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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Sufficient condition for a function to be affine

If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form ...
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2answers
135 views

Convexity of $\log \det(I+(P-\text{tr}(X))\cdot X)$

where $P-\text{tr}(X)>0$ and $X$ is a diagonal matrix with diagonal elements are $(x_1,x_2,\dots x_m)$
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2answers
92 views

An eigen problem

$K$ is a symmetric positive semidefefinit matrix. $K1 = 0$ (i.e. The sum of elements in each row is $0$. Or in other words matrix $K$ is centered. From this we conclude the smallest eigenvalue of $K$ ...
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0answers
51 views

Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
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1answer
614 views

Why is this composition of concave and convex functions concave?

Please forgive my ignorance. I have a quick silly question about a statement given without proof in Convex Optimization by Boyd and Vandenberghe (page 87). Suppose $\mathbb{R}_+^n$ is the set of ...
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1answer
58 views

Is the set of all concave functions a convex set?

How can I prove this? I saw a similar question here: (But this was only for when g(x) is ≥0) Prove that a set defined by concave functions on $R^n$ is convex
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2answers
100 views

straightforward way to determine if this set is convex?

straightforward way to determine if this set is convex? $Z=\left\{x\in\mathbb{R}^2:3x_1^4-x_1x_2+x_2^4\le x_2,x_1>2,x_2>2\right\}$ I know I can try by manipulation of linear combination of two ...
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How $x^4$ is strictly convex function?

I hear that $f(x)=x^4$ is a strictly convex function $\forall x \in \Re$. However, strict convexity condition is that the second derivative should be positive $\forall x \in \Re$. For the mentioned ...
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1answer
96 views

Are these sets not convex?

Definition of convex set says: an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. From: ...
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0answers
84 views

Cyclically monotone sets on four points

A subset of $\mathcal{X} \times \mathcal{Y} \subset \mathbb{R}^d \times \mathbb{R}^d$ is cyclically monotone if $\sum_{i=1}^n \langle x_i,y_i\rangle \ge \sum_{i=1}^n \langle x_i,y_{i+1}\rangle$, where ...
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1answer
299 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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Need advice: what should be my next step?($2$) (does Cauchy-Schwarz help here?)

This question is based on the question that I asked here Need advice: what should be my next step? I did a little more progress and wanted to share with you. As this is a new question, without any ...
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1answer
80 views

$L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$.

How can I prove that $L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$. Here $\mathbb{T} = \mathbb{R}/\mathbb{Z}$
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1answer
171 views

Convex hull is the minimal convex set containing $X$

How one can prove that convex hull is the minimal convex set containing $X$? We need to show that for each convex set $M$ if $X\subseteq M$ then $conv(X)\subseteq M$. I am thinking of proof by ...
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2answers
142 views

Property of convex functions

I am trying to show that if a function $f$ defined in $\mathbb R^n$ is differentiable and convex then $f(y)-f(x)\ge \nabla f(x)(y-x).$ for each $x,y\in\mathbb R^n$ Using differentiability of $f$ I ...
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2answers
82 views

About the convexity of Ky Fan's norm

As we know, the Ky Fan norm is convex, and so is the Ky Fan k-norm. My question is, does this imply that the difference between them is a non-convex function, since it results from "difference between ...
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2answers
142 views

$ \text{int}(A) = \text{int(cl}(A))$, where $A$ is convex

How can one prove that $ \text{int}(A) = \text{int(cl}(A))$, where $ A \subseteq \mathbb{R}^n$ is convex?
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1answer
93 views

Is this function convex

Is function $$f(x, y) = \left(\frac{x}{y} - a\right)^2 \left(\frac{y}{x} - \frac{1}{a}\right)^2$$ convex on the domain $$\{(x,y): x, y \in \mathbb{R}, x >0, y >0 \}\quad?$$ Now I think that it ...
9
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2answers
313 views

On the convexity of element-wise norm 1 of the inverse

Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$ \|A\|_1= \sum_{i,j} |A_{i,j}|. $$ Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
4
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1answer
86 views

$ \text{cl}(\text{int}(A)) = \text{cl}(A)$ when A is convex

How can one prove that $ \text{cl}(\text{int}(A)) = \text{cl}(A)$, where $ A \subseteq \mathbb{R}$ is convex? I know that if $A$ is convex, $\text{int(A)}$ and $\text{cl(A)}$ are convex too. I need ...
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1answer
150 views

Is it possible for a Strongly Convex function to be unbounded below?

