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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An inequality involving expectation and a concave function

I have a real-valued random variable $X$ and a function defined on the real line $x \mapsto U(x)$ which is concave and strictly increasing. What I am wondering is whether the following is true. For ...
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When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
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Concave up theorem for $f:A \to\mathbb R, A \subseteq \mathbb R$ - True or false?

I am a 1st year-2nd semester student of the department of Applied Mathematics @ NTUA. Some days ago, I had a really interesting conversation with one of my old school mathematics teacher about a ...
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Good text book recomendation

I would like to do some reading about a technique called sequential convex programming. There is a lot of material about sequential quadratic programming out there, including books (Nocedal & ...
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Alaoglu & Krein-Milman to show bounded convex weak-$*$ closed subset has extreme point

Let $X$ be a normed linear space and $K$ a bounded convex weak-$*$ closed subset of $X^{*}$. I need to show that $K$ possesses an extreme point; however, I am not entirely sure how to do this. I ...
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Are odd functions that are concave and increasing everywhere necessarily linear?

The title gives away pretty much the entire question except for the fact that the class of functions I am interested in is a subset of twice continuously differentiable functions. I think that if an ...
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What value of 'a' will be the function is convex, concave or not either?

$$f(x,y) = -6x^2 + (2a+4)xy - y^2 + 4ay$$ The solution has to be : $$-2-\text{gyök}(6) \leq a \leq -2 + \text{gyök}(6)$$ I tried to define the derivation of the function accordance with $x$ , and ...
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Caratheodory's theorem for a point in boundary

I am wondering whether the following holds: if $x$ in $\mathbb{R}^d$ lies in the boundary of the convex hull of a set $P$, then $x$ can be expressed as a convex combination of $d$ points in $P$. We ...
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How to get inequality representation of cone give by generators

To be as specific as possible, I have a subspace $I \subset R^{14}$ of dimension $3$ and an $11\times14$ matrix $A$ with linearly independent rows spanning the orthogonal complement of $I$. I have an ...
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Property of Conical Hull

Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$\operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}.$$ (it is a cone in ...
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Dual of a point which is in the convex cone of a set, contains the dual cone of that set

Let $\Lambda\subseteq R^n$ contains $m$ elements, where $\lambda_i$ is the $ith$ element, and $co(\Lambda)$ is the smallest convex cone contains $\Lambda$. Also, consider any point $u\in R^n$. Now, I ...
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Under which conditions is $f(x)=\frac{1}{2}x^TPx+q^Tx+r$ convex?

I am given the function $$f(x)=\frac{1}{2}x^TPx+q^Tx+r$$ and am asked to establish under which conditions $f(x)$ is a convex function. I have to use the definition of a convex function where we look ...
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How can show the following function is log-concave?

Suppose that $g(x)$ is an increasing function and $0\leq g(x)\leq1$. I was working on a problem and it reduced to show that if $1-g(x)$ is log-concave then $$f(x)=(1-g^a(x))^b, a\geq 1, b,x>0$$ is ...
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How to deal with an $xy\le 1$ constraint?

I have to solve the following optimization problem: \begin{align*} \min_{x,y} &\{-x-y\} \\ \text{such that} \\ y &\ge 3 \\ y &\le 30 \\ x &\ge 0 \\ xy &\le 1 \\ \end{align*} ...
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A bound (dominated function) for $\cosh^2\left(t\sqrt{1-\gamma^2}\right)$

I would like to bound $$\cosh^2\left(t\sqrt{1-\gamma^2}\right)$$ for all $t\in\mathbb{R}$, where $\gamma^2\leq1$. How can I do such thing? This inequality maybe useful cosh x inequality
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Showing that $f$ is convex given that $(\nabla f(x) - \nabla f(y)) \cdot (x-y) \geq c(x-y)\cdot(x-y), \;\;\forall x,y \in \mathbb{R}^n$

Assume that $f$ is continuously differentiable and that for some constant $c > 0,$ the gradient $\nabla f$ satisfies, (\nabla f(x) - \nabla f(y)) \cdot (x-y) \geq c(x-y)\cdot(x-y), ...
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How to prove that Bezier(t) polynomial lies in convex hull of points (i/n,ai) for i from 1 to n

I think i should prove firstly that: Bn,$x(t)$ for t between $0$ and $1$ lies inside the convex hull of the points $(k/n, xk)$. I know only that$k/n$ = max between $0$ and $1$ and i found that Bezier ...
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(M,N) J-convex functions

During my analysis course, our teacher told us about (M,N) J-convex functions and quasi-arithmetic means. Do you know any article I could find out more information? Thank you!
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How to show $\partial A = \varnothing \Rightarrow A=R^n$ [closed]

Let $A\subset R^n$ and dim$A=n$, $\partial A$ is the relative boundary of $A$. If $\partial A=\varnothing$ how to show $A$ is $R^n$ ? Picture below is from XX page of Schneider R.-Convex Bodies_ ...
Suppose that $X$ is compact and metric and let $g:X\to\mathbb R$ be a Borel map. Let $\mu$ be a Borel probability measure on $X$. Then it seems that $\int_Xgd\mu$ is a member of the closure of the ...
Here is a simple property of a concave function from $\mathbb{R}$ to $\mathbb{R}$, Given $x,x'\in \mathbb{R}$ with $x'>x$, if $\exists \kappa'\in (0,1)$ such that c(\kappa'x+(1-\kappa')x')=\...