All Questions
685 questions
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What is the relation between different generalizations of linear programming?
Linear programming subsumed by each of
Semidefinite programming (SDP)
Convex programming (CXP)
SOS programming (SSP)
Is there any relation between each pair in the three?
Are all three equivalent in ...
- 1,826
1
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0
answers
78
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Reference for the algorithm to find the intersection between a subspace and positive orthant
I came across this algorithm, in this question Algorithm for the intersection of a vector subspace with a cone of non-negative vectors ;
Is there any reference for the algorithm described in the ...
0
votes
1
answer
139
views
Linear programming with exponential inequalities and rational variables
If we are given a set of real linear inequalities then using elimination theory or just linear programming we can decide. If the program also has inequalities of form $2^x\leq g$ in addition to linear ...
- 1,826
2
votes
1
answer
591
views
Intersection of a vector subspace with a cone
Given a set of vectors $S=\{v_1, v_2,...,v_d\} \subset \mathbb{R}^{N}, \, N>d$, is there any algorithm to decide if there exist a vector with all coordinates strictly positive in the generating ...
1
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0
answers
67
views
What are the corners of this polytope?
Let $f$ be a non-negative function on the positive integers such that $f(s+t)\geq f(s) + f(t)$ for all $s,t\in\mathbb{Z}^+$. Consider the polytope consisting of all $x\in \mathbb{R}^n$ such that $$\...
- 11
3
votes
1
answer
261
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When is the optimum of an optimization problem affine in the constraint parameter?
While working on a variational problem I have reached to the following question:
Let $f:(0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $f(1)=0$. Suppose that $f(x)$ is strictly increasing ...
- 6,741
7
votes
1
answer
403
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Does the plane clustered to minimize sum distances^2 to clusters centers ( inertia / "K-means") produce hexagonal clusters / hexagonal lattice?
"K-means" is the most simple and famous clustering algorithm, which has numerous applications.
For a given as an input number of clusters it segments set of points in R^n to that given number of ...
- 24.9k
13
votes
3
answers
421
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Maximal distance between $2d+1$ points on the $(d-1)$-sphere
If one arranges $2d$ points on the sphere $\mathbf S^{d-1}\subset\Bbb R^d$ at the vertices of the crosspolytope, then one can achieve a minimal spherical distance of $\pi/2$ between any two points, ...
- 13.6k
0
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1
answer
226
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Fractional values in linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $p$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
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1
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320
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Sub optimal algorithm for linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $c$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
1
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1
answer
195
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Is the minimum of a constraint optimization problem differentiable in the constraint parameter?
Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$.
Let $s&...
- 6,741
2
votes
0
answers
66
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Proving the existence of a dual for an infinite linear program
I am concerned with proving the existence of the dual of an infinite linear program. In addition to the writings of Rockafellar, Luenberger, and Boyd & Vandenberghe on: subdifferentials, Legendre-...
- 121
0
votes
1
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144
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Maximize function on rotation matrices [closed]
Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\...
2
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0
answers
406
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Pros and cons of using integer programming alone or combined integer and global optimization?
First, I am not sure if this is the right question to ask in this forum. But I have been looking for answers for a long time and I have been also asking my university's "engineering" professors but I ...
2
votes
1
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636
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When does $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$ hold for a real function $f(x,y)$?
Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality:
$$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$
For ...
- 884
3
votes
0
answers
163
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A new "adversarial" Wasserstein distance?
Let us consider $\mu_1, \mu_2$ and $\mu_3$ three probability measures living on $[0,1]^{k_1}, [0,1]^{k_2}$ and $[0,1]^k$respectively, with $k_1 +k_2=k$. Let us denote by $\Gamma(\mu,\nu)$ the set of ...
- 555
11
votes
2
answers
963
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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
2
votes
1
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485
views
Odd cycle transversal
Suppose we have a graph G. Say B a fundamental basis of the cycle space of G. Say LP a linear programming problem where there is a variable for each vertex of G, each variable can take value $\geq 0$, ...
1
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2
answers
958
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Integer linear programming (ILP) formulation of connectivity of induced subgraph
Can anyone assist me to find out what should be the ILP formulation of a case when I try to label vertices by say $0$, $1$ and $2$ and want the subgraph of graph $(V,E)$ made by same vertex set but ...
2
votes
1
answer
871
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Interior point of a convex polytope
Suppose the convex polytope is the set of feasible solutions $\mathbf{x}\in\mathbb{R}^n$ for the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}\,,\; \mathbf{A}\in\mathbb{R}^{m\times n}$ subject to ...
