All Questions
1,018 questions
35
votes
4
answers
5k
views
Why are optimization problems often called "programs"?
Why are optimization problems often called programs?
linear programming
geometric programming
convex programming
Integer programming
...
- 461
30
votes
4
answers
4k
views
Elementary applications of Krein-Milman
This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try.
Recall that the Krein-...
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 ...
- 391
27
votes
5
answers
2k
views
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,121
25
votes
2
answers
2k
views
An Interesting Optimization Problem
You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
- 363
25
votes
3
answers
2k
views
Is the Ford-Fulkerson algorithm a tropical rational function?
The Ford-Fulkerson algorithm
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
- 33.8k
24
votes
3
answers
1k
views
Average measure of intersection of a convex region with its translate
Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.
My question is about
$$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$
How ...
- 343
23
votes
2
answers
2k
views
Are such functions differentiable?
In my recent researches, I encountered functions $f$ satisfying the following functional inequality:
$$
(*)\; f(x)\geq f(y)(1+x-y) \; ; \; x,y\in \mathbb{R}.
$$
Since $f$ is convex (because $\...
- 995
21
votes
2
answers
2k
views
A strange variant of the Gaussian log-Sobolev inequality
Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function, and assume that it grows at most linearly at infinity for simplicity. Denote by $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$, ...
- 562
20
votes
3
answers
2k
views
Convergence of convex functions
I can prove the following result.
Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
Then ...
- 28k
19
votes
4
answers
1k
views
Applications of linear programming duality in combinatorics
So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
- 997
19
votes
2
answers
2k
views
Is the tensor product of polyhedra a polyhedron?
Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
- 33.8k
18
votes
3
answers
3k
views
Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
- 667
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 ...
- 385
18
votes
3
answers
997
views
Convex functions in convex sets
Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...
18
votes
1
answer
3k
views
How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
- 409
17
votes
3
answers
6k
views
The cone of positive semidefinite matrices is self-dual? (reference needed)
I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long time,...
- 1,513
17
votes
3
answers
2k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
- 2,288
17
votes
1
answer
861
views
Extreme points of convex compact sets
Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an ...
- 109k
16
votes
3
answers
1k
views
Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 163
15
votes
1
answer
1k
views
Second order differentiability of convex functions
Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...
- 28k
14
votes
1
answer
4k
views
Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension
By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
- 193
14
votes
1
answer
2k
views
An intuition for three different types of subgradients (proximal, regular, limiting)
I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking.
These subgradients are (assume $x \in$ ...
- 249
14
votes
0
answers
310
views
How large are the smallest-area projections of a high-dimensional convex body?
Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that
$$\intop_B x \, dx = 0$$
$$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$
a ...
- 4,000
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
13
votes
2
answers
664
views
Complexity of a weirdo two-dimensional sorting problem
Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...
- 19.2k
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
13
votes
1
answer
1k
views
Aleksandrov's proof of the second order differentiability of convex functions
Aleksandrov [A], proved a remarkable property of convex functions.
Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...
- 28k
13
votes
3
answers
834
views
Famous theorems that are special cases of linear programming (or convex) duality
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any ...
13
votes
2
answers
720
views
Regularity of convex sets in $\mathbb{R}^n$
The following result is Proposition 2.4.3 in [1]:
Theorem. Let $K\subset\mathbb{R}^n$ be a bounded convex set with the non-empty interior. Then $\partial K\in C^{1,1}$ if and only if
there is $r>0$...
- 28k
13
votes
1
answer
2k
views
Minimize sum of $\ell_2$ norm and linear combination, on simplex
Let $\Delta_n := \{x \in \mathbb{R}^n | x \ge 0, \sum_{1 \le i \le n}x_i = 1\}$ be the $n$-simplex. For $a, b \in \mathbb R^n$, with $\Delta_n \not \ni a$, consider the problem of computing the ...
- 6,853
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
12
votes
2
answers
752
views
Geometric applications of Ekeland's variational principle
I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself:
Definition. Let $(X,d)$ be a ...
- 13.5k
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 \\
\...
- 1,608
11
votes
2
answers
963
views
Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?
Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...
- 1,826
11
votes
3
answers
6k
views
Random Sampling a linearly constrained region in n-dimensions...
Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...
- 113
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 ...
- 128k
10
votes
4
answers
3k
views
Convexity and Lipschitz continuity
It is probably an easy question, but somehow I am stuck.
Question Is the following statement true? If yes, how to prove it?
Suppose that $f\in C^1(\mathbb{R}^n)$ is convex and
$$
\langle\nabla f(x)-\...
- 28k
10
votes
2
answers
3k
views
How do you tell if a system of linear inequalities has a solution?
A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
- 693
10
votes
2
answers
923
views
Constructing an independent uniform random variable from two independent ones
Does there exist a continuous (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable $...
- 355
10
votes
2
answers
2k
views
Continuous functions with convex level sets
Assume that $f:\mathbb{R}^{2}\to \mathbb{R}$ is a continuous function such that each level set $f^{-1}(c)$ is a convex set.
To what extent such functions are studied?
In particular:
Is there a ...
- 356
10
votes
3
answers
6k
views
Solving a system of linear inequalities -- what is the dimension of the solution set?
It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$?
For the applications I have in mind the ...
- 7,857
10
votes
3
answers
1k
views
Efficient computation of "discrete infimal convolution"
This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...
- 8,491
10
votes
1
answer
4k
views
Uniform convergence of convex functions
It is a well-known result that if a sequence of convex function $f_n(\cdot)$ converges on a dense set $C'$ of an open set $C$, then the limit function $f$ exists on $C$, and the converge is uniform ...
- 599
10
votes
1
answer
2k
views
Sum of difference moduli vs. sum of modulus differences
This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.
Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...
- 33.8k
10
votes
1
answer
411
views
Network flows with capacities on pairs of edges
Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$.
Now add edge-pair ...
- 37.7k
10
votes
0
answers
722
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
9
votes
1
answer
6k
views
Proving that a binary matrix is totally unimodular
I'm working on a set of problems for which I can formulate binary integer programs. When I solve the linear relaxations of these problems, I always get integer solutions. I would like to prove that ...
- 213
9
votes
1
answer
3k
views
Inverse of a totally unimodular matrix
A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...
- 93
9
votes
2
answers
843
views
How did they come up with the MRRW bound?
Among the good asymptotic bounds in coding theory in the MRRW bound. It is obtained by using the linear programming problem of Delsarte's and providing a solution. The LP problem is
Suppose $C \...
- 885