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.

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Variant of orthogonal Procrustes problem

The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
Matt's user avatar
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2 answers
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Multiplicative gradient descent?

The normal gradient descent is additive: $w_{t+1}=w_t-\lambda_t\nabla f(w_t)$, but is there a multiplicative gradient descent that looks something like $w_{t+1}=w_t[-\lambda_t\nabla f(w_t)]$? I know ...
thinkbear's user avatar
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2 answers
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Continuous Linear Programming: Estimating a Solution

I have a "continuous" linear programming problem that involves maximizing a linear function over a curved convex space. In typical LP problems, the convex space is a polytope, but in this case the ...
David S-D's user avatar
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3 answers
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Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi, I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
Miroslav Korbelar's user avatar
5 votes
3 answers
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Linear programming - uniqueness of optimal solution

Is it possible to build such an objective function for a given set of constraints, so that there will be only one optimal solution? My general problem is to get any vertex of a polytope formed by a ...
Michael's user avatar
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1 answer
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Examples of Polyhedra with Large Shadows

Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...
Christopher's user avatar
5 votes
1 answer
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Resource Constrained Routing with Refueling

What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity? Especially modeling ...
Manfred Weis's user avatar
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Bounding the minimal maximum norm of a solution of a linear system.

I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...
user3645's user avatar
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2 answers
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Applications of minmax theorem(s)

Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions, $$ \inf_Y \sup_X f = \sup_X \...
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Continuous Transportation Problem

Hi all, I'm trying to formulate an infinite linear program to prove optimality (via duality) for the Continuous Transportation Problem, e.g. the Kantorovich-Wasserstein distance. This is the ...
Carrie Nuttall's user avatar
5 votes
1 answer
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Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$

The problem of giving an explicit formula for $A_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A_q(n,d)$ is given by the maximum number of codewords in a q-ary ...
Max Hopkins's user avatar
5 votes
1 answer
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Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?" More formally and generally, what I'm looking for ...
Xoph's user avatar
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1 answer
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Feasibility of linear programs

It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to deciding whether the intersection is non-empty?
Vinayak Pathak's user avatar
5 votes
1 answer
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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 \...
Mikhail Ostrovskii's user avatar
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1 answer
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What is the LP gap of vertex cover in planar graphs?

What is the LP gap of vertex cover in planar graphs? The LP I refer to is min $\sum_{e \in E } c_e x_e \ \ $ subject to $ \ \ x_v + x_u \geq 1 \ \ \ \forall uv \in E $ $ c_e \geq 0 $ are ...
Hao S's user avatar
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1 answer
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How does one go from convexity to submodularity?

If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular? It would be helpful is someone can share some specific ...
user6818's user avatar
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5 votes
2 answers
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relation between solution of a linear program and its perturbation

I have a linear program over a finite set of points $(x_1, x_2,\ldots, x_m)\in\mathbb{R}^n$: $$ \max_j c' x_j $$ Suppose the solution of this LP is obtained at a point $x_{j_1}$, which is a vertex ...
Abhishek Kumar's user avatar
5 votes
0 answers
161 views

Homogeneous linear and quadratic inequalities

I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that $$...
Fetchinson0234's user avatar
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A specific case of the $p$-center problem

Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
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A linear optimization problem on a graph

Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
TOM's user avatar
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A polytope associated with the Hadamard Transform

In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in http://arxiv....
Victor Miller's user avatar
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559 views

When is polytope compatible with network flow?

A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...
David Harris's user avatar
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4 votes
2 answers
268 views

Connecting $2n$ points in $\mathbb R^2$ with line segments s.t. each point belongs to exactly one line segment

I'm trying to do a certain simulation related to the toric code and I'm looking for an algorithm that connects $2n$ points ($n \in \mathbb Z_+$) in $\mathbb R^2$ with line segments with the following ...
Sanchayan Dutta's user avatar
4 votes
3 answers
921 views

Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$: $$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in \mathcal{...
Adrien's user avatar
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4 votes
4 answers
689 views

efficient way to compute the inversion of the following matrix

Hi, there I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...
Nancy's user avatar
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4 votes
2 answers
1k views

Set Cover:Greedy vs LP

Hi Both, the greedy and the LP approach for Set Cover give a O(log n) approximation. Is there some inherent difference on the two approximation approaches? thanks
user695652's user avatar
4 votes
2 answers
211 views

combinatorial and linear duality

Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual $\...
Jeremy Kahn's user avatar
4 votes
1 answer
850 views

When is a triangular matrix totally unimodular?

I have a {0,1}, invertible, triangular matrix, that I would like to show is totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?
Michael Biro's user avatar
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1 answer
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Under what conditions does an Integer Programming problem run in polynomial time?

