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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80 views

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
43 views

Is closed convex set with finite number of extreme points convex polyhedron

I have this simple question related to convex set and convex polyhedron. As the content in the title, it's basically my question: Is closed convex set with finite number of extreme points convex ...
0
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1answer
19 views

How to interpret the regular condition in this theorem about cones in convex analysis?

Theorem:Let $K_1,\dots, K_m$ be convex cones in $R^n$ and let $K = K_1 \cap K_2 \cap \dots K_m$. If $K_1 \cap int(K_2) \cap \dots \cap int(K_m) \neq \emptyset$(regularity assumption), then $K^\circ = ...
0
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0answers
38 views

Why is half space not affine?

I've read that half spaces are convex but not affine. I'm trying to understand this geometrically. Does it mean that if I connect any 2 points in the half space, it may result in a line that extends ...
2
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2answers
83 views

proving 1/x is convex (without differentiating)

I know that $\frac{1}{x}$ is convex when $x \in (0,\infty)$, this can be proven easily by showing that the second derivative is positive. However, I am finding difficulty showing it using the ...
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0answers
24 views

Stronger Condition for Strict Convexity?

As I understand, a strictly convex function $f: D \rightarrow \mathbb{R}$ is one that satisfies the property: $\forall x, y \in D, x \neq y, \forall t \in (0,1), f((1-t)x+ty)<(1-t)f(x)+tf(y)$. A ...
1
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1answer
49 views

Is $-\ln(1+e^x)$ a convex function?

Is $-\ln(1+e^x)$ a convex function? My answer book says no because its second derivative is $-\dfrac{e^{2x}}{(1+e^x)^2}$ but I am sure that it is incorrect. I have that the second derivative is ...
0
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1answer
27 views

How to prove that the Statistical Entropy $S_{BG}$ is concave

So I am for the moment studying the properties of the Boltzmann-Gibbs statistical entropy \begin{equation} S_{BG}=-k_B \sum_{i}p_i\ln p_i, \end{equation} where of course $k_B$ is the Boltzmann ...
1
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1answer
28 views

How to prove $f$ is 1-strongly convex convex if and only if $f - \frac{1}{2}\|\cdot\|^2$ is convex?

I am trying to prove that a function $f:Z \mapsto \mathbb{R}$ is 1-strongly convex if and only if the function $f - \frac{1}{2}\|\cdot\|^2$ is convex. Assuming that $f$ is strongly convex, I have by ...
1
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1answer
26 views

Polytopes in binary field

So I just stumbled across something kind of interesting. Say we're in $\{0,1\}^3$ with modulo 2 addition. The convex hull of this is the unit cube. Now, if we want to define a polytope on our cube, ...
2
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1answer
70 views

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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0answers
8 views

Approximating disjoint convex sets by subsets with positive separation

If $A$ and $B$ are disjoint convex sets, is it possible to write $A=\bigcup_{n\in\mathbb{N}}A_n$ where: 1) each $A_n$ is a convex set and 2) The distance between $A_n$ and $B$, $d(A_n, ...
1
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1answer
18 views

dimension of space of origin-symmetric ellipsoids

I wonder how I can compute the dimension of the space of all origin-symmetric ellipsoids? What is the best approach?
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1answer
26 views

What kinds of infinite sequences have accumulation points?

I am actually asking about one particular sequence. $$ y^k = \frac{x^k - \Pi_{\bar{X}}(x^k)}{\|x^k - \Pi_{\bar{X}}(x^k)\|} $$ Here $x^k$ is a sequence converging to $x^\star$, $x^\star$ is not ...
0
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1answer
27 views

Does the non-expansion property of the projection operator hold for all definitions of norm?

For convex problem, of course. I vaguely remember this holds for weighted norm also. But I am curious if there are some general conclusions about what kinds of norm will fit in this framework?
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16 views

Lower bounds on lattice points on a convex curve

I was just reading this paper on the number of integral points on a convex curve of arc length l. The paper begins: In 1926, Jarnik [4] proved that a strictly convex arc y = f(x) of length l ...
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1answer
52 views

Question about definition of separating hyperplanes (theorem)

Let $A,B$ be two sets. We say the hyperplane $\langle a,x\rangle =c$ separates $A,B$ if $A\subset H^-$ and $B\subset H^+$, that is $$ x\in A \implies \langle a,x\rangle \leq c\\ x\in B \implies ...
4
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135 views

How to prove this inequality (already verified by numerical simulation)?

