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.

learn more… | top users | synonyms (1)

1
vote
0answers
5 views

Example for a Schur-convex function that is not convex

Let $x \succ y $ be the majorization pre-order on real vectors. (Wikipedia link) We say a function from real vectors to the reals is Schur convex if $x\succ y$ implies $f(x) ≥ f(y)$. With the result ...
0
votes
1answer
25 views

$f:\Bbb R\to\Bbb R$ increasing and convex $\Rightarrow f(x_0)\le f(x)-c(x-x_0)$

Let $f:\Bbb R\to\Bbb R$ such that $f',f''\ge0$ on the whole real line. Then for every $x_0$ fixed, $\exists\; c\in\Bbb R$ s.t. $$ f(x_0)\le f(x)-c(x-x_0)\;\;,\;\;\forall x\in\Bbb R. $$ Now ...
0
votes
0answers
14 views

finding the polar set

the question say find the polar-duals of the following sets in $R^2$ 1) {(x,y):x>=2} 2) {(x,y):x<=2} 3) {(x,y):x=2} the answers are {(x,0):x<=0} , {(x,0):0<=x<=1/2} , {(x,0):x<=1/2} ...
1
vote
0answers
28 views

Matrix convexity!

Given $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, if $\mathsf{rank}(M+Q_i)=\mathsf{rank}(Q_i)$ where $i\in\{1,2\}$ with $Q_i\in\Bbb R_{\geq0}^{n\times n}$, then if $\forall ...
0
votes
0answers
30 views

On the weak* compactness of subdifferentials

Let $X$ be a normed vector space over $\mathbb R$ and $X'$ its dual space (the set of norm-continuous linear functionals on $X$). Let $f:X\to\mathbb R$ be a convex function. Consider the ...
0
votes
0answers
19 views

Is set defined by concave function convex [on hold]

Suppose I have a set defined as follows: \begin{align*} S=\{ x: f(x) \le c\} \end{align*} where $f(x)$ is non-negative function and strictly concave. Is the set $S$ convex or concave? I would really ...
1
vote
1answer
52 views

$f$ convex, $\lim_{x\to\infty}\frac{f(x)}{x}=0$, then $f$ is constant

Let $f$ be a convex function of $\Bbb R$ and suppose $\lim\limits_{x\to\pm\infty}\frac{f(x)}{x}=0$. How we can prove that $f$ is constant function?
0
votes
0answers
22 views

Calculating the Convex hull of a specific set in $\mathbb{R}^3$

I have to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup \{(k,k^2/4-1,-k^2/4-k)|l\le -2\}$. I am aware with the Fenchel-Bunt theorem, so I just have to consider every (closed) ...
-1
votes
0answers
9 views

Minkowski Weyl theorem for integer sets

It states that all points in a polyhedron can be represented as convex combinations of extreme points and conical combinations of extreme rays. This theorem is useful for doing column generation etc ...
0
votes
0answers
18 views

Convex Constraint on Sine Wave Simularity

So lets say you have a vector X = [x1 x2 x3 ..... xn] You want to optimize a cost function over X. However you want to constrain the vector X to look like a sine wave. Say you can parameterize a ...
3
votes
3answers
98 views

Why are convex polyhedric cones closed?

Let $V = \mathbb{R}^n$, $v_1, \dots, v_s \in V$ and let $\sigma = \text{Cone}(v_1, \dots, v_s) = \{r_1v_1 + \dots + r_sv_s \mid r_i \geq 0\}$ be the associated convex polyhedric cone in $V$. Why is ...
0
votes
1answer
13 views

Is multiplication of monotonically decreasing convex functions convex?

I'm aware that if $h(x)$ and $f(x)$ are convex functions, $g(x) = h(x)f(x)$ may not necessarily be convex. I'm curious whether $g(x)$ is convex if both $h(x)$ and $f(x)$ are also monotonically ...
0
votes
1answer
29 views

What is the definition of convexity from $f : \mathbb{R}^2 \rightarrow \mathbb{R}$?

$f(\lambda x + (1-\lambda y) \leq \lambda f(x) + (1- \lambda) f(y)$. This is the definition of convexity I am used to. If $f$ is a convex function, then $f : \mathbb{R} \rightarrow \mathbb{R}$. What ...
0
votes
0answers
17 views

Infinite dimensional convex cone

For every infinite set $I$, the closed convex cone $S:=\{f\in \mathbb{R}^{(I)}:f\geq 0\}$ in $\mathbb{R}^{(I)}$, equipped with the finest locally convex topology, has empty interior. How do I ...
1
vote
1answer
30 views

Proving a function is convex

From the Defintion of convex: Theorem to be proven: If $f$ is differentiable and $f'$ is increasing, then $f$ is convex. Use Proof by Contradiction. Consider, $I = (a, b)$ with $a < x < ...
1
vote
1answer
32 views

How to proof that a straight line can split a convex to at most two regions?

