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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Proof that this set is convex

I need a help with prooving that a given set is a convex set: $\{ x \in R^n | Ax \leq b, Cx = d \}$ I know the definition of convexity: $X \in R^n$ is a convex set if $\forall \alpha \in R, 0 ...
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129 views

Is $S^\circ$ convex if $S$ convex?

Suppose $S\subset \mathbb{R^n}$ and $S^\circ$ denoted as the interior of $S$.Is $S^\circ$ convex if $S$ convex? $S$ is Convex mean $ \forall x,y\in S, kx+(1-k)y\in S, k\in [0,1]$ I know how to prove ...
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0answers
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Concave lower bound on inner product of two distributions.

Given two discrete distributions $p, q$ which lie in an $m$-dimensional simplex, is it possible to provide a concave lower bound on the inner product between these distributions. That is we wish to ...
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267 views

prove that m(a,y) is non decreasing in a and y and concave in a.

I am a university student specializing mathematics for economists. I am in a preparation for my final exam. My prof gave me some questions that might be on the exam. One of the question dragged me ...
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1answer
69 views

A consequence of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$. Studying the behaviour of the difference quotient, it is clear that $$f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$$ ...
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1answer
768 views

Show this function is convex.

Could someone point me in the right direction for proving the following? Given that $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is an affine map given by $f(x)=A\mathbf{x}+\mathbf{b}$, ...
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1answer
190 views

Gradient of log softmax in matrix form

Suppose $J(\mathbf{A})$ is defined as follows $$J=\text{tr}(\log \mathbf{P})$$ $$\mathbf{P}=\frac{e^\mathbf{A}}{\mathbf{1} \mathbf{1}' e^\mathbf{A}}$$ where division, exp and log are taken pointwise, ...
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Find a convex combination of scalars given a point within them.

I've been banging my head on this one all day! I'm going to do my best to explain the problem, but bear with me. Given a set of numbers $S = \{X_1, X_2, \dots, X_n\}$ and a scalar $T$, where it is ...
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2answers
226 views

proof of any line joining two points lying in opposite half-spaces determined by a hyperplane in $\Bbb{R}^n$ intersects the hyperplane.

I am a student specializing in mathematics for economists. I have been struggling with proof question regarding hyperplane and was wondering if you could please give a helpful hand. The question is: ...
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1answer
324 views

Second order approximation of log det X

I'm trying to follow the derivation of second order approximation of log det X from Boyd's "Convex Optimization", p.658 in http://www.stanford.edu/~boyd/cvxbook/ How is the last step derived? IE, ...
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35 views

can we find general form for elements of intersection of positive semidefinite matrices with convex cones of other matrices?

On sep 8, 2011 a question was asked about cones of positive semidefinite matrices that can be generated by rank 1 matrices. A respondent answered "any convex cone in Rn×n is defined by a collection of ...
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1answer
159 views

Mathematical applications of the subgradient

Do you know mathematical results which can be nicely proved using subgradient? For example, Jensen's inequilaty can be proved like that: Let $\varphi : \mathbb{R}^n \to \mathbb{R}$ be a convex ...
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1answer
104 views

How to prove that the space created by pointwise Bernoulli random variables are compact

I have a function $$\delta:\mathbb{R}\rightarrow [0,1].$$ We obtain this funtion pointwise as follows: For each point $y\in\mathbb{R}$, $\delta(y)$ is a real number in $[0,1]$. More explicitely, ...
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1answer
60 views

Convex function plus $v e^{-x}$

If $f(x)$ is strictly convex, and $$\lim_{x\to \infty}\left(f(x) - x - ue^{x}\right) = w$$ for some $u\ge 0$ and $w$ then what can be said about: $$g(x) = ve^{-x} + f(x)$$ on $x\ge0$ where $v$ is ...
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1answer
241 views

Global Min-Max Optimization

When is \begin{equation} \min_X \max_Y f(X,Y) \end{equation} globally solvable? (i.e. we can find global solution for the optimization problem?) I am not looking for reformulations. Is it only when ...
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2answers
82 views

A Quadratic Problem (which looks very simple)

This arises as a part of my work. \begin{align} \min_{x^{H}x=1}~&x^{H}A_1x \\ subject~to~&x^{H}A_2x=0 \end{align} $A_1$ and $A_2$ are $N\times N$ hermitian matrices and $x$ is a unit norm ...
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3answers
112 views

