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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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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15 views

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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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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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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19 views

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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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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17 views

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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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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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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20 views

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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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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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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26 views

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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Intersections of convex hulls

Given a set of $n$ points $\{A_1, \ldots , A_n\}$ of the plane and every possible triangle formed with $3$ points $A$, I would like to describe the intersections fo theses triangles. By intersection, ...
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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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43 views

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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40 views

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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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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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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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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33 views

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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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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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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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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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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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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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 ...
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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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Representation of a point in terms of extreme points and extreme directions in a LP feasible region

We know that every point in a feasible region (X) of a Linear Program can be represented as a convex combination of the extreme points of X plus a non-negative combination of the extreme directions of ...
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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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Conjectured characterization of a set relative to a convex cone

Let $X\subset \mathbb{R}^N$ be a convex cone (i.e., for all $x,y\in X$ and $\alpha,\beta\geq 0$ scalars, $\alpha x+\beta y\in X$). Define the set $$A(x)=\{a:x+a\in X \wedge x-a\in X\}.$$ Then, ...
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59 views

In finite dimensional normed space, every convex set contains a basis

I've been reading Lemma 5.60 here: http://epge.fgv.br/we/MD/TeoriaEconomicaAvancadaI/2009?action=AttachFile&do=get&target=Aliprantis-Infinite-Dimensional-Analysis.pdf (p.g 200, =217 on the ...
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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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How nuclear norm is convex whereas weighted nuclaer norm is not?

In (http://nuit-blanche.blogspot.in/2014/05/wnnm-weighted-nuclear-norm-minimization.html), it is stated that nuclear norm of a matrix $\mathbf{X}$, given as $||\mathbf{X}||_{*}=\sum_{i} ...
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Convex conjugate of a function of sum of norms

I am trying to find the conjugate of function $f(x) = \|x\|_2 + \frac{1}{2} \|x\|_2^2$ i.e., $f^*(v) = \sup_x (v^Tx - f(x))$ where $x \in\mathbb R^n$ Although $f(x)$ is convex, I am stuck as the ...
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How to check convexity of a composition when some properties of inner and outer functions are known?

If $g(x)$ function is concave in $x$, and we want $g( f(x) )$ (where $f(x)$ is another function) to be convex in $x$, what are the required properties of $g(x)$ and $f(x)$? It would be appreciated ...
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50 views

Normal Cone of $\mathbb{R}^n_+$ and $S^n$?

I'm trying to solve the problem $\min_x \{f(x) + \delta_X(x)\}$ where $f$ is a differentiable function and $\delta$ is the indicator function $\delta_X(x) = \begin{array}{l}0, x \in X \\ ...
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Eliminating equality constains

The following text derived from book convex optimization by Boyd, page 143. For a convex problem the equality constraints must be linear, i.e., of the form $Ax = b$. In this case they can be ...
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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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Surjective bilinear map

Let $Q$ be a CONVEX quadrilateral in $R^2$ with vertices $a_1,a_2,a_3,a_4 \in R^2$. Consider the bilinear map $f: [0,1]^2 \to Q$ $$f(x,y)=a_1+(a_2−a_1)x+(a_4−a_1)y+(a_1+a_3−a_2−a_4)xy$$ Note that $f$ ...
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Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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73 views

Continuous functions of minimal norm

Let $C$ denote the set of continuos functions on $[0,1]$ with the supremum norm. $M\subset C$ such that $$\displaystyle\int_{0}^{1/2}f(t)\, dt-\int_{1/2}^{1}f(t)\, dt=1,\; \forall f\in M$$ Show ...
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Intersection of compact convexes

Let $C_1,C_2,C_3,C_4$ be compact convexes of $\mathbb{R}^2$ such that $C_1\cap C_2\cap C_3\neq\emptyset,C_1\cap C_2\cap C_4\neq\emptyset,C_1\cap C_3\cap C_4\neq\emptyset,C_2\cap C_3\cap ...
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A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
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55 views

Show that the set $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex

Let $f:\mathbb{R}^N\rightarrow \mathbb{R}$ be a $C^1$ convex function. Show that $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex (we assume that empty set is convex). Any hint?
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35 views

Convex function and expectation

I was wondering: if f is a convex function and X a random variable, what does E(f(X)) = f(E(X)) implies? Thanks a lot, David
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68 views

Proving that quadratic form is convex in (vector, matrix) arguments

I'm studying with the quadratic form $$ F( (x,Q) ) = \langle x,Q^{-1}x\rangle $$ considered over $\mathbb{R}^n\times\mathbb{R}^{n\times n}_+$, where $\mathbb{R}^{n\times n}_+$ is the set of all ...
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Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...