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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Concavity of the $n$th root of the volume of $r$-neighborhoods of a set

Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Is the function $f(r) = \mu(A_r)^{1/n}$ ...
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3answers
29 views

Convexity of a non linear optimization problem

I have a non linear optimization problem, namely: $$\min {\sqrt{(x-u)^2 + (y-v)^2 + (z-w)^2)}}$$ How can i show that the above function is convex. Doing via Hessian is a difficult task.
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Is the function $\frac{f(x)}x$ increasing, if $f(x)$ is convex? [on hold]

Suppose $x$ is real and positive. $f''(x) > 0$ (f is convex) is the function $h(x) = \frac{f(x)}x$ increasing? If so, is it strictly increasing? If yes, why? Thank you.
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What is the solution for this optimization problem?

I have an optimization problem in the form: $$\max (a-\bar{a})(b-\bar{b}) \qquad \text{subject to} \qquad a+b=1.$$ Here $\bar{a}$ and $\bar{b}$ are known values and both of them are positive. Let ...
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1answer
16 views

Collecting terms in an example of checking concavity/convexity

$z=x_1^2+x_2^2$ $u=(u_1,u_2)$ $v=(v_1,v_2)$ Height of arc: $f[\theta u+(1-\theta )v]= f[\theta u_1+(1-\theta)v_1,\theta u_2 + (1-\theta)v_2 ]$ $= [\theta u_1+(1-\theta)v_1]^2 + [\theta u_2 + ...
3
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1answer
107 views

A connectivity-preserving function from a connected set onto an interval

Let $C$ be a connected set in the plane and $I$ the unit interval interval. Call a function $f$ from $C$ onto $I$ Connectivity-preserving if the following is true for every subset $I'\subseteq I$: ...
5
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2answers
197 views

Set is Convex regardless of b

Let the function $f$ be convex, $f :\Bbb R^n \rightarrow \Bbb R$ and let $$S = \{x : f(x) \le b\}$$ The proposition states that the set $S$ is convex regardless of $b$. Can someone explain to me how ...
2
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1answer
24 views

Continous family of $n$-gons

The statement of this problem asks to show that if $A$ and $B$ are two distinct convex $n$-gons there is a continous family of convex $n$-gons such that the first in that family is $A$ and the last is ...
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1answer
24 views

Which functions are the composition of convex functions?

The composition of convex functions is not necessarily convex or concave: For example, composing $f_1(x) = x^2-1$ and $f_2(y) = y^2$ gives $f_2(f_1(x)) = (x^2-1)^2$. Or consider $f_1(x) = x^2$ and ...
4
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28 views

What class of functions is characterized by the property $f[\operatorname{conv} A] \subseteq \operatorname{conv} f[A]$

It is well-known that the inclusion $f[\overline A] \subseteq \overline{f[A]}$ (for every subset $A$) characterizes continuous functions.1 Asking similar questions for other closure operators seems ...
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2answers
25 views

SVD with Sparisty Penalties

I am trying to solve a problem like $$\min_{u,v} \Vert X-uv^T \Vert_F^2 + \beta \Vert v \Vert_1 \mbox{ s.t. } \Vert u \Vert_2 \le 1$$ where $X \in \mathbb{R}^{N\times N}$, $u, v \in ...
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0answers
48 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
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0answers
19 views

Convex optimization of a fractional objective function involving matrix determinants

I am interested in convex representation of the following fractional optimization problem. I have also described my approach in the following. However, as I am new to convex optimization, I am not ...
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4answers
54 views

for $I = [0,1]$, is $I\times I$ convex in $\mathbb{R} \times \mathbb{R}$?

for $I = [0,1]$, is $I \times I$ convex in $\mathbb{R} \times \mathbb{R}$? The definition of convex seems to be that $Y \subset X$ is convex in $X$ if $\forall a < b $ in $Y$ whole of $(a,b)$ ...
1
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1answer
35 views

Prove Jensen Inequality holds for a function

Given function $$f:\mathbb{R}^n_{+} \rightarrow \mathbb{R}, \ f(x) = \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{a_{2k}} \cdots x_n^{a_{nk}}$$ Show that for any $x, y \in \text{dom} \ f, \theta \in [0,1]$, ...
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0answers
19 views

Negative eigenvalue of a hessian matrix entails a local decrease in function value?

I was reading up on non-convex optimization, and I can across this sentence: "Since Hessian(f(w)) has a negative eigenvalue, there is always a point that is near w which has smaller function value" ...
0
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1answer
19 views

Convexity of an exponential function.

A random variable $Y_i$ is given such that, $\mid$Y$_i\mid$$\leq$ $c_i$ where i ranges from 1,.....,t and t is some constant. Now, $Y_i$ is expressed as : $Y_i = ((Y_i - c_i) + (Y_i + c_i))/2$ $= ...
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1answer
23 views

Is there a name for dividing a set into pieces, some of which may be empty?

