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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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
22 views

How would you show that this fraction function is convex and decreasing?

Show that $$ f(\vec{x}) = \frac{1}{x_1 - \frac{1}{x_2 - \frac{1}{x_3 - \frac{1}{x_4}}}} $$ is convex when all denominators are greater than $0$.
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41 views

Prove that $f(x,y) = x/(y^2+1)$ is convex

Suppose $f(x,y) = x/(y^2+1)$. I was trying to prove that this function is convex. So I took partial double-derivative and constructed the Hessian for this function. Here the Hessian is a 2 by 2 matrix ...
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19 views

normal cone inclusion and non-symmetric matrix and optimization problem

I have the following normal cone inclusion $$-(A x + b) \in \mathcal{N}_\mathcal{C}(x) \qquad (1)$$ where $\mathcal{N}_\mathcal{C}$ denotes the normal cone to the convex set $\mathcal{C}$ at the ...
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1answer
39 views

A Combination of decreasing functions

I have a strictly decreasing convex function $f$ (at least over $\Bbb R^+$ ), and the non negative numbers $a_1 , a_2$ and $b_1 , b_2$. Is the following a decreasing function ( at least on $t \in \Bbb ...
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1answer
21 views

Unit ball with p norm in $\mathbb{R}^3$ space

I know unit ball for $p$-norm with $p = 2$ is a square, my confusion is how does it look like in $\mathbb{R}^3$ space. In $\mathbb{R}^3$ space it looks like a cuboid, is this correct ?
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1answer
21 views

Is the following fractional function convex?

Is $\displaystyle f(x_1,x_2) = x_1 - \frac{1}{x_2}$ a convex function? What if we restrict the values of this function to the positive reals?
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3answers
84 views

Show that $f(x) = -\ln(x)$ is convex (WITHOUT using second derivative!)

In the lecture notes for a course I'm taking, the definition of a convex function is given as follows: "a function $f$ is convex if, for any $x_1$ and $x_2$, and for any $\alpha$ $\in$ [0,1], $\alpha ...
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0answers
17 views

For a convex function, why does lower semicontinuity imply weak lower semicontinuity?

I have come across the statement that, for a convex function, all notions of lower semicontinuity are equivalent. That is: weak lower, sequential lower, and weak sequential lower semicontinuity are ...
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1answer
19 views

Sequentially weakly lower semicontinuity on reflexive Banach spaces

Let $J$ be a functional over a reflexive Banach space $X$. Is it true that the sequentially weakly lower semicontinuity is equivalent to convexity and continuity for the functional $J$? I think the ...
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1answer
21 views

How to show the Hessian matrix of such functions are positive semi-definite?

Let $f:R\to R$, $g:R^n\to R$. Thus $f\circ g:R^n\to R$. Now suppose $f$ is non-decreasing and convex while $g$ is convex. In additon, $f,g$ are of $C^2$. I want to show that their composition is ...
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8 views

Submodularity definition

I am currently looking at a paper whose submodularity definition is different from whatever I thought I knew. In this paper, the authors consider a function $\Pi_2(q;a^r)$, where $q$ is composed of ...
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0answers
19 views

Projection of hyper-cubes via multiple variable elimination

I am not a mathematician but I do use some tools from geometry in robotics. So, I apologize if what I am writing here is not mathematically consistent but I really do need your help. I have a linear ...
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1answer
15 views

Optimization problem for non-symmetric normal cone inclusion or antiderivative of Ax+b

I have the following equation: $$ -(\mathbf{A}\mathbf{x} + \mathbf{c}) \in \mathcal{N}_{C}(\mathbf{x}) \qquad (1) $$ where $\mathbf{A} \in \mathbb{R}^{n\times n}$ is symmetric and positive definite, ...
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1answer
1k views

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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2answers
32 views

How to describe $\lbrace \mathbf{x}\in \mathbb{R^n}: |x_j|\le1 $ for$ 1\le j\le n \rbrace $ in terms of $x_j=x_j^+-x_j^-$

How to describe the set $A$=$\lbrace \mathbf{x}\in \mathbb{R^n}: |x_j|\le1 $ for$ 1\le j\le n \rbrace $ in terms of $x_j=x_j^+-x_j^-$ where $x_j^+\ge0$ and $x_j^-\ge0$ The answer says: $B$=$\lbrace ...
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0answers
21 views

Moreau Decomposition with Bregman Distance

I am working with a non-Euclidean proximity operator defined by a Bregman distance function $D(\cdot, \cdot)$: $$ \operatorname{prox}_f(x) = \operatorname*{argmin}_u \{ f(u) + D(u, x) \} $$ Is ...
2
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2answers
32 views

How to show $f(x)=(e^x-1)/x, x>0$ is convex?

How does one show that $f(x)=(e^x-1)/x$ is convex on $(0,\infty)$? I plotted the curve and it looks clearly convex. However, when I tried differentiating it, I cannot show the second derivative, $$ ...
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1answer
598 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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1answer
912 views

convex hull function in matlab

Is there any way 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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1answer
29 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 ...
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1answer
17 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 = ...
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1answer
45 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 ...
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27 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 ...
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2answers
82 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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21 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 ...
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0answers
28 views

Proving the global minimum of a continuous piecewise-convex function on a closed interval $[a,b]$

How can prove this clear fact that the global minimum of a continuous piecewise-convex function on a closed interval $[a,b]$, always happens at either extreme points ($a$ or $b$) or the minimum of ...
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1answer
23 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 ...
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1answer
47 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 ...
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1answer
30 views

Convergence of second derivatives of uniformly convergent convex functions

Set-up: Let $\{f_n \}_{n=1}^\infty$ be a sequence of smooth convex functions on $(0,1) \subset \mathbb R$ that converge uniformly to the continuous (not necessarily differentiable) convex function ...
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1answer
19 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 ...
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1answer
400 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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1answer
24 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, ...
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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, ...
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1answer
22 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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1answer
16 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 ...
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35 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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0answers
13 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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123 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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1answer
132 views

Derivative of intersection volume

Let $K$ be a convex body in $\mathbb{R}^n$ and set $f:\textrm{SL}(n)\rightarrow \mathbb{R}$ as $f(T)=\textrm{Vol}_n (TB\cap K)$ where $B$ is the Euclidean unit ball. How can we find extreme points of ...
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1answer
416 views

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
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31 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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2answers
60 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} ...
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1answer
30 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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1answer
32 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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8 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
59 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 ...
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1answer
132 views

Obtaining an estimate for the Lagrangian $L=H^*$ from the Hamiltonian $H$

This is from C. Evans' PDE book, page 130. The convex function $H:\mathbb{R}^n\to\mathbb{R}$ is $C^2$ and satisfies $$ H\big(\frac{p_1+p_2}{2}\big) \leq \frac{1}{2}H(p_1) + \frac{1}{2}H(p_2) - ...
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1answer
50 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) = ...