Proof of Convexity

Is the function $Trace(AX^TBX)$ a convex function in $X$ or not ? Here, $X$ is a rectangular matrix and $A,B$ are square, symmetric, p.s.d matrices. The entries in $X,A,B$ are real valued.
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1answer
75 views

about convexification

Let $f: \mathbb{R}^{n} \rightarrow \overline {\mathbb{R}}$. Called conjugate in the sense of Young-Fenchel of $f$, the following function: $$f^{*}(x^{*})=Sup\lbrace \langle x,x^{*}\rangle -f(x) : x ...
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1answer
242 views

Joint Convexity

Is the problem \begin{equation} \min_X \max_Y -\operatorname{tr}(X^TY)-\operatorname{tr}(Y^TYX) \end{equation} Jointly convex in $X$ and $Y$? Can we solve it globally? Why or Why not? $X$ and $Y$ ...
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253 views

Convex function from Hessian

Am I correct to say that the following function is convex? $$\begin{align} & f(x,y)=-\sqrt{xy} \\ & x>0,y>0 \\ \end{align}$$ After computing the Hessian: $$ Hf =\left[ \begin ...
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1answer
119 views

Convex Sets Versus Convex Functions

Can we specify all convex sets, in terms of convex constraints (convex inequality functions) on a variable?
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59 views

Consultation on point extreme

Let $C$ be a convex subset of a vector space $X$. A point $x\in C$ is called an extreme point if and only if whenever $x=ty+(1−t)z$, $t\in (0,1)$, implies $x=y=z$. It is known that the boundary ...
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1answer
95 views

Fixed point of a non linear contraction in a convex set

Hi I'm stuck on the following problem of Haim's functional analysis book. Let $C\subseteq H$ ($H$ a Hilbert space) be a non-empty closed convex subset and let $T:C\rightarrow C$ be a non linear ...
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1answer
114 views

How do you describe a convex hull for the set of points in $y = \sin x$

More specifically in, the set notation for the original set is: $$ S = \begin{Bmatrix} \begin{bmatrix} x\\ y\\ \end{bmatrix} : y = \sin x \end{Bmatrix} $$ So, how would you find the convex ...
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1answer
503 views

What is a support function: $\sup_{z \in K} \langle z, x \rangle$?

I want to ask that, what is a support function intuitively. It is defined as: $$\sup_{z \in K} \langle z, x \rangle$$ where $z \in K$, $K$ is a nonempty set. In this formulation, $\langle \cdot, ...
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1answer
332 views

What Stopping Criteria to Use in Projected Gradient Descent

Suppose we want to solve a convex constrained minimization problem. We want to use projected gradient descent. If there was no constraint the stopping condition for a gradient descent algorithm would ...
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43 views

General properties of an optimal solution of a convex program

How do we seek certain properties for a solution of a convex minimization problem. For example we want to make sure if the below objective has a symmetric optimal solution: \begin{equation} \min_X ...
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1answer
33 views

Approximating from the interior of a convex set

In a problem I'm working on I found myself with a point $y\in \mathbb{R}^m$ lying at the boundary of a non-closed convex set $K$. I'd like to express it as as "infinite convex combination" ...
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Why such a function is convex?

Let $f:(a,b)\rightarrow \mathbb R$ be a continuous function satisfying: $$ f(x) \leq \frac{1}{2h} \int_{x-h}^{x+h} f(t) dt $$ for all $x,h$ such that $a\leq x-h<x+h \leq b$. How to show that $f$ ...
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72 views

lower bound of a special type of convex functions

Suppose $f$ is a convex, differentiable and $\|\nabla f(x)-\nabla f(y)\|\leq L\|x-y\|$. The minimum of $f$ is $0$. ($f$ may not be twice differentiable.) How to show $f(x)\geq\frac{1}{2L}\|\nabla ...
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53 views

Is the closure of a set contained in the convex hull?