Suppose that $X$ is a set and $V_{0}$, $J$, and $V_{1}$ are pairwise disjoint subsets of $X$ whose union is $X$. If the three subsets were nonempty it would be a partition of $X$. However, I wish to ...
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1answer
28 views

Inequality with three unknowns

Consider: \begin{equation} \Big(e^x-1\Big)\mathbb{1}_{(x\geq0)} \leq \lambda_1+\lambda_2e^x + \lambda_3x^2 \end{equation} where $\lambda_1$, $\lambda_2$, $\lambda_3$ are three unknown constants. By ...
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1answer
18 views

Prove that there exists some $y \in \mathbb{R}^n$ in the boundary of $C$ satisfying $\{z \in \mathbb{R}^n : (x-y)^T(z-y) = 0\}$.

Let $C \subset R^n$ with $C \neq \emptyset$ be a closed convex set. Consider some $x \in \mathbb{R}^n$ satisfying $x \notin C$. Prove that there exists some $y \in \mathbb{R}^n$ in the boundary of $C$ ...
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1answer
43 views

Are posynomial functions convex?

I know that you can transform a posynomial function into an exponential function, which is convex. Does this imply that all posynomial functions are convex?
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1answer
29 views

Partitioning a convex object without cutting existing convex subsets

$C$ is a convex object in the plane. $D_1,\dots,D_n$ are pairwise-disjoint convex subsets of $C$ such that $D_1\cup\dots\cup D_n \subsetneq C$, like this: Is it always possible to partition $C$ to ...
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The join of two convex sets is convex?

Let A and B be convex subsets of $\Bbb R^n$. The join of A and B is the set of all $\vec x$ such that $\vec x$ lies on a line segment with one endpoint in A and the other in B. I am wondering how to ...
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0answers
20 views

Verifying a startegy to prove convexity on partial domain

Assume you have the multivariate function $$f(x_1,x_2,..,x_n)$$ where: $x_i>0 \forall i$, and $\sum_i x_i = 1$. I need to show that $f$ is a convex function. My plan is to show that it is ...
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1answer
56 views

Number of ways to separate $n$ points in the plane

Say you are given $n$ points such that no three are colinear. Show the number of ways to separate them into two subsets by drawing a straight line depends on $n$ but not the position of the points.
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When a set is convex, how does the polar set of its polar set equal the original?

I have read the following proposition, and haven't been able to connect the convexity of $X$ to the statement's main equality. Any guidance would be much appreciated. Define $X^\text{o}$, the polar ...
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Is the Covariance of Two Random Variables Convex or Concave or Neither?

Are there any standard results established regarding the behavior of the Covariance of two random variables? For example, whether it is a convex or concave functions and so on and under what ...
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How to prove that unnormalized neg entropy is strongly convex with respect to 1-norm?

the unnormalized negative entropy of $\mathbf{x} \in \mathbb{R}^n_+$ is $$ g(\mathbf{x}) = \sum_i (x_i \log(x_i) - x_i) $$ it is stated that $g(\mathbf{x})$ is strongly convex with respect to 1-norm, ...
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1answer
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$f$ convex strictly decreasing function , is $f'(x+\delta)-f'(x)$ convex

Assume you have a strictly decreasing convex differentiable function $f(x)$, $x \in \Bbb R^+$, I am wondering if the increment of the first derivative is also convex; i.e., $$g(x) = f'(x+\delta) - ...
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1answer
23 views

True or false? “sum of an m-strongly convex and a convex function is m-strongly convex”

I would like to know if the following conjecture is true or false? If $f(x) = g(x) + h(x)$ where $g$ is m-strongly convex and $h$ is convex, then $f$ is m-strongly convex. NOTE: For a ...
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1answer
22 views

Determining whether or not these spaces are convex

Consider $\{(x,y) \in \mathbb{R}^2: |x| +y^2\leq 5\}$ and $\{ (x,y) \in \mathbb{R}^2: y\geq x^2,y\leq e^{-x^2}\}$. Determine whether or not these two are convex sets. I have used the visual ...
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1answer
25 views

Prove that $A - B$ is closed if $A$ or $B$ is bounded (or both)

Suppose that $A$ and $B$ are two nonempty convex closed sets in $\mathbb{R}^n$, with $A \cap B = \emptyset$. Further, define $A - B = \{a - b \space | \space a\in A, b \in B\}$. Prove that $A - B$ is ...
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When is $f(x,y,z)= \frac{x \cdot y}{z}$ convex?

