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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Convex Extremal
I was asked whether this is true in a question paper: If p is a subset of q (where both p & q are convex), then an extreme point of q is also an extreme point of p. Ans.: Yes statement is correct.
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3answers
82 views
Simple proof? If $x$ lies outside a compact convex set, there exists a $y$ closer to every point in the set than $x$.
This seems rather obvious intuitively, but I can't find a simple proof.
If $C$ is a compact, convex subset of $\mathbb{R}^n$ and $x \not \in C$, then there exists a point $y$ such that, for every ...
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1answer
86 views
Is this function convex or non-convex?
Let $$f(a,b,c,d)=\frac{(a-b)*(c-d)}{\sqrt{(a-b)^2+(c-d)^2}},$$
where $a,b,c,d$ are variables.
Is this function convex or non-convex?
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0answers
28 views
Is this question stated wrong?
I'm trying to check whether this question might be worded wrong, and here it is:
Show that if $A$ is a convex subset of a topological vector space $X$, $u \in A^o$ (the interior of $A$), $v \in ...
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1answer
43 views
Questions regarding internal and interior points for a convex subset of a topological vector space
Suppose that $X$ is a topological vector space, with a convex subset $A$. How do we show that if the vector $u$ is in the interior of $A$, then $u$ is an internal point of $A$ and if the interior of ...
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1answer
89 views
How to determine whether a function of many variables is convex or non-convex?
A function $f$ is convex if $$f(\theta x + (1 − \theta)y) \leq \theta f(x) + (1 − \theta)f(y)$$ for all $x, y \in \mathcal{D}(f)$, the domain of $f$, and $\theta \in [0, 1]$.
How do I determine ...
3
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1answer
85 views
Intersection of two $n$-dimensional quadratic inequalities?
I have two quadratic inequalities of the form
$$
a_1x^TAx + b_1^Tx + c_1 \le 0\\
a_2x^TAx + b_2^Tx + c_2 \le 0
$$
where $A\in\mathbb{R}^{n\times n}$ is positive semidefinite, $x\in\mathbb{R}^n$, ...
2
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2answers
93 views
Sets whose intersection with line segments have finite components
Certain subsets $A$ of $\mathbb{R}^n$ satisfy the property that given any line segment $L$, $A \cap L$ has a finite number of connected components. For instance, $A$ can be any finitary union of ...
2
votes
1answer
57 views
Convex functions and uniform convergence of derivatives
Let $f_n:[0,1]\to\mathbb{R},\ n\in\mathbb{N}$ be a sequence of convex analytic functions.
Consider the sequence of derivatives $(f_n')_{n\in\mathbb{N}}$. Suppose that ...
3
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1answer
55 views
A particular ILP where the existence of a relaxed solution implies the existence of an integer solution
This question emerged from a discussion on my previous question Determining quickly whether this Integer Linear Program has any solution at all, which I would like to elaborate separately.
I am ...
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0answers
73 views
Numerical Methods for minimizing a Non-Differentiable Convex Function of Several Variables
I have a multi-variable convex continuous function which is not differentiable. I am interested to know about different numerical techniques, possibly also references to them, used for this.
Read ...
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1answer
80 views
Edges of the convex hull of a finite point set
I'm looking for a formal proof of a statement that seems obvious (and yet may be wrong) to make sure one of my algorithms is correct.
Given a set S of N points in $\mathbb{R}^3$, suppose we have a ...
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1answer
54 views
Direction of recession, convex analysis
Hi how to show the following:
Let $C$ and $D$ be two non-empty closed convex sets
with no common direction of recession. Then $C - D$ is closed.
Thanks a lot...
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1answer
25 views
Intersection of a 2-Dimensional body and a Line given west-most point and south-most point
I have a 2-Dimensional Closed Convex Compact Body $\mathbb{S}$(a set, for eg, a circular disc)). Assume, it has a non-zero intersection with the all-negative quadrant. Consider the following 2-D point ...
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0answers
23 views
Literature suggestion for “strong convexity”
Does anyone know of a reference that discusses strong convexity and strong smoothness of proper convex functions over Banach spaces?
All the references I find only deal with the finite dimensional ...
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0answers
65 views
Convexity of affine function.
Can someone help me with a proof that affine function preserves convexity?
Given that $f$ is convex, $A$ is in $\mathbb{R}^{M\times N}$ and $b$ is in $\mathbb{R}^m$ then show that $g(x) = f(Ax+b)$ is ...
