y}{1+e^x}$ where $0<y<1$ and $x \in \mathbb{R}$. If you plot this function, it looks like this: Also note that for a given value of $y$ the function $f$ ...