I have a conjecture which has been verified extensively by simulation. The conjecture is as follows: $\forall t \in [0, 1], \alpha \in [0,1]$, and positive real sequences $\{p\}_{i:1,\dots,n}, $, ...
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0answers
33 views

Lieb convexity theorem

So I am currently working my way through Rajendra Bahtia's book matrix analysis. For the proof of the Lieb convexity theorem on page 271 he proofs following Lemma: Let $R_1, R_2, S_1, S_2, T_1, T_2$ ...
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0answers
41 views

Scaled proximal operator for proximal Newton method

The scaled proximal operator was introduced as an extension of the (regular) proximal operator: $prox^H_h(x) = \arg\min_y h(y) + \frac{1}{2}\|y-x\|^2_H$. (See ...
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1answer
35 views

Convexity of the natural exponential fuction - directly from the definition

Without using the Second Derivative Test, can the convexity of the natural exponential function be shown directly from the definition of convexity? The expression \begin{equation*} e^{t} = ...
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0answers
11 views

Lower hemicontinuity of intersections

This old question (with answer) is about how to prove that the intersection of two lower hemicontinuous multifunctions is lower hemicontinuous in the case that the intersection always has nonempty ...
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1answer
35 views

Is this convex minimizer a continuous function?

Consider the function $g: \mathbb R^n \rightarrow \mathbb R$ given by: $$ g(x) = \arg\min_{y\in\mathbb R} \sum_{i=1}^n f_i(|y - x_i|) $$ where $f_i$ are convex, strictly increasing and continuous. ...
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3answers
86 views

Is there an efficient way to evaluate the proximal operator of $f(x) = \|x\|_2 + I_{\geq 0}(x)$?

Is there an efficient way to evaluate the proximal operator of the function $f:\mathbb R^n \to \mathbb R \cup \{ \infty \}$ defined by \begin{equation} f(x) = \| x \|_2 + I_{\geq 0}(x), \end{equation} ...
1
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1answer
33 views

Proof convexity of Logarithmic function

Prove that: $\ln(e^{x+y} +1 )$ is a convex function. I have tried to used $F \circ G$, while $F = \ln (t+1)$ and $G = e^{x+y}$ $G$ is convex but I need to prove that $F$ is convex and growing, ...
2
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1answer
61 views

Uniqueness of the Convex Combination of Positive-Definite Matrices

I am trying to connect the matrices $X$ and $Y$ with a curve defined by the convex combination of $X X^T$ and $Y Y^T$. If I define $Z Z^T = c(X X^T) + (1-c) (Y Y^T), \ c \in [0,1]$, it is true that ...
0
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1answer
53 views

How to prove Clarkson's inequality?

I do not know how to prove one of the four Clarkson's inequalities: let $u,v \in L^p(\Omega)$, if $1 < p < 2$, then $$ \bigg\lVert \frac{u+v}{2} \bigg\rVert_p^p + \bigg\lVert \frac{u-v}{2} ...
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43 views

Existence of steepest descent curves of convex functions

Preliminaries. Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then for any point $x\in\mathbb{R}^n$ and direction $v\in\mathbb{R}^n$ the directional derivative $\nabla_v f(x)$ of $f$ at $x$ ...
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30 views

$\overline{\mathrm{conv}} \{x_i : i \in \mathbf{N} \}$ is compact if $x_i \rightarrow 0$

I want to show that the set $\overline{\mathrm{conv}} \{x_i : i \in \mathbf{N} \}$ is compact in a Banach-space $X$ if $(x_i)_{i \in \mathbf{N}}$ is a sequence in $X$ converging to the origin. My ...
0
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1answer
58 views

Properties of increasingly convex function

Suppose $f : \mathbb{R} \to \mathbb{F}$ is strictly increasing, convex, and twice continuously differentiable function. Define $g$ and $h$ as $$ g(x) = \frac{f(x+1)-f(1)}{f(x)-f(0)}, h(x) = ...
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0answers
44 views

Quasiconcavity of a product of ratios

Given $f(x_1\ldots x_k) = \dfrac{x_1x_2\cdots x_k}{(x_0+c_1)(x_0+c_2)\cdots(x_0+c_k)}$ where $x_i > 0$, the $c_i > 0$ are constants, and $$x_0 = \sum_{i=1}^k x_i$$ is it true that $f$ is ...
5
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1answer
49 views

Functions where the pre-image of convex sets is convex

For functions $f:\mathbb R\to\mathbb R$, I've noticed an interesting property: $f$ is monotonous exactly if the pre-images of convex sets are convex. Now the latter condition can of course be defined ...
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2answers
111 views

Strictly increasing, strictly convex function: is the second derivative positive?

