All Questions
1,018 questions
1
vote
0
answers
28
views
Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
-1
votes
0
answers
41
views
Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
2
votes
1
answer
875
views
Interpreting mincost flow dual variables
Consider the task of finding flow of size $b$ with minimum possible cost.
It may be formulated as linear programming in a following way:
$$\boxed{\begin{gather}
\min\limits_{f_{ij} \in \mathbb R} &...
CommunityBot
- 1
2
votes
1
answer
128
views
Reference request for elementary convex geometry property
I need to use the following lemma for a proof. It is an elementary result, which I am sure is well known, just I am not familiar enough with the relative literature to find a direct reference. Is ...
- 128k
0
votes
1
answer
104
views
Optimality condition for strongly convex function under sparsity constraint
Let $f: \mathbb{R}^p \to \mathbb{R}$ be a $2s$-sparse strongly smooth, $2s$-sparse strongly convex and twice differentiable function. In other words, there exists positive constants $\alpha, L >0$ ...
CommunityBot
- 1
0
votes
2
answers
530
views
Any idea of solving an optimization problem with cubic constraints?
I have the following optimization problem with cubic constraints, which is hard to solve. Are there any ideas, or related references, of solving such a problem?
$$ \begin{array}{ll} \underset {y, z} {\...
CommunityBot
- 1
2
votes
1
answer
91
views
Does approximately null gradient imply approximately global minimum for convex functions?
Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a non-negative and differentiable convex function which vanishes in a non-empty convex set $\Omega$ - possibly unbounded. Usually, when one ...
- 128k
12
votes
1
answer
627
views
A function with unexpectedly simple Legendre transformation
Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...
CommunityBot
- 1
2
votes
0
answers
67
views
Two-terms Euler-Maclaurin formula for concave functions over polytopes
Let $P\subset\mathbb{R}^{n}$ be a lattice polytope (vertices are
in $\mathbb{Z}^{n}$). Set $P_{k}=P\cap k^{-1}\mathbb{Z}^{n}$ for
$k\geq1$. Given a concave function $\phi:P\rightarrow\mathbb{R}$
(...
- 73
0
votes
0
answers
21
views
Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
1
vote
1
answer
350
views
How to prove the convexity of a simple function involving a ratio of two polygamma functions?
Let
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0
\end{equation*}
and
$$
\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}.
$$
In the literature, ...
CommunityBot
- 1
7
votes
2
answers
242
views
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
- 109k
4
votes
1
answer
318
views
Does smoothing a non-log-concave distribution make it more log-concave?
Suppose that $p$ is a density on $\mathbb{R}^d$ that is $C^2$ and nonzero everywhere, and such that the Hessian of its negative logarithm is lower bounded:
$$-\nabla^2 \ln p\succeq L$$
for some matrix ...
CommunityBot
- 1
1
vote
1
answer
115
views
$\mathrm{LP}$ formulation for $\mathrm{k}$-$\operatorname{opt}$ moves
$\mathrm{k}$-$\operatorname{opt}$ moves are an idea to improve non-optimal Hamilton cycles in weighted symmetric graphs by exchanging $\mathrm{k}$ tour-edges with $\mathrm{k}$ edges that do not belong ...
CommunityBot
- 1
1
vote
0
answers
60
views
Behaviour of the solutions of parametrized multivariable non-linear (non polynomial) system of equations
The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying
$$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
- 745
18
votes
2
answers
1k
views
The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions
I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty ...
2
votes
4
answers
212
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 63.9k
1
vote
1
answer
40
views
Envelopes of functions with respect to some convex cone $\mathcal{F}$
Let's say we are given a function $f:\mathbb R ^d\to \mathbb R$ continuous. Assume that $\mathcal F$ is a convex cone of continuous functions ($\mathbb R^d$ to $\mathbb R$) closed under maxima. I am ...
- 1,714
1
vote
1
answer
106
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 128k
1
vote
1
answer
331
views
Finding a special solution in a solution set over F2
Given a solution set of a linear system of the following form
$$
\{ \begin{bmatrix}
x_{1} \\
\vdots \\
x_{n}
\end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
CommunityBot
- 1
2
votes
1
answer
132
views
Points of differentiability of convex functions
Let $U$ be an open neighbourhood of $0 \in \mathbb{R}^2$ and $f\colon U \to \mathbb{R}$ a convex (and bounded) function. Denote by $D \subset U$ the set of points on which $f$ is totally ...
- 128k
11
votes
2
answers
430
views
On the convex cone of convex functions
$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of ...
- 91.8k
7
votes
0
answers
248
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
2
votes
0
answers
60
views
Test probability distributions increasing in convex order on $\mathbb R^2$?
Two probability distributions $\mu, \nu$ on $\mathbb R^d$ are said to be increasing in convex order if
$$\int_{\mathbb R^d} |x|\mu(dx) + \int_{\mathbb R^d} |x|\nu(dx)<\infty$$
and
$$\int_{\mathbb R^...
- 1,334
3
votes
1
answer
368
views
Lot sizing problem: how to add these cuts efficiently
Consider the set of constraints of the uncapacitated lot sizing problem:
$$
\{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
CommunityBot
- 1
0
votes
1
answer
114
views
Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
CommunityBot
- 1
0
votes
1
answer
396
views
What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
CommunityBot
- 1
6
votes
0
answers
304
views
Distribution class closed under convolution counterexample?
