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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Showing the intersection/union of a cone is a cone

Defining a set $C \subset \mathbb{R}^n$ as a cone if for ever $x \in C$ and $\alpha \geq 0$ we have $\alpha x \in C$. ie they are closed under scalar multiplication. How can I show that the ...
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47 views

When the closure of a convex set contains a ball

Suppose $C$ is a convex set in $\mathbb{R}^n$ whose closure contains the open ball $B(x,r)$. Is it true that $C$ contains $B(x,r)$? Motivation: I am asking this because something like this seems to ...
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definition of open set and is gauge function is well defined?

Q1 :I know that a set A $ \subset$ X is open if it contains an open ball about each of its points i.e. for all x in A ,there exists $ \epsilon $>0 s.t. $ B_\epsilon $(x) $\subset $ A But then does ...
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Hessian to show convexity - check my approach please

I need to check the convexity of $f(x)$ for these two questions, using the Hessian matrix. I am aware the function can be said to be convex if over the domain of $f$ the hessian is defined and is ...
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37 views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and ...
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Dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$

This slide shows the dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$ is $K^{*} = \{(x,t) \mid \| \boldsymbol{x} \|_{\infty} \le t\}$. Is it right? How is it proved?
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Help with this convex set proof

Take $C ⊂ \mathbb{R}^n$ a convex set. Fix $x_0 ∈ C$ and a nonzero vector $v ∈ \mathbb{R}^n$ . Define the set $I(x_0,v) := \{t ∈ R : x_0 + tv ∈ C \}$. Prove that $I_(x_0,v)$ is a convex subset of ...
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31 views

the continuity of argmin

how can I show the minimum of a converging sequence of continuous functions is converging to the minimum of the limit of that sequence.i.e. $$\lim_i \text{argmin}_{x} f_{i}(x)=\text{argmin}_{x} \lim_i ...
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20 views

How to prove the following convex cone property

Suppose $A,B$ are closed cvx cones. $A^*,B^*$ are their dual cones respectively. How to show $$ (A^*+B^*)^*\subset A \cap B$$ My idea is: $$A^* = \{x_1|x_1^Tx \geq 0, \forall x \in A\}$$ ...
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Any example of strongly convex functions whose gradients are Lipschitz continuous in $\mathbb{R}^N$

Let $f:\mathbb{R}^N\to\mathbb{R}$ be strongly convex and its gradient is Lipschitz continuous, i.e., for some $l>0$ and $L>0$ we have $$f(y)\geq f(x)+\nabla f(x)^T(y-x)+\frac{l}{2}||y-x||^2,$$ ...
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improper convex function

In Rockafellar's convex analysis there was an example of improper convex function: $$ f(x) = \begin{cases} -\infty & \text{if } ||x||<1, \\ 0 & \text{if } ||x|| =1, \\ +\infty ...
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66 views

Non-subdifferentiable convex function on a normed space

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?
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how to construct the sup convex function with fixed values

Set $\mathcal{S} = \{(x_1, f_1),(x_2,f_2),...\vert x_i \in \mathbb{R}^d, f_i \in \mathbb{R}\}$, we assume convex functions $f(x): \mathbb{R}^d\rightarrow \mathbb{R}, $ which passes those points exist. ...
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When is a pseudoinverse of a matrix non-negative?

Consider a matrix $A \in \mathbb{R}^{n \times m}, n > m$ with independent columns and non-negative entries. Consider the oblique pseudo-inverse of $A$, i.e. the matrix $A^\dagger_B = (B ^\top ...
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459 views

convex hull function in matlab

Is there anyway to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
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$\{x_n\} \to x$ iff $\bigcap_{n=1}^\infty K_n = \{x\}$

Let $\{x_n\}$ be a sequence in $\mathbb{R}^k$ and let $K_n$ be the intersection of all closed convex sets that contain $x_m$ for all $m \ge n$. How do I show that $\{x_n\}$ converges to $x$ if and ...
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793 views

generalized inequalities defined by proper cones

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...
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44 views

What is the motivation behind the, convex and concave closures of submodular functions?

