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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What is “Exposed Face”?

So a face is either an extreme point/vertex, an edge or a facet depending on $dimF$. But what does it mean to an an exposed face?
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Prove $(1+x)^p+(1-x)^p \ge 2(1+x^p)$ for $0\le x\le1$ and real number $p\ge2$.

I don't know how to prove the following questions: If $p\ge2$ is real, then $$ (1+x)^p+(1-x)^p \ge 2(1+x^p) \quad \text{for } 0\le x\le1; $$ if $1\le p<2$, then opposite direction of the inequality ...
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2answers
34 views

Two decreasing, convex functions agreeing on a closed set

Fix a closed subset $B$ of $[0,\infty)$ and assume that $0\in B$. I am striving to construct two functions $f,g:[0,\infty)\to\mathbb R$ such that $f(0)=g(0)=1$; $f(x)\geq 0$ and $g(x)\geq0$ for each ...
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1answer
12 views

Convex subsets of a topological vector space

I'm trying to prove: Let $X$ be a topological vector space and $A \subseteq X$. $A$ is convex if and only if $$\forall s,t \in \mathbb{R}_{+}, (s+t)A = sA + tA $$ Let $sx + ty \in sA + tA$. How ...
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28 views

Compute the edges of P

Let $P=\{v \in \mathbb R^2 | Av \leq b\}$, where $$ A= \begin{pmatrix} -1 & -1 \\ 2 & -1 \\ -1 & 2 \\ 1 & 2 \end{pmatrix}, b= \begin{pmatrix} 0 \\ 1 \\ 1 ...
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1answer
22 views

Dual of the Minkowski Sum

Suppose $X$ and $Y$ are convex sets in $\mathbb{R}^d$ such that the origin is in each of their interiors. Then the dual of $X$, $X'$ is defined as the set of linear functionals $\alpha$ such that ...
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9 views

Convert this problem with 2-norm cost to an SOCP?

I am solving a non-convex problem via sequential convex optimization. Here is a minimal example of my problem at iteration $i$: $$ \begin{align*} \min_{\Delta t,F_i}&\left( \Delta ...
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33 views

Positivity of the last component of non negative least squares based on active set method

I have followed the instructions given in Lawson and Hanson book for non-negative least squares using active set method. I am having a trouble in justifying one of the statements they have made about ...
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1answer
22 views

Proving that the tangent to a convex function is always below the function

Consider a real-valued convex function f defined on an open interval $(a,b) \subset \mathbb{R}$. $x,y \in (a,b)$. I want to prove that \begin{equation} f((1-\lambda)x + \lambda y) \leq ...
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+100

Why is the Barycenter operation in Hadamard spaces Lipschitz continuous?

I am looking into exercise 9.2.22 of "A course in metric geometry" by Burago-Burago-Ivanov. For a Hadamard space $H$ (a complete simply connected metric space of nonpositive curvature in the ...
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Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
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1answer
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Proving convexity from 2-dimensional convexity

I have a function $f(x_1,x_2,\ldots,x_m):\mathbb{R}^m\rightarrow \mathbb{R}$ ($m\geq 2$) that is jointly convex in $x_i$ and $x_j$ for all $i$ and $j$. Can I prove that this function is convex in ...
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27 views

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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1answer
25 views

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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23 views

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, \begin{equation} (\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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28 views

How to say $\text {log}\ \ a^{-1} \geq 1-a$ from the concavity of $\text{log}(\cdot)$

I am reading a paper and confront the following small trick: $\text {log}\ \ a^{-1} \geq 1-a$, where $0\leq a \leq1$. By the concavity of $\text{log}(\cdot)$. From the formula: ...
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1answer
20 views

Convexify $x\le a+by^2$

I have the following non-convex constraint: $$ x\le a+by^2\quad\text{where}\quad a,b>0,\,y\in[0,y_{max}]\text{ and }a\approx by_{max}^2 $$ On a drawing, it looks something like this: The above ...
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1answer
16 views

How to prove that the right derivative of a convex function is right continuous?

let $f$ be a convex function and $D^+f$ be the right-derivative of $f$, I want to show that $D^+f$ is right continuous. first I want to use the $\epsilon-\delta$: by the definition of $D^+f$, ...
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4 views

(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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1answer
57 views

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_ ...
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26 views

Convex optimization qualifying exam [closed]