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1answer
73 views
affine set convex set
How to show the following:
Let $C$ be a convex subset of $\mathbb R^d$. Then $\operatorname{int} C \neq \emptyset$ if and only if
$\operatorname{aff} C = \mathbb R^d$ where $\operatorname{aff} C$ is ...
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0answers
27 views
affine subset closedness
How to show an affine subset $C$ of $R^d$ is closed (yes convergence in norm sense)
I thought the following: Let $x_0$ be an accumulation point $C$ define
$C - C + x_0$ then $C-C$ being closed $C - C ...
2
votes
1answer
238 views
Prove every local minimum is a global minimum
Let $Q\in\mathbb{R^{dxd}}$ and $A\in\mathbb{R^{d'xd}}$ be two matrixes and $b\in\mathbb{R^d}$, $c\in\mathbb{R^{d'}}$. Suppose $d'\lt d $. For $x\in\mathbb{R^d}$.
Minimize
$$f(x)= ...
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2answers
95 views
proving this inequality related to conjugate functions
For $x \in \mathbb{R}^n$ let us denote $x_{[i]}$ the $i$th largest component of $x$ s.t
$$
x_{[1]} \geq x_{[2]} \geq x_{[3]}\ge\cdots
$$
The function
$$
f(x)= \sum_{i=1}^r x_{[i]}
$$
is the sum of ...
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vote
1answer
75 views
Continuous boundary of a convex set
Is the boundary of a compact convex set in Rn continuous? Seems like the answer should obviously be yes, but I cannot find any such result in the literature. Can somebody provide a reference (or a ...
4
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1answer
159 views
The composition of two convex functions is convex
Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
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0answers
29 views
Mid-Point Convexity Implies Convexity in Banach Spaces? [duplicate]
Possible Duplicate:
Showing that $f$ is convex
Let $X$ be a real Banach space and $f:X\rightarrow \mathbb{R}$ a continuous function. We say that $f$ is Mid-Point convex if for all $x,y\in ...
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2answers
86 views
Is $f(x_1,x_2)=x_1x_2$, with $x_1,x_2 \in \mathbb{R}_+$, a convex function?
Let $f(x_1,x_2)=x_1x_2$, with $x_1,x_2 \ge 0$. Is $f$ a convex function? Why?
$\ \\$
Edit (in view of the comments below)
The Hessian matrix is $H=[0\, 1; \,1 \,0]$, which is indefinite (in ...
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0answers
17 views
Formula for finding integer points in fundamental parallelepipeds — reference for proof?
Does anyone know of a text where I can find a proof like the one found in Lemma 5.2 of here?
It's a formula for the integer points inside the fundamental parallelepiped of a simplicial integral cone. ...
1
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1answer
98 views
Jensen's inequality and a estimate in $L^p$
In problem 3 we have:
If $f:\mathbb{R} \longrightarrow\mathbb{R}$ is mensurable, $E:=\mathrm{supp}\ f$ and
$$\int_E e^{|f(x)|}dx =1,$$
then $f\in L^p(\mathbb{R})$, for all $p\in(0,\infty)$ and
...
2
votes
1answer
42 views
How to test the convexity of mutual information using leading principal minors?
I read from textbooks that the mutual information function $I(X;Y)$ is a concave function of $p(x)$ for fixed $p(y|x)$ and a convex function of $p(y|x)$ for fixed $p(x)$.
I tried to test the ...
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votes
2answers
86 views
A standard quadratic minimization problem
Consider the "Complex" Quadratic minimization problem
\begin{align}
\min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1
\end{align}
...
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1answer
54 views
Mathematical expectation is inside convex hull of support
Let $\xi$ be a random variable supported in some set $A \in \mathbb{R}^n$: $\xi \in A$ a.e. How to show that $\mathsf E \xi \in \mathop{\mathrm{conv}}{A}$?
Let $s(x)$ be a support function of set ...
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votes
2answers
108 views
How should I prove a set is convex?
Given a set $$ \mathbf{S} = \{ \mathbf{x}\:|\: \mathbf{x}^T\mathbf{V}\mathbf{x}=1 \} $$ where $\mathbf{V}$ is a positive semidefinite matrix. How to prove this set is convex?
I tried in the ...
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1answer
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Convex subset of Hilbert space as intersection of closed balls
How does one prove that any closed, convex, and bounded subset of a Hilbert space is the intersection of the closed balls that contain it?