What is the motivation behind the , convex and concave closures of submodular functions? Also, my understanding is that the submodularity condition is somewhat like concavity which makes it counter ...
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Convexity of $I(X;Y)$: why $H(Y)$ convex in $p(y)$ $\Rightarrow$ $H(Y)$ convex in $p(x)$

I would like to understand the proof that mutual information $I(X;Y)$ is concave in $p(x)$ - as presented in Elements of Information Theory by Cover & Thomas, theorem 2.7.4. Here's the proof from ...
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How to prove the following cone theorem

If $K \subset R^n$ and $0 \in K$. Define $K'=\{u|\langle u,x\rangle \leq1, \forall x \in K\}$. Note: $\langle u,x\rangle = u^Tx$ Prove: If $K$ is a cone, then $K' = -K^*$, where $K^*$ is the ...
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Legendre transformation of a convex function and primal minimum

I am having trouble in proving following property: If $f$ is convex (and consequently $f^{**} = f$) and minimal in set $X$ exists, i.e. there is $x^* \in X$ such that $f^* = f(x^*) = \inf_{x \in X} ...
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26 views

How can I show the following statements are equivalent?

Let $C ⊂ \mathbb{R}^n$ Prove that the following statements are equivalent. (i) $C$ is an affine set (ii) For every $x_0 ∈ C$ , the set $C − x_0 := \{ z − x_0: z ∈ C \}$ is a subspace. (iii) There ...
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27 views

closure of a convex set in a normed linear space is convex ?

Is it true that if $A$ is a convex set in a normed linear space $V$ , then the closure of $A$ is also convex ? (I know that the interior is convex )
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An example for A will not included in B

Consider three sets $X,Y,Z \subset R^n$ and $t \in R$. They satisfy: $X+Z \subset Y+Z$, where $X$ and $Y$ are convex, $Y$ is closed, and $Z$ bounded.Define: $X+Y = \{x+y | x \in X, y \in Y\}$, and ...
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Disjoint Convex Sets That Are Not Strictly Separated

Question 2.23 out of Boyd and Vanderberghe: Give an example of two closed convex sets that are disjoint but cannot be strictly separated. The obvious idea is to take something like unbounded sets ...
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319 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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On existence of extreme points of special type of non-empty closed convex sets of $\mathbb R^n$ [closed]

Let $A \subseteq \mathbb R^n $ be a non-empty closed(w.r.t. usual Euclidean metric of $\mathbb R^n$) convex set such that for some $x \in \mathbb R^n$ and $r>0$ , $B(x,r) \cap A=\phi$ , then must ...
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Image of convex hull

I came across a problem that I could simplify, if I knew that this is true: Let $A:= conv(x,y,z)$, where $x,y,z \in \mathbb{R}^n$ and $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear map. Does ...
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29 views

Is the biconjugate of a continuous functions also continuous?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be given and assume that $|f(x)|\leq C|x|^2$. Is it true that the bi-(convex/Fenchel)-conjugate $f^{**}$ is also continuous. It was claimed in a book without a ...
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About vertices of the convex hull of any finite set of points in $\mathbb R^n$

Let $S$ be a finite subset of $\mathbb R^n$ , we know that $x \in S$ is a vertex of $Conv (S)$ , the convex hull or convex polytope of $S$ , iff $x \notin Conv\Big(S$ \ $\{x\}\Big)$ ; then is the no. ...
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Convex set with empty interior is nowhere dense?

Suppose $C\subseteq\mathbb R^n$ is a convex set and $C^o=\varnothing$. Is it necessarily true that $(\overline C)^o=\varnothing$? In general, is this true if $\mathbb R^n$ is replaced by a topological ...
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Showing convexity proof

Let $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be an affine function, i.e., $F (x) = L(x) + b$, with $L : \mathbb{R}^n \rightarrow \mathbb{R}^m$ linear and $b \in \mathbb{R}^m$ Then for every convex ...
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A condition for mid-convex implies convex

Let I an open interval, and $ f: I \rightarrow \mathbb{R} $ such that: $\forall (x,y) \in I^2 ; f(\frac{x+y}{2}) \le \frac{f(x)+f(y)}{2}$ There exists an interval $[a,b]$ such that $a<b$ and ...
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Is epi(max(f,g)) the intersection of epi(f) and epi(g)?