I am self-studying the book "Concrete Mathematics". The authors state the statement: "A straight line can split a convex region into at most two new regions, which will also be convex" 1) How can one ...
0
votes
0answers
8 views

Multivariate Normal Density Concavity

For this variance compunent model $Y$~$N(X\beta, \Omega)$, where $\Omega=\sum_{i=1}^m\sigma_i^2V_i$, the log likelihood function is $(\beta, \sigma_1^1, ..., \sigma_m^2)=C+\frac12\log ...
0
votes
1answer
23 views

Convexity and equality in Jensen inequality

Theorem 3.3 from W. Rudin, Real and complex analysis, says: Let $\mu$ be a probabilistic measure on a $\sigma$-algebra of subsets of a given set $\Omega$. If a function $f:X \rightarrow \mathbb R$ ...
1
vote
1answer
25 views

Does having positive second derivative at a point imply convexity in some neighborhood?

Suppose that I have a real valued function of a single variable $f(x)$ which is twice differentiable in some open interval $I$. Then, I know from calculus that if $f''(x) >0 $ on $I$, then $f$ is ...
0
votes
1answer
27 views

is this a convex optimization problem?

Can someone clarify is this a convex optimization problem or not. $min \| X-UV\|_{F}\quad $ s.t $ \quad U \geq ,V\geq0$ . If not , what makes the problem non-convex?
-1
votes
0answers
19 views

Establish Function is Convex [closed]

I need to establish that the following function is convex with respect to $k$ when $p+q>1$. $$\sum_{n=0}^k \binom k n p^n q^{k-n}$$
-1
votes
0answers
20 views

convex hull of a set in Banach

I know that the closure of the convex hull of a set in a Banach space is a subset of the closed convex hull of it, would you please give me the proof of the inverse ?
1
vote
1answer
22 views

Elements of a Convex Set

Let $ S \subset \mathbb R^n $ be a convex set. Given $ \vec x, \vec y, \vec z \in S $ and three positive numbers such that $ a+b+c=1 $, show that $a\vec x+b\vec y+c\vec z$ is in $S$ also. Ok, so, I ...
1
vote
0answers
15 views

Proof sketch for a convex function, help. [duplicate]

Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) ...
1
vote
1answer
45 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
0
votes
2answers
39 views

Non-linear analysis

1) I am looking for a book which would give the proof of the following theorem(see below). I didn't find any book who does it: in infinite dimension (in Rockafellar Convex analysis book we are in ...
2
votes
0answers
21 views

Preimage of Legendre-Fenchel transform

Let $X$ be a Banach space with dual $X'$, and let $f : X'\to (-\infty,+\infty]$ be a convex lower semicontinuous function. Does there exist some characterization or some nontrivial results concerning ...
0
votes
0answers
25 views

When are the extreme points of a set the bondary?

Let $X$ be a convex compact set. When is the set of extreme points equal to the boundary of $X$? NOTE: by boundary I mean $\overline{X} \setminus \mbox{Int}(X)$.
1
vote
0answers
26 views

sup is bounded or not?

The sup is as following: $c_f = sup_{x,s\in D} \ f(y) - f(x) - (y-x)^Tb$ where $y=x+\alpha(s-x)$, $\alpha \in (0,1 )$ is constant and $b$ is a constant vector. $D$ is a convex compact set and ...
0
votes
0answers
29 views

Proof sketch for a convex function, help.

Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) ...
1
vote
1answer
20 views

Proving a function of matrix is convex

I have a function of a matrix and a vector $f(A,b)=y^\top (I-A)^{-1} b$ and I want to know the conditions under which it is convex. For functions of a vector, the positive definiteness of the Hessian ...
1
vote
0answers
15 views

Term for a Convex Function whose derivative is also convex

Let $f(x)$ be a monotone non-decreasing convex function such that its derivative $\frac{d}{dx}f(x) = f'(x)$ is also a convex function. Is there a term in literature that is used to refer to such ...
0
votes
1answer
25 views