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About the problem 20 chap 3 (functional analysis, Walter Rudin) [duplicate]
Possible Duplicate:
I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.
Let $\{u_1,u_2,u_3,\dots \}$ be sequence of pairwise orthogonal unit vectors in Hilbert ...
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1answer
122 views
Convex hull of extreme points
Let $P\subseteq \mathbb{R}^{n}$ is convex hull of finite points: $P=conv(x_1,x_2,\ldots,x_m)$.
I need to show that $P$ is convex hull of its extreme points.
I am thinking about such proof.
Let ...
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1answer
131 views
Do projections onto convex sets always decrease distances?
Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that:
...
2
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1answer
97 views
Why a closed bounded convex set in $\mathbb{R}^{n}$ always has an extreme point?
Let $X\subseteq \mathbb{R}^{n}$ is closed, bounded convex set. How to prove that $X$ contains such point $x$ that we can't represent as $x=\frac{1}{2}x_{1}+\frac{1}{2}x_{2}$ where $x_1\in X$ and ...
3
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0answers
51 views
On the duality gap for quasiconvex optimisation problems
This stack exchange question got me thinking about quasiconvex analysis.
Given a compact,convex subset $X\subset \mathbb{R}^n$ and a quasiconvex function $f:X\rightarrow \mathbb{R}$
Define the ...
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1answer
144 views
Prove that the spaces have the same homotopy type
This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima.
"Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space ...
3
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0answers
44 views
Two convex cones
Consider a convex cone $C$ (in some real vector space), with $C_1$ a convex subcone. I'm led to study elements $v\in C_1$ with the following "relative form" of extremality: if $v=x+y$, with $x,y\in ...
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1answer
209 views
What is the probability of having a pentagon in 6 points
If the probability that $5$ random points in the plane whose horizontal
coordinate and vertical coordinate are uniformly distributed on the
interval $\left(0,1\right)$ occur to be the vertices of a ...
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1answer
214 views
Expected size of subset forming convex polygon.
If there are $4$ random points in the plane whose horizontal coordinate
and vertical coordinate are uniformly distributed on the interval
$\left(0,1\right)$, what is the expected largest size (or ...
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1answer
221 views
Approximating the area below average of a concave function
Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define
\begin{align*}
F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\}
...
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0answers
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Convex analysis books and self study.
I have taken some courses in Convex optimization. Now I would like to know a little bit more about the pure mathematical side. Is there any good books in convex analysis?
I have read and worked with ...
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2answers
101 views
Codimension of the largest subspace of a closed convex cone in a normed vector space
Please, help to prove (or disprove) the following statement.
Let $K \neq V$ be a closed convex cone ($K+K=K$, $\alpha K \subseteq K$ for any $\alpha \geq 0$) in a real normed vector space $V$. If $-K ...
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0answers
38 views
non-concave function with all local maxima equals global maxima
Is there a special name for the functions in which all the local maxima are also the global maxima?
For e.g. sinusoidal wave is non-concave and it has multiple maxima. But each of the maxima are ...
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0answers
41 views
Minimize a complex quadratic subject to two convex quadratic constraints
I have the following the optimization problem
\begin{align}
\min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\}
\\\ ...
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1answer
50 views
Confusion regarding the convexity of a function
I want to know how come the function f(y)=1/y is convex?
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1answer
38 views
Showing that $T+S$ is firmly nonexpansive
Show that $T+S$ is firmly nonexpansive considering that $T$ and $S$ are firmly nonexpansive mappings from $\mathbb R^n$ to $\mathbb R^n$.
Definition: We say that $F$ is firmly nonexpansive if: ...
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0answers
33 views
Parameterized convex optimization
I'm trying to formulate a game so that at Nash equilibrium I achieve supply equales demand. Then I ran into this problem.
For all $i,$ $v_{i}\left(x_{i}\right)$ is concave
in $x_{i}$. The value ...
1
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1answer
74 views
proving compactness and convexity of a set
Suppose functions $f(x)$ and $g(x)$ are continuous with domain $X \subset \mathbb{R} $ which is nonempty, convex and compact, can we show that
$$S \equiv (f(x), g(x)) $$ for all $x \in X$ is ...
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0answers
47 views
Closed form for Lagrange dual
Can Lagrange dual always be computed in closed form? Can you give me a simple example where the dual is not analytically computable?
