Questions tagged [linear-programming]

Optimization with a linear objective function, subject to linear equality and linear inequality constraints.

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Knapsack Problem Via Column Generation

If I were to solve the linear relaxation of a knapsack problem via column generation how could I model the master problem and pricing subproblem? Given a set $N$ of items with value $v_{i}$ and weight ...
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On the hardness of satisfying K number of linear constraints

Background: Normally in linear programing we have some objective function $$\text{maximize}\sum_{i = 1}^n a_i x_i $$ $$\text{subject to} \sum_{i =1}^n b_{ji}x_i \leq c_j \text{ for all } 1 \leq j \leq ...
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About Steiner tree problem in graphs

In the paper (p. 3) and the slides presents the formulation of the Steiner problem on graphs via so called Steiner cuts. But according to the definition, the number of Steiner cuts and so the ...
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Minimal subset of rows that generate smaller polyhedron

Given a matrix $[A|B]$ I want to find a minimal matrix $[A'|B'] \subseteq [A|B]$ (i.e. the rows in $[A'|B']$ are also in $[A|B]$) such that $A'x < B' \Rightarrow Ax < B$. Geometrically, I want ...
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Linear Programming with constrained sum of rows and sum of columns

Is there a structure to the solution of the following linear program? $\min_{x_{ij} } \sum_{i,j} x_{ij} \mu_{ij}$ $s.t. \forall j, \sum_{i} x_{ij} = \beta_j,$ row sum $\forall i, \sum_{j} x_{ij} D_{...
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1answer
47 views

Can an optimization algorithm be “universal”?

I am wondering if a Bayesian Optimization framework (e.g. Google's Vizier) can be used in lieu of a traditional solver like Gurobi or CPLEX. In trying to answer this question, I realized that I don'...
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1answer
59 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....
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42 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{...
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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:
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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 ...
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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 (...
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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 ...
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1answer
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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 \...
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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 ...
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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&...
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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 ...
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1answer
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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 ...
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118 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 ...
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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 ...
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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}\...
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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 ...
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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=...
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1answer
54 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: ...
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1answer
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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46 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?
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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 ...
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1answer
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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 ...
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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} \...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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1answer
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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)...
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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 ...
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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&...
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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.
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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?
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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 ...
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1answer
81 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 ...
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“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 \...
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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 ...
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1answer
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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 ...
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473 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 ...
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1answer
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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 ...
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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 ...
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1answer
128 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 ...
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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 ...