Questions tagged [linear-programming]
Optimization with a linear objective function, subject to linear equality and linear inequality constraints.
271
questions
3
votes
1answer
42 views
Linear programs with strict inequalities and supremum objectives
Linear programming can solve only problems with weak inequalities, such as "maximize $c x$ such that $A x \leq b$". This makes sense, since problems with strict inequality often do not have a solution....
0
votes
1answer
37 views
“Greater than AND smaller than” condition in integer linear program with a binary variable
I found this related question, but that's not quite it
Is it possible to model this with integer programming:
$$A = \begin{cases} 1 & \text{if } B \geq C \geq D \\ 0 & \text{otherwise}\end{...
1
vote
0answers
21 views
Does a decision variable that's just an upper bound on other decision variables make an integer program harder to solve?
I have several decision variables $max_s \in \mathbb{N}, s \in \mathcal{S}$ that are just upper bounds on other decision variables: $$x_{i,s} \leq max_s \forall s \in \mathcal{S}, \forall i \in \...
0
votes
0answers
14 views
Given a primal LP p, and another LP d, how can i formally prove that d is the dual problem of p?
Given a primal LP p, and another LP d, how can i formally prove that d is the dual problem of p?
Specifically, i'm talking about the shortest s-t path:
where:
And the dual LP:
0
votes
0answers
7 views
How to input random values (constraints) of any variables in case of formulating Linear Programming Problem?
Suppose,
Min 2x+3y
Subject to, x=2,x=5,x=7
y=5, y=9
is a linear program. Where x holds the values 2 or 5 or 7 and y holds the values 5 or 9. Then what should the correct ...
0
votes
0answers
19 views
How is ellipsoid method a polynomial-time algorithm for LP?
I have always thought that the ellipsoid algorithm is an algorithm which can be used to solve LP in polynomial-time. However, what confuses me is the dependence on the ratio of volumes of the balls (...
0
votes
0answers
18 views
Creating a waste-optimizing algortihm for cutting a 1d block
I have a one-dimensional block of material. I run an analysis that divides the material into usable and unusable regions.
In a manufacturing process, said material is cut and the unusable regions ...
1
vote
1answer
27 views
Relaxation of the knapsack constraints
A set $\mathcal{A}$ is the relaxation of another set $\mathcal{B}$, if $\mathcal{B} \subseteq \mathcal{A}$.
I have a set of points defined as the knapsack constraint
$$
\mathcal{X} = \{x \in \...
0
votes
0answers
21 views
How can I quickly find the dual of a linear program?
In linear programming, the standard maximum form of a program (which we will call the "primal") is
max $c^{tr}x$ subject to
$Ax \le b$, $x \ge 0$
and the standard minimum form, the dual, is
...
1
vote
0answers
26 views
Solving a linear program with simplex algorithm, matrix not full rank
I need to solve the following LP
$$\begin{array}{ll} \text{minimize} & c^T x\\ \text{subject to} & A x = b\end{array}$$
where
$$A = \begin{bmatrix} 1 & 3 &1&0&0 \\ -2&...
4
votes
1answer
120 views
Mistake in a proof of termination phase of Simplex algorithm in CLRS?
There is a pseudo-code for Simplex algorithm in CLRS:
The proof consists from three-part loop invariant:
Proof We use the following three-part loop invariant:
At the start of each iteration ...
1
vote
1answer
25 views
Linear programing, objective function. variable depending on the sign of another variable
I have the variable Si. How to express a variable Di in LP that satisfies:
Di=100*Si if Si>=0
...
3
votes
0answers
84 views
Minimum clique cover
How can the problem of finding the minimal clique cover be solved using linear/integer programming in a reasonable amount of time?
Having an undirected graph, I am trying to partition all its ...
1
vote
0answers
26 views
Goemans' Extended Formulation of the Permutahedron And Comparator Networks that are not Sorting Networks
I am interested in using Michel Goemans' extended formulation first developed for the permutahedron to study comparator networks that are not sorting networks. In his paper "Smallest Compact ...
1
vote
0answers
35 views
An LP with two covering constraints - how to round
I came across an LP with two covering problems, and I wonder how to
find a good approximation. For the relevant part of the LP: We have
a set $E$ , for each $e\in E$ we have a corresponding set $Y_{e}\...
4
votes
2answers
56 views
Does real linear programming produce bipartite perfect matching using maxflow reduction?
Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram:
We can formulate max flow as an linear programming problem with integer variables in ...
2
votes
1answer
29 views
Linear programming IFF with equality constrain
Is it possible to write the following logical constrain in linear programming?