Given $AX\leq B$ where $A\in\Bbb Z^{m\times n}$,$B\in\Bbb Z^m$ finding $X\in\Bbb Z^n$ where $m\geq n$ is the integer programming problem. If $A$ is totally unimodular then the problem is solvable in ...
Turbo's user avatar
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4 votes
1 answer
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Solving for Hamiltonian path with constraints on allowable routes through vertices

Suppose you have a complete graph with N vertexes, with a distinguished vertex $n=1$ ("start"), and you wish to find a route traveling exactly once through each vertex so that the distance along the ...
Daniel Cartin's user avatar
4 votes
1 answer
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solving multiple linear programming problems with the same set of constraints

Hi, I need to solve a set of linear programs of the form: Problem $i$: $\quad \max c_i \cdot x$ s.t. $ A x \leq b$. The $c_i$'s are different vectors so each problem has a different objective ...
Or Zuk's user avatar
  • 560
4 votes
1 answer
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Find the minimum distance between two convex hulls

We work over $\mathbb{R}^N$. Let $\mathbf{P}_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}_2$ the convex ...
tam's user avatar
  • 223
4 votes
1 answer
843 views

Basic result in semi-infinite linear programming

Consider a standard linear program of the form $$\textrm{minimize}_x~~~~ c^Tx~~~~ s.t. \\ Ax = b \\ x \geq 0$$ with $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. It is well known that, if ...
Tom Solberg's user avatar
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4 votes
1 answer
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optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint. $D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter. $x$ and $v$ are two known p-dimensional vectors. The ...
Bing's user avatar
  • 49
4 votes
2 answers
2k views

Simplified knapsack problem

There is a problem that I can not solve. Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
Leo-guest's user avatar
4 votes
1 answer
272 views

Symmetry of the integer gap

Are there results that bound the asymmetry of the duality gap of an integer program? That is to say, if the difference between the LP solution and the IP (primal) solution is $a$, is there a function ...
Chris Caragianis's user avatar
4 votes
1 answer
202 views

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 ...
Till's user avatar
  • 469
4 votes
2 answers
674 views

Minimum number of rectangles in a polygon

Given a polygon and dimension $d$, find a minimum partition of rectangles that has either of its dimensions equal to $d$. Example: Consider the following diagram: I want to cover maximum shaded ...
ubaabd's user avatar
  • 175
4 votes
1 answer
574 views

How to implement linear constraints that include several absolute values

Dear all, I am trying to implement a linear constraint that includes several absolute values in the form: Abs(A) + Abs(B) + Abs(C) + Abs(D) + ... = 1 Since the minimization problem includes quite a ...
Hugo's user avatar
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4 votes
1 answer
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How to find which subset of bitfields xor to another bitfield?

I have a somewhat coding-oriented problem. I have a bunch of bitfields and would like to calculate what subset of them to xor together to achieve a certain other bitfield, or if there isn't a way to ...
Bram Cohen's user avatar
4 votes
1 answer
316 views

Existence of Nonnegative Solutions of Linear Systems of Equations and Inequalities with particular constraints

Suppose we have an $n \times m$ nonnegative matrix $A$, where each row sums to $1$. I wonder whether there exists an $m \times n$ nonnegative matrix $X$ that satisfies the following constraints: ...
Ann's user avatar
  • 49
4 votes
1 answer
719 views

submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
mohi's user avatar
  • 859
4 votes
1 answer
270 views

Equivalent method for maximum likelihood estimation of covariance parameters

My goal is to estimate the parameters of a covariance matrix $\Omega$, by maximizing the following log-likelihood function: $$\log L(\vec\tau, \rho, \sigma \mid W, X) = -m\ln(\left | \Omega \right |) ...
Ruben van Bergen's user avatar
4 votes
0 answers
172 views

$\ell^1$-norm minimization duality

I am looking for an explicit description and discussion of the dual of the $\ell^1$-norm minimization problem $\lVert A x\rVert_1\to\min$, where $A$ is a matrix, and $x$ belongs to the $n$-simplex $\...
R W's user avatar
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4 votes
0 answers
495 views

Using Linear Programming as an iterative procedure

Suppose, we have a linear program and an optimal solution to it. Suppose now, we get a new constraint. We want to obtain an optimal solution to the given linear program extended by that new constraint....
D. Rusin's user avatar
  • 381
4 votes
1 answer
2k views

Maximizing supermodular functions

I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k . I am wondering if anyone can give me more information about a ...
Majid Yazdani's user avatar
4 votes
0 answers
784 views

Is it possible to use linear programming to solve this problem?

I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group. Could someone comment on whether this is ...
user6546's user avatar
3 votes
1 answer
7k views

Detection of Redundant Constraints

Suppose I pose the following query to a constraint logic programming system: ?- Y <= 6 - X, Y <= (- 4) + 4 * X, Y <= 4 + X / 3. Are there systems that would recognize the last inequality as ...
user avatar
3 votes
4 answers
4k views

Existence of nonnegative solutions to an underdetermined system of linear equations

Similar questions have been asked elsewhere, but I think this is sufficiently different to warrant a new post. I have a particular matrix $A$ and would like to know when the system $Ax = 0$ has at ...
bandini's user avatar
  • 481
3 votes
2 answers
425 views

convex polytope integer points

is there a simple proof for the following lemma: An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.
heinz's user avatar
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