Let $X$ be a non-reflexive Banach space and $f:X\rightarrow\mathbb{R}$ a $C^1$ function that is Strongly Convex, i.e. $$f(u)-f(v)\geq\langle f'(v),u-v\rangle+c\|u-v\|^2$$ where $c>0$ is constant. ...
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0answers
29 views

epigraph lsc they are equivalent

Let $f$ be taking values from a topological vector space to extended real line. Then f is lower semicontinous IFF epigraph f is closed. How to show this? Thanks a lot!
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A query about convexity in $L^p$ spaces

It defines the set $H^{p}_{\varepsilon}=\lbrace f \in L^{p}(0,1):\Vert f\Vert _{p}=(\int \vert f\vert^{p}dm)^{\frac{1}{p}}<\varepsilon\rbrace$ with respect to the measuring space ...
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2answers
80 views

computational strategy for solving convex-concave minmax problem

Assume f(x,y) is convex in $x$ and concave in $y$. Then \begin{equation}\min_x \max_y f(x,y)\end{equation} is globally solvable, because f is convex in x (max of convex is convex.) But can we find a ...
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0answers
240 views

Find a tight upper bound of the following expectation.

I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
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0answers
276 views

What is the relation between convex metric spaces and convex sets?

Here's another question that came to mind when I was reading the article on convex metric spaces in Wikipedia: According to the article, "a circle, with the distance between two points measured along ...
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1answer
64 views

Convex relaxation for the complement of Lorentz cone

Is it possible to obtain a convex relaxation for $$ \{ (x,t): t \le \|x\|_2\} \in \mathbb{R}^{d+1} $$ where $x \in \mathbb{R}^d$ and $\|x\|_2$ is the usual Euclidean norm, by moving to higher ...
4
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1answer
139 views

Convexity of a Given Function

Is the following function convex or concave? $$f(\mathbf{x}) = \frac{1}{N}\sum_{i=1}^N\sqrt{1+\frac{x_i^2}{\left(\frac{1}{N}\sum_{i=1}^{N}x_i\right)^2}},$$ $\mathbf{x} = [x_1,x_2,...,x_N]^T, x_i \ge ...
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There is a ray from each point of unbounded convex set that is inside the set.

Let $A$ be a non-empty convex, unbounded set in $\mathbb R^n$. Prove that for each point $a \in A$, there is a non-zero vector $h \in \mathbb R^n$ such that $l = \{x \in \mathbb R^n \mid x=a+th,\ ...
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1answer
47 views

Is this a linear programming problem

If $x \in R^n$, then $\min \|x\|_{\infty}$ sub to $Ax = b$, $x \geq 0$ where $\|x\|_{\infty}$ is the infinity norm which is $\max\{\|x_1\|,\|x_2\|,\ldots,\|x_n\|\}$. If not then how can ...
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2answers
127 views

Convexity and minimum of a vector function

Prove that the function $f:\mathbb{R}^n\to \mathbb{R}$ given by $f(x)=x^T \cdot x$ is strictly convex. Use this result to find the absolute minimum by equating the derivative to zero. I am not sure ...
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$(d-1)$-rectifiability of a boundary of compact convex set

Let us have a compact convex set $A\in \mathbb{R}^d$. Then $\delta A$ should be a $(d-1)$-dimensional rectifiable set. I don't seem to be able to show that it can be covered by a countable union of ...
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1answer
35 views

Is $\{x\in\mathbb{R}^4: x\ge 0, \, x_1x_2+x_3x_4\ge\alpha\}$ convex?

Is $\{x\in\mathbb{R}^4: x\ge 0\, \mbox{ and }\, x_1x_2+x_3x_4\ge\alpha\}$, for $\alpha>0$, a convex set? A related question is this one: Is $f(x_1,x_2)=x_1x_2$, with $x_1,x_2 \in \mathbb{R}_+$, a ...
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1answer
52 views

non convex optimisation

\begin{eqnarray} {\textbf{maximise}} \hspace{2mm} Ar^{-(a+b)} + Br^{-(a+b+c)}-C \nonumber \end{eqnarray} such that, \begin{eqnarray} c= l(h-m_{0}) \nonumber\\ m_{1} \leq h \leq m_{2} \nonumber\\ ...
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1answer
265 views