Let $v$ be an integer variable and $k$ an integer constant. Let $y$ be a binary variable. The logical constraint is
$y=...
1
vote
1answer
47 views
If-Then with disjunctions (OR) in Integer Linear Programming (ILP)
I have the following constraints I'm trying to model in Linear Integer Programming. I will try out diverse solvers for this later, but first I need to model the problem.
Given the integer variables: ...
1
vote
1answer
47 views
Partitioning a boolean circuit for automatic parallelization
tl;dr: I have a problem where I have a Boolean circuit and need to implement it with very specific single-thread primitives, such that SIMD computation is significantly cheaper after a threshold. I'm ...
0
votes
1answer
33 views
How to model equality in Integer Linear Programming
How to implement v=(a==b) using Linear Programming?
$$
v=
\begin{cases}
True, a=b\\
False, a≠b\\
\end{cases}
$$
Until now I tried the big M-Method.
To show a≤b:
$$a-b+Mv≤M$$
$$-a+b-Mv≤-1$$
To show ...
1
vote
1answer
22 views
Is there any advantage of using an Integer Linear Program over Backtracking in a combinatorial optimization problem?
Is there any advantage of using an Integer Linear Program over Backtracking in a combinatorial optimization problem?
I saw this Gurobi post that uses Integer Linear Programming to solve the traveling ...
1
vote
1answer
40 views
What is the most efficient way to test whether a set $X \subset \{0, 1\}^n$ and its complement $\{0, 1\}^n \setminus X$ are linearly separable?
I am interested in algorithms that have optimal running time, and ideally which are also very easy to implement. If you can also give some tips on how to implement the algorithm(s) you mention in the ...
2
votes
1answer
45 views
Logic of multiple variables in ILP
Is there a better way to represent an AND of $n$ variables together other than creating $O(n)$ new variables and constraints?
0
votes
0answers
35 views
Traveling Salesman Problem with profit and time limit as ILP formulation
How to formulate the following problem?
The salesman gains a profit $p_{i}$ when visiting a city i, trip between city i and city j costs $c_{ij}$ and takes $t_{ij}$ time. The trip must not exceed a ...
1
vote
1answer
51 views
ILP representation of the number of maximal 1 sequences in a row
I am currently using an ILP to model events which occur on some input sequence from $1...n$. These events modify the input sequence in order to obtain a desired sequence. Each event can happen on some ...
0
votes
0answers
15 views
Linear Programming Formulation for Weighted Max-Cut
I am wondering if this Integer Linear Programming model I came up with as an exercise for my algorithms class is correct.
The Problem
Given a graph $G=(V, E)$, with a set of weights $W = \{w_{ij} \...
2
votes
1answer
21 views
Determine image of hypercube under linear map
Let $A$ be an $3\times N$ matrix (where $N$ is large) with nonnegative real entries. I'd like an algorithm for determining when a vector $v\in\Bbb R^3$ can be written as $Aw$ for some vector $w\in\Bbb ...
1
vote
0answers
28 views
Time complexity: Using linear programming to solve a system of linear equations
As far as I know, most direct methods for solving linear systems of equations have time copmlexity $O(n^3)$ (where $n$ is the number of variables), with the few methods being faster having huge ...
1
vote
1answer
46 views
Where's the flaw in my algorithm? (Linear program to solve NP-hard problem)
The problem (copy-pasted from this question on cs.stackexchange):
Given a connected, directed graph $G=(V,E)$, vertices $s,t \in V$ and a coloring, s.t. $s$ and $t$ are black and all other vertices ...
2
votes
1answer
67 views
Approximation algorithm for weighted set cover, using multiplicative weights
It is known that the problem of fractional set cover
can be rephrased as a linear programming problem and be approximated using the multiplicative weights method, for instance this lecture note shows ...
0
votes
1answer
27 views
Sanity check about a linear programming problem
given the linear program:
minimize $x+y$
subject to,
$ax+by \leq 1$
$x,y \geq 0$
I need to find real numbers $a,b \in \mathbb{R}$ such that the program (a) is infeasible, (b) is unbounded, and (c)...
0
votes
1answer
51 views
Minimal paths as solution of a linear program of a special network flow
Let $G= (V,E)$ be a given directed weighted graph, and $s,t$ two specified nodes, so that there is no negative cycle reachable from $s$, and $t$ is reachable from $s$.
We're looking for the shortest ...
0
votes
0answers
27 views
Handling $AND$ and $OR$ cases in MILP?
