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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Are there scenarios under which feasibility bilinear programming is easy?
Given $c\in\Bbb R^{n_1},d\in\Bbb R^{n_2}$, $E\in\Bbb R^{n_1\times n_2}$, $A\in\Bbb R^{m_1\times n_1}$, $B\in\Bbb R^{m_2\times n_2}$ $a\in\Bbb R^{m_1}$, $b\in\Bbb R^{m_2}$ and $t\in\Bbb R$ we know ...
2
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1
answer
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What does the basis of the null space of the constraint matrix of a flow problem look like?
Consider a directed graph $G=(V,\mathbb{A})$ and a set of flow constraints of the following form:
$$ \sum_{(u,v)\in\mathbb{A}}x_{u,v} - \sum_{(v,u)\in \mathbb{A}}x_{v,u} = 0 \forall v \in V$$
...
2
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0
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How to find a positive solution to an under-determined linear system (if such a solution exists)?
Like the title says, if an under-determined system of linear equations does have at least one positive solution, how to find it efficiently?
Suppose we have an under-determined system:
$$Ax = b$$
...
1
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0
answers
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Calculating Cost-Optimal 1-Factors in Digraphs
I need to find a cost-optimal 1-factor in a positively weighted, directed, regular graph $G(V,A)$ without antiparallel arcs, i.e. given $$\text{deg}_{\text{in}}(u)=\text{deg}_{\text{in}}(v)=\text{deg}...
2
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0
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Making a polyhedron integral by selecting value for a specific co-ordinate of constraint vector
I am currently trying to solve a binary integer programming(maximization) problem, where the first row of the constraint matrix corresponds to the constraints on the total number of 1's in the vector ...
2
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0
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Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
1
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1
answer
111
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Optimal "Generalization" of Polylines
This question is inspired by a lossy compression technique for polylines, namely to identify a subset of the points of polyline $\mathcal{P}$, whose removal yields a polyline $\mathcal{Q}$ within a ...
14
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3
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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:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
10
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0
answers
655
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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 ...
2
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0
answers
331
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Linear programming with an infinite matrix
I would like to solve the following infinite linear system subject to $x_i \ge 0$ that minimizes $x_3$.
The third column contains no additional nonzero values beyond what is shown. Though the first ...
1
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2
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Algorithm for a linear optimization problem
For the vectors $X=(x_1,\cdots, x_n),~ Y=(y_1,\cdots, y_n)$ and $\alpha=(\alpha_1,\cdots,\alpha_n),~ \beta=(\beta_1,\cdots, \beta_n)\in\mathbb R^n_+$ s.t. $\sum_{k=1}^n\alpha_k~~=~~\sum_{k=1}^n\beta_k~...
1
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0
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Family of functions which satisfies $f(\boldsymbol{x}) = 0$ if $\nabla f(\boldsymbol{x})=0$? [closed]
I have a Lagrangian of which I want to find the supremum in the primal variable $\boldsymbol{x}$:
$\mathscr{L}(\boldsymbol{x},\boldsymbol{\lambda})=f(\boldsymbol{x})^T\boldsymbol{a} + \boldsymbol{\...
1
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0
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Linear programs [closed]
Can the optimal value of the primal problem of a linear program ever be less then zero?
An example is: minimize $C=2x_1 +3x_2$ Subject to: $3x_1+4x_2 \leq 5$. Obviously, $x_1$ and $x_2$ are free ...
1
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0
answers
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quick hull algorithm detail
When using quick hull algorithm to find the polytope for half space intersection, we are required to provide an interior point to the solver qhalf.
In other words, providing
$$Ax \le b$$
is not ...
1
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0
answers
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a question about probabilities on spaces of digraphs
Let $G$ be a directed graph with fixed nodes $s$ and $t$. Assume that each edge $e$ in the graph comes with a number $n(e)\in[0,1]$.
We consider probability spaces $S$ whose points are directed ...
3
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2
answers
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Equality constraints in mixed-integer optimization
Suppose I have a linear mixed-integer optimization problem of the form
$$MIP: min_{(x,y) \in \mathbb{R}^n \times \mathbb{Z}^m} c^\top x + d^\top y \hspace{0.2cm} \text{s.t.} \hspace{0.1cm} Ax+By \leq ...
0
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1
answer
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algorithms and tools available for a particular polytope computation
Let me define each half space i as:
$${H_i}:{c_i}{\bf{x}} \le {b_i}$$
The intersection of all such ${H_i}$ gives a polyhedron (bounded or not). Suppose I am interested in if ${H_i}$ is active (...
1
vote
1
answer
70
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minimize number of unique elements in a vector
I was wondering if there is a simple or known way to minimize the number of unique elements in a decision variable (vector). Note that I'm not asking for minimization of nonzero elements (rank ...
3
votes
1
answer
343
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Lot sizing problem: how to add these cuts efficiently
Consider the set of constraints of the uncapacitated lot sizing problem:
$$
\{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
2
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2
answers
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Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative
$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.
What should $A$ satisfy to guarantee the equation set have only zero solution?
2
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1
answer
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Fast algorithm for large-scale, asymmetric transportation linear program
I have a large-ish instance of a transportation problem that is very asymmetric, say of dimensions $100\times10000$. I am currently solving it with a stock LP solver, but obviously something like the ...
1
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1
answer
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Show $0-1$ Knapsack is polynomially reducible to this problem
I have already posted this question here but have not received an answer so I am cross-posting with hope to reach a larger amount of mathematicians:
Let $T=\{1,\cdots,n\}$ and consider the ...