Let $A\subseteq\mathbb{R}^d$. Do we have $\bar A\subseteq \text{conv}(A)$?Counterexample?
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66 views

$\varepsilon$-balls and closed convex sets

I arrived at the following problem during the day. For $\iota\in I$, let $A_\iota\subseteq\mathbb R^d$ be non-empty and closed. For $\varepsilon>0$ let $B_\varepsilon(A_\iota)=\bigcup_{x\in ...
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closure, convex hull and closed convex hull

Is the closure of the convex hull of some set $A\subseteq\mathbb R^d$ equal to the convex hull of the closure of $A$, i.e. $$\text{cl}(\text{conv}(A))=\text{conv}(\text{cl}(A))?$$ If not, what are ...
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A question related to Helly's Theorem on convex sets

I have one question related to differential geometry. Initilally, I am giving some background and my question is after that. Helly's Theorem Let C be a finite family of convex sets in $R^n$ such ...
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178 views

Convex Functions on 2 variables over an interval

It is required to show that $f(x) = x_1x_2$ is a convex function on $[a,ma]^T$ where $a\ge 0$ and $m\ge1$.To show convexity we need to show that for $\lambda \in [0,1]$: $f(\lambda x + (1-\lambda ...
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102 views

Does every convex-linear map have an affine extension?

There is one step in a proof which I don't manage to show, although it seems to be very easy. Let $A, B$ be real vector spaces, let $S \subset A$ be a convex set and let $\text{aff}(S)$ be its affine ...
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1answer
20 views

What is the coX of the following

What is the coX of {(x,y) $\in$ $\mathbb R^2$ : y = $1\over1+x$, $x \ge 0$ } ? coX is the convex hull. I couldn't figure out. coX should be the smallest convex set that contains the set but in ...
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1answer
61 views

Minimal point of a intersection of N convex sets

I would like to prove that the minimal point of a intersection of $N$ convex sets in $\mathbb{R}^2$ is also the minimal point of the intersection of two of the aforementioned sets. Rephrasing the ...
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1answer
54 views

Are all polytopes also convex hulls?

It seems, at least in the 2-D case, that all polytopes are going to be convex. Does this hold if the dimensions are increased?
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Prove that $\text{int}(\text{dom}(f))$ is a convex set.

Let $f$ be a convex function. I have to prove that $\text{int}(\text{dom}(f))$ is a convex set. (Be careful with $-∞$ )
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Farkas Lemma proof

I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia: Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the ...
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One to one correspondence in faces of convex sets

Let $C$ be a nonempty convex subset of $\mathbb{R}^{n}$, and let $L$ be a nonempty subspace contained in lin$C.$ Define $C_0$ tobe $C \cap L^{\perp}.$ Show that the faces $F$ of $C$ are in one-to-one ...
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Weakly convex functions are convex

Let us consider a continuous function $f \colon \mathbb{R} \to \mathbb{R}$. Let us call $f$ weakly convex if $$ \int_{-\infty}^{+\infty}f(x)[\varphi(x+h)+\varphi(x-h)-2\varphi(x)]dx\geq 0 \tag{1} $$ ...
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Recognizing binomial mixtures

I'd like to know a procedure to recognize whether a given probability distribution over outcomes $\{0, \dots, n\}$ can be expressed as a mixture of $n$-trial binomial distributions with different ...
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2answers
266 views

fail of cantor intersection property on closed , bounded , convex sets of integrable functions

This is from my recent homework. I am asked to find a descending nested sequence of closed , bounded , nonempty convex sets $\{D_n\}$ in $L^1(\mathbb{R})$ such that the intersection is empty , where ...
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1answer
240 views

cantor intersection theorem in banach space

Here is part of the question in my HW. Let$\ \{C_n\}\subset X $ be a bounded nested decreasing sequence of closed and convex sets. I am asked to show that $\bigcap C_n \not= \emptyset$ iff X is ...
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1answer
523 views

Continuity of a convex function

I'm trying to solve the following problem: Let $f:K\rightarrow \mathbb{R} $, $f$ convex and $K \subseteq \mathbb{R}^n$ convex. Then $f$ is continuous on $K$. I have proved the only case $n=1$, but ...
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132 views

Prove that f is continuous

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a convex function on $\mathbb{R}^n$. How to prove that $f$ is continuous?
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117 views

Show the existence and uniqueness of a closed ball containing a bounded subset of a Hilbert space

The problem: Assume $A$ is a bounded subset of a Hilbert space $H$. Let $r$ be the infimum of the radii of closed balls containing $A$, so $r = \inf \{s \geq 0 $ $\vert$ there exists $x \in H$ such ...
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579 views

How to prove that the closed convex hull of a compact subset of a Banach space is compact?

Can anyone help me with this problem? Prove that if $K$ is a compact subset of a Banach space $X$, then the closed convex hull of $K$ (that is, the closure of the set of all elements of the form ...