Suppose I want to have an integer program for handling the cases
$x_1>1\wedge x_2>1\wedge x_3>1\wedge\dots\wedge x_n>1\iff\delta=1$
$x_1>1\vee x_2>1\vee x_3>1\vee\dots\vee x_n&...
0
votes
0answers
12 views
Please help to convert conditional expressions to linear [duplicate]
I have an expression:
if A≤X1-X2≤B, than Y=1, Otherwise Y=0
where A and B are constants, and X1, X2 are variables.
Please help to convert this to linear expressions.
1
vote
0answers
30 views
general question about linear algebra [closed]
have new discoveries been made in linear algebra since the dawn of AI or was it all "laid out" for the computer scientists by former mathematicians?
1
vote
1answer
35 views
Check a variable within a range with a binary variable [closed]
I have a value, a, it can be any value from 0 to 1. In an integer linear program, how can I formulate a constraint that uses a binary variable, y, to determine whether a is within a range of 0 and 1 ...
2
votes
1answer
65 views
Boolean variable that captures whether an inequality holds
I have an integer linear program with variables $x_1,\dots,x_n$. I have an inequality $a_1 x_1 + \dots + a_n x_n \ge b$ that I care about; it may or may or not hold.
I want to introduce a boolean ...
1
vote
0answers
102 views
“Greater than 0” condition in integer linear program with a binary variable [duplicate]
How can one model the following condition in an integer linear program?
$$
y = \begin{cases} 1 & \text{if } x > 0\\ 0 & \text{otherwise}\end{cases}
$$
Where $y \in \{0,1\}$ and $x \in \...
1
vote
0answers
61 views
Confusion about the geometric interpretation of the simplex method for linear programming
In Section 7.6.2 of the textbook "Algorithms" by Sanjoy Dasgupta, Christos Papadimitriou, and Umesh Vazirani, the authors provide a geometric interpretation of the two main tasks of each iteration of ...
1
vote
1answer
55 views
What is the intuition behind the way of reading off a dual optimal solution from simplex primal tabular in CLRS?
Section 29.4 "Duality" of CLRS (3rd Edition) describes the way of reading off an optimal dual solution from the last slack form of the primal as follows:
Suppose that the last slack form of the ...
4
votes
1answer
340 views
Maximum matching using linear programming
In a bipartite graph $G = (V,E)$, there is a neat algorithm for finding a maximum matching (or even a maximum-weight matching) using linear programming. It is explained here. The first step is to ...
2
votes
1answer
44 views
How to efficiently specify a MILP constraint with nested AND and ORs
Let's say I want to set x1=1 if (x2=1 AND x3=1 AND x4=1) or (x5=1 and x6=1) or (x7=1) else x1=0
All of the xs are binary ...
0
votes
0answers
19 views
When does total unimodularity implies integrality of solutions?
In Gartner & Matousek's book on linear programming, they prove the following lemma (Lemma 8.2.4, page 145):
Let us consider a linear program with $n$ nonnegative variables and $m$ inequalities ...
2
votes
1answer
111 views
Using LP to prove the max matching - min cover theorem
Konig's theorem says that, in a bipartite graph, the size of the maximum matching equals the size of the minimum vertex cover. This theorem has several proofs; I would like to know if the following ...
2
votes
0answers
36 views
Relaxations for MILP with logical constraints
I have an LP with a (non-fixed) number of logical constraints in the form of $X_1 \rightarrow X_2$ (where $X_1$ and $X_2$ are linear functions inequalities of the $n$ input variables). To express ...
1
vote
0answers
18 views
Given a set of solutions, find an IP formulation with the same solution set
Input:
A list of integer variables $x_1, ..., x_n$.
A finite set of feasible solutions $S \subset \mathbb{Z}^n$.
Task:
Find an integer linear program (IP) on the integer variables $x_1,...,x_n$ ...
1
vote
0answers
38 views
7
votes
2answers
162 views
Reducing linear programming to positive linear programming
Suppose we have an oracle that solves problems of the form
\begin{align*}
\text{maximize} ~~ & c^T x
\\
\text{subject to} ~~ & A x = b, x\geq 0
\end{align*}
when $c\geq 0$ (all coefficients ...
0
votes
0answers
34 views
Branch and bound Dual Gap
Let's suppose we want to solve the knapsack problem using the Branch and Bound algorithm.
I know that the algorithm ends when the optimality gap is = 0. However i have not understood how the dual ...
0
votes
0answers
16 views
How to formulate initial costs in linear programming?
Consider this example problem:
Suppose the cost for setting up a factory to generate a pencil is 1000 and to generate a pen is 2000. The profit for each pencil is 10 and the profit for each pen is 12....