6
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2
answers
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Linear programming is continuous
Consider an arbitrary linear program:
$$\max \vec c \cdot \vec x$$
subject to:
$$\textbf{A}\cdot \vec x = 0, \quad \vec a \le \vec x \le \vec b$$
Assume that this program is feasible and bounded. ...
0
votes
1
answer
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Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
0
votes
0
answers
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A seemingly easy integer programming question
Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
1
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1
answer
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Are convex combinations of 0-1 Pareto efficient vectors efficient?
Let $Y$ be any subset of $\{0,1\}^n$ for $n\geq3$. A vector $\alpha\in$ $Y$ is Pareto efficient if there is no $\beta\in$ $Y$ such that $\beta_i$ $\geq$ $\alpha_i$ for each $i\in\{1,...,n\}$ and $\...
15
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3
answers
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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?
...
2
votes
1
answer
462
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Under what condition does Courant–Fischer–Weyl min-max principle hold in general?
From Wikipedia:
Let $A$ be an $n \times n$ Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient $R_A :
\mathbf C^n \setminus \{0\} \to \...
26
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5
answers
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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
$$\begin{pmatrix}6 & 20 & 6& 0 ...
2
votes
2
answers
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Linear programming with infinitely many constraints
I wish to study the following linear program
$$\begin{array}{ll} \text{minimize} & \mathrm c^{\top} \mathrm x\\ \text{subject to} & \mathrm A \mathrm x = \mathrm b\\ & \mathrm x \geq 0\...
0
votes
2
answers
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Reference request: dependence on linear constraints
Excuse me if my question is stupid. I'm seeking the references on the dependence of the (linear) optimization problem on (linear) constraints. Namely, consdier the following optimization problem:
$$P(...
3
votes
0
answers
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Dependence of optimization problem on the linear constraints
Let $I=\{x_1,\cdots, x_n\}\subset \mathbb R$ be fixed. Given two probability distributions $\alpha=(\alpha_i)_{1\le i\le n}$ and $\beta=(\beta_i)_{1\le i\le n}$ on $I$, and a matrix $c=(c_{i,j})_{1\le ...
2
votes
1
answer
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Complementary slackness for approximately optimal Dual solution
Given a Primal LP (P) and it Dual LP (D) we know that the optimal solutions to P ($x_{opt}$) and D $(y_{opt})$ satisfy complementary slackness condition, i.e. under optimal solutions either a ...
0
votes
0
answers
880
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Maximum shortest path problem
I have the following problem. You have a graph and every edge has a certain set of possible weights. The question is to find the assignment of those weight which will maximize the shortest path.
In ...
2
votes
0
answers
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Unveiling hidden structures
One way to unveil a hidden structure of a undirected graph - given as an adjacency matrix - is to permute the rows and columns until a pattern with a maximal geometrical symmetry is found. (The ...
19
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4
answers
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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 ...
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Number of different combinations in a 0-1 knapsack problem with integer weights [closed]
My question is actually very similar to this other one: Given a vector of positive integers, count the number of combinations which have a sum that produces a different value. But, since this previous ...
3
votes
1
answer
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Properties of one dimensional null space
Let $\mathcal{G}$ be denote the set of all $3 \times 3$ real symmetric matrices and let $\mathcal{G}^+$ denote the set of all $3 \times 3$ positive semidefinite matrices (see definition).
Let $S: \...
4
votes
2
answers
675
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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 ...
1
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0
answers
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Analytic formula for minimizing the maximum inner product of a set of vectors
Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find
$$
\widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|.
$$
I am also interested in the special case where we further ...
1
vote
1
answer
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Do doubly infeasible Linear Programming problems always have doubly infeasible bases?
Consider a Linear Programming problem in dictionary form,
$$\max\Big\{f^\pi+\!\!\!\sum_{j\in D(\pi)}\!\! d^\pi_jx_j~\Big|~\forall~i\!\in\!B(\pi)~~~ b^\pi_i+\!\!\!\sum_{j\in D(\pi)}\!\! G^\pi_{ij}x_j\...
0
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1
answer
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Recursive linear programming on a linear subset of a simplex
The problem I am working on is:
Given an $n$ dimensional vector $r \in \mathcal{R}^n$, and a convex set $G=\{\mu \in \mathcal{R}^n | \mu_i \ge 0, ~ \mu^T \mathbf{1}=1, ~ A\mu =0 \}$ where $\mathbf{1}...
2
votes
3
answers
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Quadratic Programming With Piecewise Linear Term
The problem I have can be defined as:
$$
\min \frac{1}{2}\mathbf{x}^T\mathbf{Q}\mathbf{x} + \mathbf{c}^T\mathbf{x}
$$
s.t. linear equality constraints:
$$
\mathbf{Ax=b}
$$
and linear inequality ...
0
votes
1
answer
190
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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!
1
vote
0
answers
177
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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 ...
3
votes
2
answers
1k
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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{...
2
votes
1
answer
167
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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
$$ ...
2
votes
1
answer
513
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Integer programming and Groebner basis
I enjoyed reading different papers about using Groebner basis to solve integer programming.
Is there any literature about the complexity and/or comparison with other (more classical) methods like ...
1
vote
1
answer
151
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Derive a vertex representation of a permutohedron from its linear-inequalities form
Let us define the $n$-permutohedron $P_n$ as the set of all $x\in\mathbb{Q}^n$ such that
$$\sum_{i=1}^n x_i = \binom{n{+}1}{2}\ \ \ \land\ \ \ \forall\,\text{nonempty}\ S\subsetneq\mathbb{N}_n\colon\ ...