All Questions
Tagged with linear-programming linear-programming or
492 questions
0
votes
1
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212
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How to find out if a polytope contains a sphere?
Given a polytope described by linear inequalities $Ax \le b, x \in \mathbb R^n$, how do you find out if there exist a (non degenerate) sphere of dimension $n-1$ contained in the polytope?
Thanks!
- 225
1
vote
0
answers
187
views
Strong Duality of Mixed Integer Linear Program
The problem at hand is to optimize a mixed-integer linear program closely related to the maximum flow problem. I would like to reformulate the problem with its dual and I'm concerned with the ...
- 11
3
votes
2
answers
1k
views
SDP relaxation vs LP relaxation
I have a question I hope you might be able to answer.
Let's say we have an integer program for the stable set problem (or clique, not principal).
\begin{equation}
\begin{aligned}
& \text{...
- 342
2
votes
1
answer
171
views
Maximization of Binary Multilinear Fractional Function
Problem: Let $a_{i,j}$, $b_{i,j}\in\mathbb{R}$ for all $(i,j)\in\left[m\right]^2$ such that $a_{i,j}=a_{j,i}$ and $b_{i,j}=b_{j,i}$. Let $z_k\in\{0,1\}$ for $k\in\left[m\right]$. We wish to maximize,
...
2
votes
3
answers
2k
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Better tactics for removing redundant constraints than Linear Programming?
After reading:
Detection of Redundant Constraints
It appears that linear-programming is the most commonly known way to remove ALL redundant constraints from a system of inequalities of the form
$$ ...
- 2,689
2
votes
1
answer
529
views
Integer programming and Groebner basis
I enjoyed reading different papers about using Groebner basis to solve integer programming.
Is there any literature about the complexity and/or comparison with other (more classical) methods like ...
- 337
1
vote
1
answer
155
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Derive a vertex representation of a permutohedron from its linear-inequalities form
Let us define the $n$-permutohedron $P_n$ as the set of all $x\in\mathbb{Q}^n$ such that
$$\sum_{i=1}^n x_i = \binom{n{+}1}{2}\ \ \ \land\ \ \ \forall\,\text{nonempty}\ S\subsetneq\mathbb{N}_n\colon\ ...
user94257
1
vote
1
answer
1k
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convert absolute form into linear programming problem [closed]
I would like to convert this problem into a Linear Programming Problem :
$\min |x|+|y|+|z|$
subject to $x+y \leq 1$
$2x+z=3$.
The solution to this problem is given chapter and here. But I still ...
- 113
9
votes
1
answer
2k
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Uniform sampling from general simplex with a twist
This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
- 221
1
vote
0
answers
64
views
Maximize discrete harmonic function at given point
Let $n>0$, and let $S_n$ denote the discrete square
$S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and $C_n=S_n\...
- 621
2
votes
0
answers
299
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Practical application of envelope theorem for linear programs
Assume that we have solved a (standard) linear program
$$
\text{minimize}_{x\in {\mathbb R^n}}\,\, c_0^Tx, \,\,\,\,\, \text{s.t. } A_0x \leq b_0,
$$
and would like to know how sensitive is the optimal ...
- 7,182
2
votes
1
answer
3k
views
max-flow at max-cost
I have a flow network with gains. In practical terms, a gain is the opposite of a cost. So, I interested in finding the maximal gain of a network flow, what could be interpreted as finding a maximum ...
- 121
1
vote
0
answers
55
views
Separation on discrete set
Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$.
Define linear functions $f(x)= a_1x_1+ \...
- 61
2
votes
0
answers
64
views
Finding orthogonal basis with constraint
Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...
- 909
2
votes
0
answers
71
views
Existence of probability distribution satisfying upper/lower bounds on events
Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...
- 21
4
votes
1
answer
345
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:
...
- 49
2
votes
1
answer
201
views
Minimum cover for sets in which each element appears in exactly 2 sets?
Is there an algorithm for finding minimal covers of a set of sets in which each element of the universe appears in exactly 2 sets? I realize that LP relaxation approximates this to within a factor of ...
7
votes
0
answers
1k
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Closed-form solution of a linear programming question
Among all the probability matrices
\begin{equation*}
P =
\left(\begin{array}{cccc}
p_{00} & p_{01} & \ldots & p_{0,J-1} \\
p_{10} & p_{11} & \ldots & p_{1,J-1} \\
\vdots & \...
2
votes
2
answers
438
views
Perturbation of Linear Programs
Consider the linear program,
$$\begin{array}{ll} \text{maximize} & c^T x\\ \text{subject to} & Ax \leq b\\
& x \geq 0\end{array}$$
I want to study the sensitivity of the optimal $x^*$ ...
- 275
0
votes
1
answer
204
views
Is the linear production game a convex game?
In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem.
Does anyone know if the LPG is a convex ...
- 17
1
vote
1
answer
390
views
Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces
Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by
$$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$
$$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$
I wish to find their convex hull, that is a ...
- 2,689
2
votes
0
answers
177
views
Formulating shortest path as submodular minimization
I'm curious about the general question of whether any combinatorial optimization problem with polynomial time solution can necessarily be reformulated as minimizing a submodular function.
The answer ...
- 21
6
votes
1
answer
263
views
Algorithm that solves every Mixed Integer Linear Program (to optimality)?
Given a Mixed Integer Linear Program with rational coefficients (both for the objective functions and all constraints), is it always possible to solve it algorithmically?