Convex Functions: Property Proof

Let $f\colon S\to \mathbb R$ be a $C^1$ function on a convex domain $S \subseteq \mathbb R^n$. Show that if $f$ is convex then $(\nabla f(x) - \nabla f(y)) \cdot (x-y) \ge 0$ for all $x,y \in S$. ...
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1answer
39 views

Proving the convexity of a set drawn by a function

I want to prove or disprove the following: For any $x_1,y_1,x_2,y_2\in \mathbb{R}$ such that $$ a \leq x_1 \leq b, \qquad c \leq y_1 \leq d$$ $$ a \leq x_2 \leq b, \qquad c \leq y_2 \leq d$$ when ...
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1answer
106 views

Fenchel conjugate of a particular function

Consider the function $$f(x,y) = \big(b-x^T A y\big)^2$$ where $x \in \mathbb{R}^{p\times 1}$, $y \in \mathbb{R}^{n\times 1}$, $A \in \mathbb{R}^{p \times n}$ and $b \in \mathbb{R}$. What is the ...
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0answers
91 views

Characterization of Convex Symmetric Balanced Sets

Let $E$ be a Vector Space over $\mathbb{K}$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}= \mathbb{C}$. We say that $U\subset E$ is balanced if $\alpha U\subset U$ whenever $|\alpha|\leq 1$, where ...
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1answer
28 views

Directions of decrease for a convex functions

Suppose $f(x,y)$ is a convex function and $$ f(x+\Delta x, y) < f(x,y), ~~~ f (x, y + \Delta y) < f(x,y)$$ Does this imply $$ f(x+\Delta x, y + \Delta y) < f(x,y)$$? I am guessing the ...
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1answer
83 views

The support function of two sets are equal iff the sets are equal

I am not sure how to approach this question from Boyd. How to show that the support function of two sets $A$ and $B$ are equal iff $A=B$. The support function for a set $A$ is defined as ...
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2answers
64 views

Is the following function differentiable?

Suppose $ C \subseteq \Bbb R^n $ is bounded, and define $f:\Bbb R^n\to\Bbb R$ by $$f(x) = \sup_{y \in C}\|x-y\|, $$ where the norm is euclidean distance. Is $f$ differentiable?
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1answer
95 views

Whether this set is convex or not?

Consider the closed disk of radius 1 at the origin. Let it be called set S. Now is the set $S'=S\setminus \{(1,0),(0,1)\}$ convex? I feel like it is convex but I am not sure how to prove. It basically ...
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0answers
176 views

how to solve the following expectation? closed-form expression or approximation

Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
2
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1answer
186 views

relative interior, convex hull, intersection

For any index set $I$, let $A_\iota\subseteq\mathbb{R}^d$ for $\iota\in I$ be closed sets. Do we have $\bigcap_{\iota\in ...
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381 views

A question dealing with the convexity of functions involving the absolute value

Just beginning to learn convex analysis and optimization, I have some inquiries to make with regard to the absolute value function $f(x)= |x|$. This function is clearly convex, but since we know that ...
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2answers
169 views

How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $y \gt 0$?

How can I prove that a function $f(x,y)= \frac{x^2}{y}$ is convex for $ y \gt 0$? I take the Hessian matrix of $\displaystyle \frac{x^2}{y}$, and I got: $$H = \displaystyle\pmatrix{\frac{2}{y} & ...
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2answers
295 views

Convex sets proof

Prove the following theorem: Let $V$ be a linear space and $D$ a convex set. Let $x_1,...,x_k$ be $k$ points in $D$. Let $a_1,...,a_k$ be non-negative scalars such that $\sum\limits_{i=1}^n a_i=1$. ...
3
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1answer
149 views

Interior point and Minkowski functional

I want to prove for a convex set that contains zero a point $x$ is interior iff the value of Minkowski functional in that point is strictly less than $1$. is there anyone to help me.
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1answer
129 views

Submodularity of the product of two non-negative, monotone increasing submodular functions

I'm trying to prove the submodularity of the product of two non-negative, monotone increasing submodular functions Formally, we have $f$ and $g$ are submodular functions, that is, ...
2
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1answer
95 views

Continuous and non-decreasing but how?

I am reading a paper and the author shows the continuity and monotonicity of a function. It seems so simple to see but I am sorry that I couldnt see the reason. I will be very happy if you can point ...