I'm studying for my qualifying exam which I'm going to take in late July and I have some problems from previous exams that I could't solve . So I would appreciate your help because I'm so stressed ...
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1answer
23 views

Integral as a member of the closure of the convex hull of the integrand

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 ...
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reference for convex function results

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 ...
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1answer
26 views

Let B be set of all twice differentialbe function $ f(0)=1, f'(0)=-1$ . .. Find supremum of $ {(f''(0):f\in B})$

Let B be set of all twice differentiable function $f$ such that $f: (-1,1) \to (0,\infty)$ and $ f(0)=1, f'(0)=-1$ . We have new function $g(x)$ such that $g(x)=\frac{1}{f(x)}$ and $g(x)$ is convex ...
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10 views

Power-Series strictly convexity

Watch the power-series $B(\beta):=\sum_{i=0}^{\infty}b_{j}e^{\beta\cdot j}$ with $b_{j}\geq 0$ for $0<\beta<r$ where $r$ is the radius of convergence. At least one $b_{j}$ for $j\geq 2$ is non ...
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1answer
14 views

Convexity of a function over a vectorial space

Consider $\mathcal{V}$ the set of vectors $X$ whose values $x_i$ are all positive. Then, consider the function f : $\mathcal{V} \rightarrow \mathbb{R} ; > ...
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Please help… is this a convex function?

Kindly help me. What can we say about the function $f$ shown in below? is it convex or non-convex over the variables $x_1, x_2,.., x_{n+1}, y_1,y_2$? \begin{align} f(x_1, x_2,.., x_{n+1}, ...
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41 views

Support function of a set: when we can replace the supremum with the maximum?

Consider a set $K\subseteq \mathbb{R}^d$ (not necessarily closed and/or convex). Let $k:=(k_1,...,k_i,...,k_d)$ be a generic element of $K$. Consider the function $h_K:\mathbb{R}^d\rightarrow ...
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Show convexity of a function via inequalities

I am stuck with deriving the convexity of the function $$ f(x) = \sqrt{1 + x^2} $$ from first principles, that is I would like to show that for any $x,y \in \mathbb R$ and $\lambda \in (0,1)$ we ...
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Building a convex set out of two convex sets where each extremal point of one set shares and edge with each extremal point of the other [duplicate]

Consider a convex set $P$ with two faces $f_1, f_2$ s.t. all extreme points of the convex set belong to either $f_1$ or $f_2$ (but none blong to both - the two faces are disjoint in the set of ...
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How do I prove this simple result for the face structure of convex sets?

I have a convex set $P$ with faces $f_1, f_2$ such that all extremal points of the convex set belong to either $f_1$ or $f_2$ (the faces are disjoint and cover $P$). How can I prove that if every ...
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1answer
19 views

Convexity of multi variate functions

Let $f:\mathbb{R}^m\rightarrow \mathbb{R}$ be a smooth function. I know $f(x)$ is convex if its Hessian ($\frac{\partial^2 f(x)}{\partial x\partial x^T}$) is positive semi-definite. Now, let ...
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1answer
24 views

How can I prove the concavity of $f(p_1,p_2,…p_n) = \sum_{i = 1}^{n}p_i(1-p_i)$?

Assume $p_n$ is the probability of being in class $n$ which mean that $f(0) = 0$ , $f(1) =0$ , and $p_1+p_2 = 1$ I need to come up with a concave function that show the relation between $p_1$ and ...
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71 views

Can you prove that this function is convex? $\sqrt{2x_1^2+3x_2^2+x_3^2+4x_1x_2+7} + (x_1^2+x_2^2+x_3^2+1)^2$.

My analysis: The second term can be proven to be convex as follows. It is basically a composition of norm with an affine transformation to the power of four: $(x_1^2+x_2^2+x_3^2+1)^2 = \|(x_1^2, ...
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5 views

Is $f(x)=-3x_1+x_2-x_3^2$ pseudoconvex at $\bar x$?

Is $f(x)=-3x_1+x_2-x_3^2$ pseudoconvex at $\bar x=[-115/588, -95/588, 5/14]^T$? Pseudoconvexity: If $\nabla f(\bar x)^T(x-\bar x)\ge0$, then $f(x)\ge f(\bar x)$ for any $x\in \mathbb{R^3}$ (in this ...
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Creating convex/monotone polygons from concave.