I know that you usually ...
- 1,282
2
votes
0
answers
112
views
Pcross-like, nonogram-like in near-linear time [closed]
I have a problem with a puzzle game like pcross in which I have a nxn square: At any index of rows and columns I have an integer that say the maximum numbers of points that I can place in that row/col....
2
votes
1
answer
531
views
Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables
I am working out an interesting problem and would like some help with this particular sub problem:
Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ a_{ij}\...
- 21
0
votes
1
answer
543
views
Convert general optimization problem to LP problem
I am trying to convert the following problem into a linear programming problem:
There are $M\times N$ matrix $T$ of real numbers between 0 and 1 and $N\times 1$ vector $w$ of real numbers between 0 ...
- 99
0
votes
1
answer
2k
views
When to use non-negative-least square and least-square [closed]
What are the typical case we need to use Non-negative least squares NNLS
$$
||Ax - B||^2
$$
instead of least-square $$ Ax-B$$ (or vice versa)?
And is there any drawback in applying them on large $A$...
- 135
3
votes
1
answer
260
views
Better alternative to solve quadratic programming for large matrices
I have the following problem. Let's say we have $x_{jk}$ it is an expression value of gene $j$ in a sample $k$. It is the average of expression levels across the cell types $s_{ij}$, weighted by ...
- 135
3
votes
0
answers
970
views
Testing if a point is inside a convex polytope formed by halfspaces in n-dimension
Assume we have a convex polytope that is formed by the intersection of $k$-halfspaces in $\mathbb{R}^{n}$.
$$
a_{0,0}x^{n-1} + {a}_{0,1}x^{n-2} + ... a_{0,n-1} \leq 0
$$
$$
a_{1,0}x^{n-1} + {a}_{1,...
- 31
0
votes
1
answer
82
views
Introducton books for $\frak{E}_p(I)$
Are there any good books different from abstract harmonic analysis by hewitt to study $\frak{E}_p(I)$. where $\frak{E}_p(I)$ is: Let $I$ be an arbitrary index set. For each $i\in I$ let $H_i$ ...
- 209
5
votes
1
answer
146
views
How does one go from convexity to submodularity?
If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...
- 1,893
1
vote
1
answer
219
views
approximate diameter of polytopes in high dimensions
I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 \...
- 75
2
votes
0
answers
154
views
Listing all Lattice Points in a Box
Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
- 173
3
votes
2
answers
792
views
Has anyone developed a technique to generate a polytope given (possibly redundant) inequality constraints? [closed]
I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm just trying to find the vertices for a feasible region, given a set of inequality ...
- 131
1
vote
0
answers
171
views
Finding all feasible solutions
Let $u$ be a $n_{max} \times m$ matrix. Let $z$ be a $n_{max} \times s_{max} \times n_{max}$ cube. Let $w$ be a $n_{max} \times 1$ vector. All the three matrices can have values from the set $\{ 0, 1\}...
- 175
2
votes
0
answers
210
views
Finding optimal linear transformation for intersection of convex polytopes
I previously posted this on MathSE and am now trying here.
I have the following situation, as shown in the following diagram:
$W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) ...
- 695
1
vote
0
answers
120
views
The column generation technique on a Train Unit Assignment Problem [Linear Programming]
I am doing an assignment where I need to implement a mathematical model that I can't wrap my head around. For the technique of column generation, one would need to my understanding, a master problem ...
- 111
4
votes
1
answer
3k
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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 ...
- 233
5
votes
1
answer
183
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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 ...
- 13.2k
4
votes
1
answer
891
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 ...
- 4,049
2
votes
1
answer
186
views
How can I find the maximum value of this function?
For given values of $A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m$, how can I find the value of:
$$
\max_{x \in [0,1]^n} \|Ax+b \|_1
$$
Or is this problem NP-hard?
- 23
2
votes
1
answer
229
views
Algorithm to find the vertices of the equidistant lines between N closed polygonal lines
I have a set $\{C_1, C_2, \ldots, C_N\}$ of $N$ nonintersecting closed piecewise linear curves on the Euclidean plane. For every point $x \in \mathbb{R}^2$ we say it belongs to a territory serviced by ...
- 329
1
vote
0
answers
85
views
Smallest sum of original column entries in 2d matrix [closed]
I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this:
...
- 111
3
votes
1
answer
1k
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Constrained vs Unconstrained Optimization
I'm currently working on an optimization problem with a linear objective with linear and nonlinear constraints, i'm facing difficulties reaching a good solution, so i was advised to move the nonlinear ...
- 31
1
vote
1
answer
207
views
Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s
This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?
So far I only figured out that I can do Monte ...
- 113
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
2
votes
0
answers
148
views
Derivation of gradient of SSE in Geodesic Regression
On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...
- 5,405
1
vote
1
answer
2k
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Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm
I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105
Unfortunately I am struggling to make the algorithm work on ...
- 11
2
votes
0
answers
71
views
Select n vectors from k vectors (in 3D) such that each component of the resultant vector >= each component of a given vector M
this was left unanswered for 1 week on MStackExchange, so I thought MOverflow would be more appropriate. Thanks :)
Let $R = (R_x, R_y, R_z)$ be the resultant vector of the n vectors and $M = (M_x, ...
- 21
2
votes
0
answers
149
views
How to solve the following generalized quadratic programming problem [closed]
I want to solve a generalized form of a quadratic programming problem
$$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$,
$$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...
- 21