Questions tagged [linear-programming]
Linear programming is the study of optimizing a linear function over a set of linear inequalities. The Simplex Method, Ellipsoid Method and Interior Point Method are popular algorithms to solve linear programs.
485
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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 ...
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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 ...
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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 ...
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1
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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 ...
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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 $$\...
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1
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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$ ...
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295
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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$ ...
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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-...
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1
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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=
\...
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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 ...
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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 ...
2
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1
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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$, ...
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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
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1
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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 ...
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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-...
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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 ...
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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 $...
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1
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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 ...
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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 ...
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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 ...
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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 ...
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2
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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 \...
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1
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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~\...
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1
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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 ...
6
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1
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648
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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 ...
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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\...
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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 ...
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1
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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
...
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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 $$\sum_{i\...
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Variant of the linear programming problem
Good afternoon, my experience in mathematical programming is low. I would like to know if there is any general method to address the following problem:
$$\text{Minimize }\sum_{i=1}^n d_i(x_j)$$
$$s.a....
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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 ...
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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{...
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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\...
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1
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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 ...
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1
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Finding the closest special orthogonal matrix in Frobenius norm sense
Given a $3\times3$ matrix $M$, if we would like to get the closest $\mathrm{SO}(3)$ matrix $R$ that minimizes
\begin{equation}
\|R-M\|_F
\end{equation}
then $R$ = $UV^{T}$ where $U$ and $V^{T}$ are ...
1
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1
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Sufficient conditions for a system of linear inequalities to admit a solution
I am looking for sufficient conditions such that a system of linear inequalities of the type $A x >0$ admits a non-negative solution $x \in \mathbb{R}^n_+$. I know a few properties of the $m \times ...
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Uniqueness of l1 minimization
Let $A \in \mathbb{R}^{m \times n}$.
Is it true that $$\min \limits_{Q \in \mathbb{R}^{n \times m}}|I - QA|_{\infty} < \frac{1}{2}$$ is criteria for the uniqueness of the 1-sparse solution to
$\...
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Prove that these linear programming problems are bounded by $O(k^{1/2})$ [closed]
The expanded partial sums of the Möbius inverse of the Harmonic numbers have two out of three properties in common with this set of linear programming problems:
$$\begin{array}{ll} \text{minimize} &...
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What do square roots as minimums have to do with Harmonic numbers?
In an earlier question where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at s=1 of a certain Dirichlet series:
$$\Lambda(m)=\lim_{s\to 1+}\zeta(s)\sum_{d\mid ...
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Uniqueness of projection under spectral norm
I am considering
$$
\min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0,
$$
where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...
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On optimal dual solutions for the minimum weight perfect matching problems in the case of metric weights
Following Lovasz-Plummer (Matching theory, North-Holland 1986, Theorem 9.2.1),
the minimum weight perfect matching problem on a complete graph
$G$ with even number of vertices and weight $w:E(G)\to
\...
3
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1
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What importance does the Hirsch conjecture have to Simplex Complexity?
The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$.
After being open for decades, Francisco Santos has ...
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Linear programming with a convergent coefficient
The following linear programming problem
$x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$
has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...
2
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1
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Does quantifier elimination help here?
Suppose we have a quantified linear program
$$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$
$$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$
$$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$
$$...
3
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1
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A question about polytopes related to linear programming
Given linear functions $f_1({\bf x}),\dots,f_n({\bf x})$ on ${\bf R}^m$, let $K = \{(a_1,\dots,a_n) \in {\bf R}^n:$ the $n$ halfspaces $\{{\bf x}: f_i({\bf x}) \leq a_i\} $ have nonempty intersection$\...
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0
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Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
0
votes
1
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38
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Gluing simplices through a common point/ realisation of a convex simplicial polytope
Given $m≥d+1$
a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that
1) they all share the common vertex M
2) the ...
1
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1
answer
126
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Quantifier elimination and where is this quantified convex program in the polynomial hierarchy?
I have a quantified convex program of the form that I need to solve
$$\exists(x_{1,1},\dots,x_{1,n})\in\mathbb R^n\quad\forall(x_{2,1},\dots,x_{2,n})\in\mathbb R^n$$
$$\vdots$$
$$\exists(x_{2t-1,1},\...
7
votes
2
answers
816
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Formula for volume of a convex polytope
So I've been searching around the internet for some answers to this, but I currently have a set of linear constraints: $Ax = b, Cx \le d$ for matrices $A \in \mathbb{R}^{n \times m}$, $b\in \mathbb{R}^...
1
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0
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Simple monotonicity property for coordinate descent and linear objective functions
Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for ...