Consider a twice continuously differentiable function $f \colon \mathbb{R} \to \mathbb{R}$. While $f''(x)>0\ \forall x$ implies strict convexity of $f$, the converse is not true (e.g. $f(x)=x^4$, ...
3
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1answer
129 views

Convexity of $x^p$ — without calculus

At least in courses in Germany, you define $\exp(x)=\sum_{n=0}^\infty x^n/n!$ much earlier than establishing basic calculus. Nevertheless, one can easily prove the convexity of $\exp$: For $t\in[0,1]$ ...
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1answer
49 views

Quasiconvexity of linear-fractional composition

In Boyd and Vandenberghe Section 3.3.4, it is stated that compositon of a quasiconvex function with an affine-fractional transformation is quasiconvex. In specific, if $f(x)$ is quasiconvex, then ...
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3answers
115 views

Why is the empty set convex?

Why is it the empty set, trivially convex? I see this results stated into a proof as something known, but I do not understand what's the idea idea behind it. How could I reason about convex ...
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0answers
35 views

Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
0
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1answer
25 views

Convex hull of $\{ \Vert x \Vert = 1 \}$ is closed in strictly convex space

I'm trying to show that the convex hull of $\{ \Vert x \Vert = 1\}$ is closed in a strictly convex Banach-space. I don't know how to tackle the problem. Are there any nice characterizations for a ...
0
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2answers
37 views

Is the logarithm of sum of multiple variables with the constraint on sum of them Concave?

I know that without any constraint $log \sum_{i=1:1:m} \alpha_i C_i $ is not Concave but I am wondering is this function Concave when we have the constraint that $ \sum_{i=1:1:m} \alpha_i =1 $ and ...
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0answers
22 views

Core points of a convex set

In the book of Gamelin "Unifrom Algebras" I found the following definition: Let $K \subset V$ be a convex set in a vector space. An element $x \in K$ is called core point of $K$ if whenever $z ...
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25 views

Analytical algorithm to obtain solution to convex optimization problem.

Assume a vector $\vec{P}$ with N elements $\in \mathbb{R}^+$ and constants $T_P$ and $\epsilon$. The vector $\vec{P}$ is arranged in a column $(N\times 1)$. Consider the problem: $$ \begin{aligned} ...
6
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3answers
197 views

Show that $\left(\int_{0}^{1}\sqrt{f(x)^2+g(x)^2}\ dx\right)^2 \geq \left(\int_{0}^{1} f(x)\ dx\right)^2 + \left(\int_{0}^{1} g(x)\ dx\right)^2$ [closed]

Show that $$ \left( \int_{0}^{1} \sqrt{f(x)^2+g(x)^2}\ \text{d}x \right)^2 \geq \left( \int_{0}^{1} f(x)\ \text{d}x\right)^2 + \left( \int_{0}^{1} g(x)\ \text{d}x \right)^2 $$ where $f$ and $g$ ...
0
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1answer
53 views

Functional analysis - check that a closed subspace of a Hilbert space is convex

Suppose that V is a Hilbert space over $F$ and $W$ is a closed subspace of $V$ . Then for every $x \in V$ , there exist unique $y \in W$ and $z \in$ (the orthogonal compliment of $W$) such that $x = y ...
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0answers
38 views

is this function an ill-shape convex function?

I have a function with parameter $\vec{{\alpha}}$ where it is formulated by the formula: $$ f(D|\alpha)=n_1{\alpha}_{1}+...+n_m \alpha_m -Nlog \sum_{i=1:m} exp(\alpha_{i}+g_i(D)) $$ where $g_i{(D)}$ ...
6
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1answer
173 views

Inf-convolution, two basic questions?

Let $E$ be a normed vector space. Given two functions $\varphi$, $\psi : E \to (-\infty, +\infty]$, one defines the inf-convolution of $\varphi$ and $\psi$ as follows: for every $x \in E$, ...
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2answers
44 views

Minimising a convex set. Is set of solutions convex?

We are minimising a convex function on a non-empty set defined by linear constraints (equalities and inequalities). $X^O$ is the set of all optimal solutions and we assume it is non-empty. Is it true ...
3
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0answers
20 views

Supporting hyperplane to a compact, convex set in Hilbert space at a given boundary point

Does one always exist? I see that it is true in finite dimensions.
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1answer
44 views

sum of two cones

A non-empty set $K$ of a vector space is called a cone if it satisfies the following: $ K +K \subseteq K,$ $\alpha K \subseteq K$ for all $\alpha \ge 0,$ $K \cap (-K) ={0}$. Let $K_{1}$ and ...
0
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0answers
23 views

All facets' coefficients contain only integers -1,0,1

Suppose a polytope $C\subset \mathbb R^{kl}$ is the $l$-product of $k-1$-simplex with extreme points containing coordinates $0$ or $1$ in each coordinate. A linear transformation is given by ...
0
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1answer
24 views

integer valued outer normal vectors

Suppose a bounded polyhedra $C$ is given by $$x\in \mathbb R^n: Ax\leq b$$ The matrix $A\in\mathbb R^{m\times n}$ contains only elements from $\{-1,0,1\}$, which implies the outer normal vectors of ...