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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Convexity of $t$-sublevel set of a convex function

I am confused about the following: Suppose $f_0(x)$ and $f_1(x)$ are both convex. and I want to solve the following simple problem: (for all $t$) \begin{equation} \begin{aligned} & ...
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1answer
628 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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Is there an efficient way to evaluate the proximal operator of $f(x) = \|x\|_2 + I_{\geq 0}(x)$?

Is there an efficient way to evaluate the proximal operator of the function $f:\mathbb R^n \to \mathbb R \cup \{ \infty \}$ defined by \begin{equation} f(x) = \| x \|_2 + I_{\geq 0}(x), \end{equation} ...
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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
18 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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472 views

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
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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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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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1answer
400 views

Simplex - Help with a proof

I am trying to prove the following (basic) claim about simplex. If you check my proof and help me with the part where I stuck, I would appreciate it very much. Let $X$ be a finite set and define ...
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1answer
41 views
+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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1answer
39 views
+50

interior of convex hull relatively open

Consider $k+1$ affinely independent vectors $\left\{p_0,p_1, \dots, p_k \right \}$ in $n$-dimensional euclidean vector space $n>k$ and consider their convex hull. It is known that each point $x$ of ...
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423 views

Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...
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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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2answers
74 views

When does it help to write a function as $f(x) = \sup_\alpha \phi_{\alpha}(x)$ (an upper envelope)?

In this question I was shown a very elegant solution based on writing a function as the upper envelope of a family of linear functions: $$f(x) = \sup_{y\in C} f(y) + \langle \nabla f(y), x-y ...
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5answers
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Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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1answer
13 views

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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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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3answers
792 views

cosh x inequality

While reading an article on Hoeffding's Inequality, I came across a curious inequality. Namely $$\cosh x \leq e^{x^2/2} \quad \forall x \in \mathbb{R}$$ I tried many ways to prove it and finally, ...
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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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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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17 views

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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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
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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4answers
53 views

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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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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27 views

interchange of convex hull operation and intersection

Let $A^{\epsilon}$ be a set. Let $\overline{co}(A)$ be the closed convex hull of $A$, i.e., the smallest convex set that contains $A$. My question is under what condition, the following is true ...
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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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(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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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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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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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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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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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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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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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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15 views

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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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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convex hull of a lower hemicontinuous correspondence is lower hemicontinuous

Let $(X, \mathcal{T})$ and $(Y, \mathcal{T}')$ be two topological spaces. We call a correspondence (set-valued function) $F: X \rightarrow 2^Y$ lower semicontinuous if for every $G \in \mathcal{T}'$ ...
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1answer
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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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2answers
70 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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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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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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0answers
14 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 ...
4
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2answers
71 views

Is a convex salient cone necessarily contained in an open half-space?

A cone $C$ in $\Bbb R^n$ is said to be salient if it does not contain any pair of opposite nonzero vectors; that is, if and only if $C \cap (-C) \subset \{0\}$. Obviously, a cone $C$ such that that ...
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33 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 - ...