Define the class of probability density functions $\mathcal{C}$: $\,p \in \mathcal{C}$ iff $p(x)=p(|x|)$, and $\log p(\!\sqrt{x})$ is convex on $[0,\infty)$.
Conjecture: if $p,q \in \mathcal{C}$, then ...
- 391
13
votes
2
answers
1k
views
Probability vector $p$ majorizes its normalized entropy vector $\small \frac{-p\log p}{H(p)}$
I guess the following inequality
$$ \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right ) \le \sum_{i=1}^n g (p_i)$$
holds for any continuous convex function $g$ and any probability ...
- 303
1
vote
0
answers
41
views
Convex 1-semiconcave functions : extreme points
Let $C$ be the set of functions $f:\mathbb{R}^n \to \mathbb{R}$ such that $x\mapsto f(x)$ and $x\mapsto \lVert x \rVert^2 - f(x)$ are both convex.
If $f$ belongs to $C$, then $f$ plus any affine ...
- 2,772
1
vote
0
answers
73
views
Convexity and subdifferential monotonicity
Do you know any reference where I can find some results in this sense:
Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
- 1,759
6
votes
1
answer
287
views
Determinantal inequality for difference of substochastic matrices
Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if
$$
\forall i,\ \ \sum_j A_{ij}\le 1\ .
$$
In the course of my ...
- 785
1
vote
0
answers
99
views
Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
- 988
4
votes
1
answer
416
views
Maximum and concavity of function
Let
\begin{align}
G(x_1,x_2,x_3)=\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi} \frac{d\theta_1 d\theta_2 d\theta_3}{(2\pi)^3} p(\vec{x},\vec{\theta}) \log\left( p(\vec{x}, \vec{\theta})\right)
\end{...
- 61.9k
2
votes
0
answers
58
views
An s-convex function lying between two convex functions
Let $f: \mathbb R_{+} \to \mathbb R_{+}$ be an $s$-function in the second sense, i.e.,
$$ f(\lambda x +(1-\lambda)y) \leq \lambda^s f(x) +(1-\lambda)^s f(y)$$ for every $\lambda \in (0,1)$. Assume ...
- 55
2
votes
0
answers
123
views
Continuity of entropies, replica trick and Hausdorff moment problem
I could not find a really appropriate title for my question (happy to revise) but let me explain.
Suppose $p(x|c)$ is a probability density function over $x \in [0,1]$ which depends continuously on ...
- 231
29
votes
6
answers
8k
views
How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 221
2
votes
0
answers
318
views
What's the number of facets of a $d$-dimensional cyclic polytope?
A face of a convex polytope $P$ is defined as
$P$ itself, or
a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
- 53
3
votes
0
answers
110
views
How many Tverberg partition are in cloud of points? [closed]
Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect.
For example, $d=2$, $r=3$, 7 points:
Let $p_1, p_2,...
- 214
0
votes
0
answers
48
views
A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,214
3
votes
1
answer
568
views
Directional derivates and unique subgradients
I have a question about the fine structure of convex functions. Convex functions behave very regular in the interior of their domain of definition (e.g. they are locally Lipschitz continuous there) ...
- 12.9k
6
votes
1
answer
366
views
An arrangement of hyperplanes [closed]
An arrangement of hyperplanes in $\mathbb{R}^d$ is simple if the hyperplanes are in general position (for every $1\leq k\leq d+1$, the intersection of $k$ hyperplanes is $(d-k)$-dimensional).
My ...
- 91.8k
2
votes
0
answers
104
views
Reference request: books on convex analysis / geometry
I am interested in convex geometry and analysis, especially in its connections with high dimensional probability theory.
I was reading the book by Pisier, The volume of convex bodies and Banach space ...
- 4,713
2
votes
1
answer
308
views
Concavity of hypergeometric function ratio
I would like to show that the function,
$$
f(x) = \frac{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c+1\,;x\big)}{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c\,;x\big)}
$$
is concave for $0 < x &...
- 391
0
votes
1
answer
150
views
When are infimal convolutions contractions?
Let $X$ be a separable Fréchet space and $\varphi,\psi:X\to \mathbb{R}$ be a lower semi-continuous and convex function with $\psi$ bounded below and coercive. Consider the infimal convolution
$$
\...
- 1,714
1
vote
1
answer
39
views
Does uniform convergence of suitable functions yield pathwise convergence of their convex envelopes?
For each $k\ge 1$, let $f_k:\mathbb R\to\mathbb R_+$ be $1-$Lipschitz, increasing such that $f_k(x)\ge x^+$ for $x\in\mathbb R$, $f_k(-\infty)=0$ and
$$\lim_{x\to+\infty} \big(f_k(x)-x\big)=0.$$
...
- 128k
3
votes
2
answers
905
views
Is a convex, lower semicontinuous function that is bounded from below, actually continuous?
While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space.
Lemma: Let $f ...
- 11
0
votes
0
answers
115
views
Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
- 1
5
votes
1
answer
176
views
Efficient counting of integer solutions to linear system
In my research, I have a particular 18x18 matrix $\mathbf{A}$ which defines the linear system $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{-1}$ over the nonnegative integers. And I'm interested in ...
- 5,429
1
vote
1
answer
94
views
Why are symmetric convex bodies with a smooth boundary and non-vanishing Gaussian curvature of particular interest in harmonic analysis?
I don't work in harmonic analysis or convex analysis, but in some literature of harmonic analysis, I often see the assumption that "let $K$ be a symmetric convex body with a smooth boundary and ...