2
votes
1
answer
237
views
Sampling algorithms on convex polytopes
Let $f=\mathbf{c}\cdot\mathbf{x}$ be the optimization objective function whose parameter vector $\mathbf{x}\in\mathbb{R}^n$ is subject to the following constraints in the very well-known linear-...
1
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0
answers
75
views
Joint optimization of the order 1 moment of a function and its Fourier transform
For the purpose of a quantum optics experiment, I come to the following problem :
Let $X,P \in \mathbf{R}^2$
\begin{equation}
J(\psi) = | \int_{-X}^{X} x |\psi^2(x) | dx + \int_{-P}^{P} p |Tf(\...
- 11
1
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0
answers
72
views
Distance between value function of deterministic and stochastic control problems
Suppose that one wants to control a diffusion process
$$
dX_t^u = \mu(X_t^u,u)dt + \sigma dW_t; \qquad X_0^u=x
$$
in order to optimize a stochastic control problem with value function
$$
V_T(u)=\...
- 5,405
0
votes
0
answers
43
views
Minimizing along independent directions, nonlinear programming
Good afternoon, I am studying the book Nonlinear Programming: Theory and Algorithms (by Mokhtar S. Bazaraa, Hanif D. Sherali, C. M.) particularly the Theorem $7.3.5$. I'm not sure I understand this ...
- 193
0
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68
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Numerically solve a specific saddle-point problem
Let $(\Omega,\mathcal E,\mu)$ be a probability space, $k\in\mathbb N$, $$W:=\left\{w:E\to[0,\infty)^k:\sum_{i=1}^kw_i=1\;\mu\text{-almost surely}\right\},$$ $G$ be a finite nonempty set and $a^{(g)}:E\...
- 167
2
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0
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47
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A linear program where coordinate descent works pretty well
I am working with a polytope $P\subset \mathbb{R}_+^n$ with the property that there are at about $n!$ minimizers of $\sum_{i=1}^n x_i$, in the following sense:
Select any coordinate $j$ and set $...
- 21
1
vote
1
answer
168
views
Perturbation of the value of a general-sum game at a equilibirium
Consider a general-sum game with $N$ players. Let $u_i(a_1, \ldots, a_N)\colon \prod_{i=1}^N A_i \rightarrow \mathbb{R} $ be the payoff of the player $i\in \{ 1, \ldots, N \}$ when each player takes ...
- 1,127
2
votes
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131
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Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?
Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find.
Let $(E,\mathcal E,\...
- 167
0
votes
0
answers
95
views
How to maximum L1 norm problem?
I have met a problem these days.
\begin{equation}
\underset{\omega}{\max} \quad \Vert \text{diag}(\mathbf{h}^H)\mathbf{G}^H\mathbf{\omega}\Vert_1 \\
s.t.\quad\mathbf{\omega}^H\mathbf{G}\mathbf{G}^H\...
- 171
0
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0
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84
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A question about multivariable calculus and optimization
Consider the function $f(x) :\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x)\geqslant 0\; \forall x\in \mathbb{R}$, and has a set of extremum points at $x_{j}$.
Consider the integral :
$$\int_{\bar{...
- 181
1
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0
answers
282
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total unimodularity of a matrix
Let G be the node-arc incidence matrix of a given directed network (rows of $G$ correspond to nodes and its columns correspond to arcs). Let $B_1,\dots, B_K$ denote a partition of the nodes of the ...
- 393
6
votes
1
answer
609
views
Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers
Let $a(n)$ be the Dirichlet inverse of the Euler totient function:
$$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$
and let the matrix $T(n,k)$ be:
$$T(n,k)=a(\gcd(n,k)) \tag{2}$$
It has been ...
- 1,183
0
votes
0
answers
44
views
Is there a multiplier rule for this minimization problem?
Let $(E,\mathcal E)$ be a measurable space, $W\subseteq\left\{w:E\to\mathbb R\mid w\text{ is }\mathcal E\text{-measurable}\right\}$ be a Banach space, $k\in\mathbb N$ and $f:W^k\to[0,\infty)$. I'm ...
- 167
1
vote
1
answer
130
views
find a PSD matrix that that verify matrices sum of equality
$A $, $ C$ $(n,n)$ are symmetric PSD matrices,
$B$ is PD symmetric matrix, and $H_i$ $\; $ $(i=[1,m])$ represent $ m $ complex matrices. $H_i$ are all one rank matrix
Our objectif is to find ...
