Questions tagged [linear-programming]
Questions on linear programming, the optimization of a linear function subject to linear constraints.
5,215 questions
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Fourier-Motzkin Elimination for Single Vertex Computation
Given a convex polytope $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$, I want to know whether we can use Fourier-Motzkin elimination (or an adaptation therefore) to compute one vertex of $P$ (or to ...
- 1
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How many big and small marbles are not used?
Translating to English from a non-English physics book about measurements:
Anif has $8$ big marbles and $15$ small marbles. The weight of the big and small marbles are $37.5$ and $12.5$ respectively. ...
- 2,491
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How do I take a second derivative of an optimal value function of an Linear Program? [closed]
Ok, before anyone responds with the fact that the optimal value function of an LP is not twice differentiable.....I know, I know. I am not sure how to write what I am asking which is a big part of why ...
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When do linear constraints form a compact? [closed]
Given a list of constraints
$$
F_i = \{ x\in\mathbb{R}^n\mid L_i(x) \leq c_i \}
$$
where $L_i\in L(\mathbb{R}^n,\mathbb{R})$ and $c_i\in\mathbb{R}$ for every $i\in\{ 0,\dots,m \}$ with $m \geq n$, ...
- 123
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Are all finite games linear programs? Why is my formulation not correct?
Let's consider a two-player finite strategic form game. Suppose $A$ is the payoff matrix for the row player playing mixed strategy $x$, $B$ is the payoff matrix for the column player playing mixed ...
- 8,701
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53
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Case Study with Problem in Optimization
I have a problem that I'm trying to solve involving a bicycle ride share program. Riders rent bikes of various kinds (electric or normal-non powered) at various stations in a city, and after they are ...
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43
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A peculiar linear optimization/programming problem with homogeneous quadratic equality constraint
I'm trying to solve the following linear optimization/programming problem:
$$
\begin{aligned}
&\textrm{maximize (or minimize)}\quad \pmb{a}^{\intercal}\, \pmb{x}
\\&\textrm{given the ...
- 941
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36
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Is vertex enumeration preferable to the simplex method in linear programming for finding all basic feasible solutions?
Context
I'm studying linear programming and trying to understand when different solution methods are most appropriate.
I understand that the simplex method is generally better than vertex enumeration ...
- 83
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54
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Formal definition of equivalence of two optimization problems
I am learning about linear programming, and I have come across several definitions of when two general optimization problems are considered equivalent. However, I find these definitions a bit ...
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Converting an LPP to a Transportation Problem.
Max z = $2x_{1} + 3x_{2} + 4x_{3} + x_{4} + 7x_{5} + 5x_{6} $
Subject to
x$_{1} + x_{2} ≤ 1 $
x$_{3} + x_{4} ≤ 1 $
x$_{5} + x_{6} ≤ 1 $
x$_{1} + x_{3} + x_{5} = 1 $
x$_{2} + x_{4} + x_{6} = 1$
...
- 101
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1
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Create a closed tour so that every number is visited maximum once
Create a closed tour visiting the numbers. The tour must start and end at the same cell. During the tour, you can only visit the neighboring cell (up,down,left,right). The tour cannot contain a number ...
- 15
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102
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Can LP calculate maximum/minimum eigenvalue of a matrix
It's not hard to show that the maximum eigenvalue of a matrix $A\in \mathbb{S}^n$ can be calculated through the following SDP:
\begin{align*}
\max&\ Tr(AX) \\
\text{subject to}& \ Tr(X) = 1\\
...
- 93
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45
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Extreme point solutions to a generalized stable set LP
Consider the linear constraints:
$$ x_{ab} + x_{a'b'} \leq 1 \quad\forall a\neq a'\in A, b,b'\in B\\ x_{ab}\geq 0 \quad\forall a\in A, b\in B. $$
It is known that any vertex of this LP is half-...
- 894
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How can I formulate this disaggregation problem as a network flow?
I am struggling to formulate this disaggregation problem as a network flow. Can anyone help me see a way to make things work? I am not fully familiar with all of the tricks and gadgets for network ...
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Upper bound on the denominators of a rational solution for feasible linear program over integers
Let $d\in \mathbb{N}$ be a fixed constant. Let $V\subseteq \{-1,0,1\}^d$ be some set of vectors and let $w\in \mathbb{Z}^d$. Consider the linear program
find $ x\in \mathbb{R}^V $ such that: $
\sum_{v\...
- 101
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33
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How can I determine the "direction of unboundedness" from the dual, if I know the primal problem is unbounded?
Apologies if the question has already been asked.
Suppose I have a primal LP problem:
$$\text{Max }z= c^Tx \text{ such that } Ax\leq b$$
which is known to be unbounded, especially in the sense that ...
- 11
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31
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Way to bound LP feasible region in n-dimensional ball?
I am working on implementing a first order method for linear programming and would like to parallelize part of it by trying the following:
bounding the feasible region in a ball
picking equally ...
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40
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Optimising selection of blocks that overlap in time
I am not sure how to express this well in words or equations, so let me give a simplified example where the solution is easy. Suppose I have a series of time intervals, and different sets of "...
- 121
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Is the following Linear system Totally Dual Integral(TDI)?
Given two integers $k$ and $n$, and a $k\times n$ matrix $A$. There are $k\times n$ variables $x_{i,j}$ for each $i\in [k]$ and $j\in [n]$. For a pair of subsets $K\subseteq [k]$ and $T\subseteq [n]$, ...
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Linear optimisation resources
I am trying to learn linear optimisation using the book introduction to linear optimisation by bertsimas. I am having trouble understanding the concepts of polyhedral representation and polyhedrally ...
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Trying to understand the dual problem of a Convex Programming Problem
Consider the problem
$$\max f(x)=-3x_1+2x_2$$
subject to
$$2x_1-x_2\geq -2\\
x_1-2x_2\leq 3\\
x_1+2x_2\leq 11\\
x_1\geq 0\\
x_2\geq 0
$$
If one let
$$A=\left[\begin{array}{cc}
-2 & 2\\
1 & -...
- 2,349
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43
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Prove or disprove a heuristic rule to find an initial basic feasible solution in LP
I encountered an implementation of simplex method where the initial basic feasible solution is found by a heuristic rule as follows:
The procedure starts with a standard form with slack variables $\...
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35
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Algorithm for collision free positioning of squares in a limited space
I'm trying to find a simple algorithm to solve the following problem: I have a set of textboxes $S$. For textbox $t_i \in S$ I have a width $W_i$ and length $L_i$ and a limited piece of line $l_i$ in ...
- 143
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A comparison of relaxations of DFJ and MTZ formulations for Asymmetric Traveling Salesman Problem (ATSP)
There are many ILPs for Asymmetric TSP two of these are the DFJ formulation and the MTZ formulation:
The cut (DFJ) formulation is the following
\begin{align*}
\min &\; \displaystyle\...
- 2,005
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77
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Why can't we transform a $\geq$ into a $\leq$ and use a slack variable instead of surplus variable?
Consider the following set of constraints:
$$ \begin{aligned}
3 x + y + 2 z &\leq 30 \\
x + y + z &\geq 8 \\
4 y + 2 z &\geq 15 \\
x , y , z &\geq ...
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