I would like to know under what conditions $f(x,y,z)= \frac{x \cdot y}{z}$ is convex, pseudo-convex, or quasi-convex. I know that $g(x,y)= \frac{x^2}{y}$ $ \text{when } y >0 $ is convex and ...
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1answer
11 views

Infimal convolution $g^\star = f_1^\star + f_2^\star$

Let $f_1$ and $f_2$ be convex functions on $R^n$. Their infimal convolution $g = f_1 \diamond f_2$ is defined as $$ g(x) = \inf \{f_1(x_1) + f_2(x_2) \mid x_1 + x_2 = x\}. $$ Prove that $g^\star = ...
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0answers
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Proof that $\nabla f(x) = \nabla f(y)$ if and only if $x = y$

Trying to tackling a convexity problem. Let $f(x)$ be a real-valued differentiable function on $R^n$. If $f(x)$ is strictly convex, prove that $\nabla f(x) = \nabla f(y)$ if and only if $x = y$.
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Minimize $\|\mathbf{x-y}\|^2 $ subject to $x \in $ set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$

We are given the set $S=\{\mathbf{x} \in \mathbb{R}^n \;\;\;\mid \;\;\; \|\mathbf{x-x_c}\|^2\leq r^2 \}$ and a point $\mathbf{y} \in \mathbb{R}^n$. Our goal is to find point $\mathbf{\hat{x}}$ ...
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1answer
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How to prove that the set of real $ n \times n $ PSD matrices of rank $\leq r < n $ and unit trace, is not contractible?

Extended Version of the Question: How to prove that the set of real $n \times n$, symmetric positive semidefinite (PSD) matrices of rank $\leq r$ ($ 1 \lt r \lt n-1 $) and unit trace, is not ...
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sub gradient of Moreau envelope

Is there an equality formula for the subgradients of the Moreau envelope, $$e_f(x) = \inf_z \frac{1}{2}||x-z||_2^2 + f(z),$$ of a lsc (lower semicontinuous) and proper function $f:\mathbb{R}^p \to ...
2
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1answer
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About a definition of “rank” of a matrix.

I am familiar with the definition of rank of a matrix as either (1) the maximal number of linearly independent rows or columns or (2) as the dimension of the image of the matrix. Another ...
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Does every collection of edge vectors of a cone span a face of the cone?

Definition 1 : A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A ...
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Convex Analysis - How To Find Non Convex Set

I have a problem regarding the following exercise (I considered to put this question on mathematica.stackexchange, but I changed my mind and though this was the right place for this particular ...
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1answer
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A strictly convex polynomial is coercive if and only if it has a positive definite Hessian

I have some difficulties in the following problem. Thank you for all comments and helping. Let $f:\mathbb{R}^n\rightarrow \mathbb{R} (n\in \mathbb{N})$ be a polynomial. Suppose that $f$ is strictly ...
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Convex closure of the support of a Levy Process

Given a Levy Process $X_{t}$ on a filtered Probability Space $(\Omega,\mathcal{F},\mathcal{F}_{t},P)$ with distribution-function $F_{t}$ for $X_{t}$. We look now for the cumulant transform $\phi_{1}$ ...
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Show that the “folium” is norm closed

A C*-algebra $\mathfrak{A}$ is a Banach algebra with an involution operation $* : \left\lbrace\begin{aligned} \mathfrak{A} &\longrightarrow \mathfrak{A} \\ a &\longmapsto a^* \end{aligned} ...
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1answer
22 views

How would you show that this fraction function is convex and decreasing?

Show that $$ f(\vec{x}) = \frac{1}{x_1 - \frac{1}{x_2 - \frac{1}{x_3 - \frac{1}{x_4}}}} $$ is convex when all denominators are greater than $0$.
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Prove that $f(x,y) = x/(y^2+1)$ is convex

Suppose $f(x,y) = x/(y^2+1)$. I was trying to prove that this function is convex. So I took partial double-derivative and constructed the Hessian for this function. Here the Hessian is a 2 by 2 matrix ...
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21 views

Can we relax the assumption of nonnegativity in this proof on convexity of a feasible region in a linear programming problem?

Is the $\color{red}{\text{non-negativity constraint (see red box)}}$ used at all in the proof? If so, where? If not, does the proof then hold for a standard LP problem without the non-negativity ...
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On the proof of corner points maximising or minimising a linear function over a bounded convex region

This proof says if $Z_P \ne Z_Q$, then $Z$ is maximised (or minimised, I guess) at one of the endpoints -- of what exactly? $\overline{PQ}$? So the maximum value of $Z$ occurs at either $P$ or $Q$? ...
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normal cone inclusion and non-symmetric matrix and optimization problem

I have the following normal cone inclusion $$-(A x + b) \in \mathcal{N}_\mathcal{C}(x) \qquad (1)$$ where $\mathcal{N}_\mathcal{C}$ denotes the normal cone to the convex set $\mathcal{C}$ at the ...