- 377
2
votes
2
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421
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Algorithm to determine if a union of half-spaces is all of $\mathbb{R}^d$
I have a collection of closed half-spaces $H_1, \dots, H_n \subseteq \mathbb{R}^d$, each given as $H_i = \{x \in \mathbb{R}^d : a_i \cdot x \geq c_i\}$ for some $a_i \in \mathbb{R}^d$ and $c_i \in \...
- 461
1
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0
answers
105
views
Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related?
I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy:
$$
\inf_{x\in\mathbb{R}^n}\quad p(x),
$$
where $p$ is a polynomial of even degree ...
- 617
0
votes
1
answer
405
views
Computing discrete optimal transport
I am trying to find a combinatorial approach to solve the following optimization problem.
\begin{align}
&\max_{x_{ij}} C_{ij} x_{ij}, \\
&\text{such that},\\
&\sum_{j} x_{ij} \leq r_i~\...
- 133
4
votes
1
answer
204
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Reference: Packing under translation is in NP
I am looking for a reference for a result that I am aware of.
Let me describe the result.
Given a polygon $C$ and polygons $p_1,\ldots,p_n$, it can be decided in NP
time, if $p_1,\ldots,p_n$ can be ...
- 479
1
vote
1
answer
189
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Fritz-John conditions: Equality-constrained case as special case of inequality constraints
In Chapter 4 of Nonlinear Programming: Theory and Algorithms by Bazarra, Sherali, and Shetty, the following claim is made after Theorem 4.3.2 (Fritz-John necessary conditions):
"Note also that these ...
- 11
6
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1
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779
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If $\ell_0$ regularization can be done via the proximal operator, why are people still using LASSO?
I have just learned that a general framework in constrained optimization is called "proximal gradient optimization". It is interesting that the $\ell_0$ "norm" is also associated with a proximal ...
1
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0
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163
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Can we reduce the maximization of this integral to the maximization of the integrand?
I would like to know whether we are able to reduce the following optimization problem to the pointwise optimization of the integrand (or how we can solve it otherwise): Maximize $$\sum_{i\in I}\sum_{j\...
- 167
0
votes
1
answer
99
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Finding dual of a scheduling LP formulation
Suppose I have an LP formulation as such:
$\min\ \ \sum\limits_{i,j,t}\ w_{ij}x_{ijt} (\frac{t-r_j}{p_{ij}}+0.5)$
$\sum\limits_{i,t}\frac{x_{ijt}}{p_{ij}}=1\,\forall\ j$
$\sum\limits_{j}x_{ijt}\leq ...
- 103
0
votes
1
answer
113
views
How do I solve this integer programming problem with non convex constraints?
I am not sure if this is the right place to post this question, please point me to the correct forum if I posted in a wrong place.
I have an optimization problem like this
...
- 103
0
votes
0
answers
101
views
How can we analytically solve this max-sum-min problem?
Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...
- 167
1
vote
1
answer
232
views
Maximize a Lebesgue integral subject to an equality constraint
I want to maximize $$\Phi_g(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)\int\lambda({\rm d}y)\left(w_i(x)p(x)q_j(y)\wedge w_j(y)p(y)q_i(x)\right)\sigma_{ij}(x,y)|g(x)-g(y)|^2$$ over all choices ...
- 167
0
votes
1
answer
61
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Variant of the linear programming problem
Good afternoon, my experience in mathematical programming is low. I would like to know if there is any general method to address the following problem:
$$\text{Minimize }\sum_{i=1}^n d_i(x_j)$$
$$s.a....
- 193
2
votes
0
answers
148
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Generalization of Farkas' Lemma to Hermitian Matrices
I recently stumbled upon a well-known version of Farkas' Lemma which, roughly speaking, I would like to generalize from real vectors to hermitian matrices, as it seems promising for something else I ...
0
votes
0
answers
35
views
Converting a vector in a cone statement to inequality constraints
I would like to convert the following condition for $x$
\begin{align}
x = N \lambda, \lambda \geq 0
\end{align}
to a pure linear inequality form, i.e. find an $L$ and eliminate $\lambda$
\begin{...
- 1
1
vote
0
answers
25
views
Weird subspace/equality-constrained LP problem/variant of change-making problem
Assume that we have a set, $\mathscr{R}$ containing $m$-dimensional vectors. Solve
$$\sum_{i=1}^n c_i\leq\delta$$
$$\text{subject to } \sum_{i=1}^n r_i c_i=x^\prime \text{ for all }x^\prime$$
where
$0\...
- 11
6
votes
1
answer
861
views
Is Binary Integer Linear Programming solvable in polynomial time?
The paper Solving the Binary Linear Programming Model in Polynomial Time claims that Binary Integer Linear Programming is in P. However, it seems that no subsequent literature in the mainstream has ...
- 221