Confusion on convex functions

I got a problem while solving a problem regarding convex functions on an interval $(a,b)$. What I had to show is if $f$ is convex then $f'$ exists except possibly at countably many points and is ...
0
votes
0answers
18 views

uniform convergences of convex functions

Let $f_n(\cdot)$ be a sequence of continuous and convex function on $\mathbb{R}^d$, and be supported on a full dimensional compact convex set $D$. If $f_n(\cdot)$ converges point-wise to $f$ in the ...
0
votes
1answer
17 views

Prove that the Cartesian Product of two Convex Sets is a Convex Subset

Here's the problem: Suppose that $S\subset \mathbb R^m$ is a convex set and $T\subset \mathbb R^n$ is a convex set. Show that the set $$S \times T = \{ (x_1 ,...,x_{m+n}\in \mathbb R^{m+n}):(x_1 ...
0
votes
2answers
31 views

Proving Convexity of an Open Disk

I need to prove that the following set is convex: $$ \{(x,y):x^2 +y^2 \lt 2\} $$ Obviously, this an open disk of radius $\sqrt2$. My intuition is to use triangle inequality for this proof because a ...
1
vote
1answer
39 views

How to derive the solution in quadratic optimization

I'm reading the book "Convex Analysis and Optimization" written by Prof. Bertsekas. In Example 2.2.1, there are the following description: I don't know how to ...
1
vote
1answer
24 views

Example of maximal monotone operators in non-reflexive Banach spaces with applications in PDE

My question is about examples of maximal monotone operators that are defined in non-reflexive Banach spaces and have applications in PDEs, variational inequalities, etc (any application actually)? If ...
0
votes
0answers
13 views

Proving disjoint sets and finding separating hyperplane

Consider \begin{equation} \mathbf{h}=\begin{bmatrix} h_0 & h_1 & \cdots & h_p \end{bmatrix} \end{equation} where $h_i \in \mathbb{R}\forall i=1:p$ and is known. \begin{equation} ...
0
votes
1answer
19 views

Projection on Epigraph of a convex function

Given a convex function $h:\mathbb{R}^n \mapsto \mathbb{R}$, and a point $(x,\alpha) \in \mathbb{R}^n \times \mathbb{R}$, how can I find a closed formula to compute the projection of $(x,\alpha)$ in ...
0
votes
0answers
18 views

Is the sum of logarithms of l1-normed positive vectors concave?

Consider the following function, $f(x) = \sum_{i} \log \frac{x_i}{\sum_k x_K}$ defined on the set of all real vectors with strictly positive entries, $x_i > 0$. The problem $\max f(x)$ has the ...
0
votes
0answers
13 views

Is there a proof for this result (about the inclusion $y \in [\partial h + T](x)$, with $T$ maximal monotone)?

I am trying to find a place (book, paper) which has a proof of this result: "Let $T: \mathbb{R}^n \mapsto 2^{\mathbb{R}^n}$ be maximal monotone, $\Omega \subset \mathbb{R}^n$ a closed convex set ...
0
votes
3answers
65 views

Is the norm of a convex function convex?

I know that the norm of $x\in R^n$, $(\sum\limits_{i=1}^n|x_i|^2)^{0.5}$ is a convex function. Also, not any composition of two convex functions is convex. So my question is: Lets say we have a real ...
0
votes
1answer
20 views

Why is a local min also a global min for convex functions?

As the title states, for an unconstrained minimizaton problem, of a convex function, why is it that the local minimum is also the global solution?
2
votes
0answers
31 views

the continuity of argmin on convex funtion

Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t ...
1
vote
2answers
58 views

Lotka-Volterra First Integral and Fixed Point

I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by: \begin{align} & ...
1
vote
1answer
40 views

intuitive meaning of sphericity

i interested in the following definition but i don't understand it because i don't understand what mean by "flat space generated by C" . the same definition is given by i have also the same ...
0
votes
0answers
28 views

How to prove that convex function has an increasing slope?

A function $f(x)$ in some domain $a\leq x \leq b$ is convex if and only if for any $x_1 < x_2 < x_3$ from domain $[a,b]$, $$\frac{(f(x_2)-f(x_1))}{(x_2-x_1)} \leq ...
0
votes
0answers
22 views

Non-differentiable regions of convex functions of more than one variable

I've heard that a convex function of a single variable is continuous in the interior of its domain, and is differentiable everywhere with the possible exception of a countable number of points. (I ...
0
votes
0answers
14 views

Prove subdifferentiable of convex lower semicotinuous function

I have read the following statement from Page 53 of Villani's book "Topics in Optimal Transportation": If $f$ is convex, lower semicontinuous, then $\partial f(x)\not=\emptyset$ for all $x\in ...