On an exam, I found the question "is max($f(x),g(x))$" convex if $f,g$ are convex? This lead me to the question in the topic. Is the intersection of epi$(f)$ and epi$(g)$ = epi($\max(f,g)$)? If so, ...
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403 views

proximal operator of infinity norm

What is the proximal operator of $||x||_\infty $? I know we have to take the subgradient and compute it but I am a bit stuck. Can anyone show me steps?
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93 views

On the decomposition of stochastic matrices as convex combinations of zero-one matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
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Sufficient condition for convexity

Let f:$ [a,b] \rightarrow \mathbb{R} $ a continous function such that $ \forall (x,y) \in [a,b]^{2}, \exists t \in ]0,1[, f(tx+(1-t)y) \le tf(x) + (1-t)f(y) $ show that f is convex
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Strictly convex set

When a function is strictly convex it has many desirable properties, most notably that it admits a unique minimum. I was wondering if there is anything desirable about a strictly convex set (meaning ...
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Minkowski sum and difference do not cancel

I need to prove that $ ( A\oplus B )\ominus$ $B$ and $( A\ominus B)\oplus$ $B$ need not equal $A$ for all sets $A$, $B$,where $\oplus$ and $\ominus$ denote the Minkowski sum and difference. As ...
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How do I see that every point inside the corresponding convex region in $\mathbb R^2$ belong to this set?

Convex set in $\mathbb R^2$. Suppose I use the convex operator $\text {conv}$ to create the convex set of $X = \{x_1, ... , x_n\} \subset \mathbb R^2$, that is $\text {conv}(X) = \{(1-\lambda)x_i + ...
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What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
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Prove that convex function on $[a,b]$ is absolutely continuous

In the book Roberts, Varberg, Convex functions, on pp.9-10 is proved that If $f:(a,b)\rightarrow \mathbb R$ is continuous and convex then $f$ is absolutely continuous on each $[c,d]\subset ...
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what is the convex hull of such a matrix cone?

A matrix cone is in the following form: $M: = \begin{pmatrix} 1 \\ x\end{pmatrix}\begin{pmatrix}1 & x^T\end{pmatrix}$ where $x\in F$ , let $F = \{x: x\in [l,u]^n\}$ How to express the convex ...
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How prove this $\frac{af(a)+bf(b)}{a+b}\ge f(a+b)$

Assume that $f(x)$ has two derivatives on $(0,2)$ and $0<a<b<a+b<2$. I have to prove that, if $f(a)\ge f(a+b)$ and $f''(x)\le 0$, then: $$\dfrac{af(a)+bf(b)}{a+b}\ge f(a+b).$$ I ...
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102 views

Prove a certain property of linear functionals, using the Hahn-Banach-Separation theorems

I've been trying to solve this question, with no luck so far: Let $X$ be a real linear space, and $\{\|\cdot \|_i\}_{i=1}^{n}$ family of norms on $X$. Let $f$ be a linear functional on $X$ such that ...
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Function on convex set is convex if all rays are convex

Consider the function $f:D\rightarrow\mathbb{R}$ for $D\subset\mathbb{R}^n$ an open convex set. Furthermore, suppose that $g(t)=f(t\boldsymbol{x})$ is convex for all $\boldsymbol{x}\in D$. Is it ...
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Proving or disproving concavity of a function

I want to prove that the following function is concave (as a part of another proof). $$f(p) = \max_{\begin{matrix}x,y\\0\le x \le 1\\0\le y \le 1 \\ x * y = p\end{matrix}} \lambda h(x) + ...
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17 views

Vector defined function is convex implies scalar defined function is convex

Let $f:\mathbb{R}^n \to \mathbb{R}$ be convex. Let $g:[0,1]\to \mathbb{R}, g(a)=f(a \cdot x+(1-a) \cdot y)$. Why does $f$-convex on $\mathbb{R}^n$ imply that $g$-convex on $[0,1]$?
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67 views

Convex function property

Let $ f_{1}, f_{2},..., f_{n} $ convex functions in the interval $[0,1]$ such that $ max(f_{1},f_{2},...,f_{n}) \geq 0 $ show that there exist positive real numbers $a_{1}, a_{2},...,a_{n} $ not ...
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30 views

Concavity of a multivariate function

Let f be a function such that f is Frechet differentiable. Prove that f is concave if and only if the following inequality holds: $$ 0\le ...