I am looking for an algorithm that creates convex or monotone polygons from a concave one. So far I found few: Seidel - it does trapezoidation but it is way too complex for me to implement. ...
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1answer
37 views

$xy \leq \frac{x^p}{p}+\frac{y^q}{q}$

I struggled to show that this is true when $x,y >0$, $p>1$ and $q=\frac{p}{p-1}$. But, I managed to show that if we assume that $x \geq 1$ the assertion is possibly false only for a finite ...
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15 views

Convexity and composition of functions

A function $g(f(x))$ is convex if $g$ and $f$ are convex and $g$ is non-decreasing, what happens if $g(f_1(x),f_2(x),...,f_m(x))$ where $x = (x_1,...,x_n)$. Is $g$ convex if each $f_i$ is convex in ...
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1answer
25 views

Is every convex cone a manifold?

Let $C \subseteq \mathbb{R}^n\setminus \{0 \}$ be a connected convex cone*. Question: Is $C$ always a topological manifold (perhaps with boundary)? A smooth one? Does anything change if we do not ...
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34 views

Log-convexity of completely montone sequences

Let $s_0, s_1, \ldots$ be a completely monotone sequence. This means that, defining \begin{align*} (\nabla s)_n &= s_{n}-s_{n+1}\quad\text{and}\\ (\nabla^{r+1}s)_n &= (\nabla^{r}s)_n - ...
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Proof of $A\ne \mathrm{conv}~ \mathrm{relbd} A\Rightarrow A$ is a flat or a half-space

Picture below is from the 16 page of Schneider R. "Convex Bodies--The Brunn-Minkowski Theory". I am fuzzy with the red line, what is the translate of $H^-$ ? And what is the effects of convexity ...
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37 views

$\big\langle\nabla f(x)-\nabla f(y),\,(x-y)\big\rangle\ge m\,\left\| x-y\right\|^2,\;m>0\;$ for strictly and strong convex function

Prove that $\,\big\langle\nabla f(x) - \nabla f(y),\, (x-y)\big\rangle \geq m\,\left\lVert x-y\right\rVert^2, \;\,m > 0\,$ for strong convex function $f: \mathbb R^n \to\mathbb R$. Is it true ...
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1answer
18 views

Lipschitz implies bounded gradient

Assume $f:\mathbb{R}^n \to \mathbb{R}$ is convex, and $L$-Lipschitz, so $|f(x)-f(y)|\leq L\|x-y\|$. I would like to show that $\|\nabla f(x)\|\leq L$. In one dimension this is a straightforward ...
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24 views

Is there a way to determine if the Convex Hull of two polyhedra is going to be huge?

So in this post: Faster Algorithms for Convex Hulls I was interested in determining if a convex hull of two $n$ dimensional polyhedra can be computed quickly, and the answer was in general: no, ...
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1answer
57 views

A consequence of the convexity of $ f(x) = x \log x $

I verified that $f:\mathbb{R_{+}^{*}} \rightarrow \mathbb{R}, f(x) = x \log x $ is convex, since it is twice differentiable and $f''(x) = \frac{1}{x}$ is positive for the domain. But my teacher asked ...
3
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1answer
37 views

Is $\Gamma(\alpha, 0, x)$ log-concave as a function of $x$ (for fixed $\alpha$)?

The incomplete Gamma function is defined as: $$\Gamma(\alpha, 0, x) = \int_0^x t^{\alpha - 1} e^{-t} dt$$ Let $\alpha \ge 1$ be a fixed number. Is $\Gamma(\alpha, 0, x)$ log-concave, as a function ...
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78 views

Some intuition on the support function of a convex set

I have some doubts on the interpretation and properties of the support function of a convex subset of $\mathbb{R}^d$. (1)Let $K$ be a convex set in $\mathbb{R}^d$. (2) The support function of $K$ is ...
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2answers
29 views

Is a convex cone a convex polyhedron?

Say that I have a convex cone $C=\{t|Ax = t, x\geq 0\}$. where $x\in R^n$, and $t\in R^m$, $A\in R^m\times R^n$. Can I say that this is a convex polyhedron? and why? EDIT: Just in order to avoid ...
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1answer
41 views

Lagrange's theorem and convex functions

Let f:U⊂ $\mathbb{R}^n$--->$\mathbb{R}$ a $C^1$ function with U being convex and an open set. Let g: U ⊂ $\mathbb{R}^n$ ---> $\mathbb{R}^m$ (with m smaller than n) an affine application. Let M={x $ ...