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Questions tagged [linear-programming]

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

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6 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.
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28 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?
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1answer
23 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 ...
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1answer
28 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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0answers
40 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 \...
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0answers
46 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 ...
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1answer
37 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 ...
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1answer
46 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
41 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 ...
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0answers
14 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 ...
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1answer
38 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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0answers
23 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 ...
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0answers
15 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 $x_1,...,x_n$ ...
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0answers
36 views

Optimizing library dimensions

Say I have a library that looks like that: ...
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2answers
134 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 ...
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0answers
28 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 ...
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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....
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1answer
24 views

How to formulate constraints in zero-one linear programming

There is a factory which produces 5 types of ice cream. If the $i_{th}$ ice cream is produced then $b_i=1$ otherwise $b_i=0$ How can I express the following constraints: The simultaneous production ...
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1answer
30 views

Complexity of linear programming with restricted quadratic constraints

A problem instance is a linear program with the following kind of quadratic inequalities allowed: For some of the variables $x_i$, there is a variable $s_i$ (intuitively for approximating $x_i^2$, and ...
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0answers
17 views

Reducing weighted linear threshold gate to unweighted one

Reading "On the power of threshold circuits with small weights" by Siu and Bruck I have faced several problems understanding how unweighted linear threshold element can be built efficiently from the ...
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0answers
35 views

Big M method for continuous variables

Is there any way to model the big M method for continuous variables? Something similar to this but $B, C \in \mathbb{R}_{\geq 0}$ and $A\in\{0,1\}$. Due to the precision problem, when the $B$ and $C$ ...
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0answers
22 views

(M)ILP overlap of two intervals

I got an ILP Model where $c_i$ represents the starting time for a visit$_i$. $c_i$ is already constraint by a number of constraints, one is $c_i > 0$. I have now outside of my model 0 or multiple ...
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1answer
124 views

What is the right term/theory for prediction of Binary Variables based upon their continuous value?

I am working with a linear programming problem in which we have around 3500 binary variables. Usually IBM's Cplex takes around 72 hours to get an objective with a gap of around 15-20% with best ...
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1answer
75 views

LP Relaxation of Maximum Coverage Problem

So I know from some research I've done that the OPT-IP <= OPT-LP for the maximum coverage problem, however I'm having some difficulty following the explanations I find. Does an example exist where ...
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0answers
9 views

How to understand the separation phrase for generalized subtour constraints?

Suppose the constraints of our integer programming model consist of two parts: the polynomial-size formulations, that have a size (number of variables and constraints) which is polynomial w.r.t. the ...
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1answer
49 views

How to write an if then logical constraint given part of the input related to a decision variable?

I am trying to solve an assignment problem-like from a bi-objective persepctive where I have a marketplace of vendors proposing different machines with different types and specs. The goal is to select ...
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3answers
60 views

Need Help Understanding MST Cutset Formulation

I just started learning about linear programming in my class, and I'm having some trouble understanding the MST Formulation Integer Linear Programming (Cutset Formulation). This is the definition: ...
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0answers
11 views

LIP - Minimum Spanning Tree Cutset Formulation [duplicate]

I just started learning about linear programming in my class, and I'm having some trouble understanding the MST Formulation Integer Linear Programming (Cutset Formulation). This is the definition: $...
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0answers
91 views

maximizing absolute value in linear programming

I know that similar questions have been answered several times, and based on the answers, I attempted a solution to my problem. But I simply do not get the right results. The problem is as follows. I ...
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1answer
21 views

Positioning items to maximize separation

Say we want to place n items on the real line. Let us denote the position of item i by $p_i$. We have interval constraints on the position $p_i$, i.e. we are given $l_i, r_i$ such that $l_i \le p_i \...
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0answers
11 views

Sparse feasible solution $|x|_0\le k$ for system of linear inequalities $A x \le b$

Suppose the set of linear inequalities $Ax\le b$, in which $A\in\mathbb{R}^{m\times n},x,b\in\mathbb{R}^n$ is given. Is it possible to determine in polynomial time with regard to $m$ and $n$ if there ...
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1answer
48 views

Is there an algorithm that can find a solution that solves the most number of equations in a linear system of equations?

My apologies if this question makes no sense. I am trying to find an algorithm that can solve a linear system of equations. Unlike most problems like this, this algorithm does not need to find a ...
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1answer
96 views

A simple way to find the feasible region of a system with simple constraints

I'm coding something... weird, and I'm running into some constraint satisfaction and graph theory problems, which are fields I'm not too experienced in. Here's the problem: I start out with this ...
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0answers
63 views

Implementing a linear programming feasibility test in 3D

I have a little problem which requires determining if a system of linear inequalities in 3D is infeasible. The constraints (or oriented planes) are added one by one, so there is an opportunity to stop ...
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2answers
40 views

Better way to formulate these constraints?

I have a binary variable $x_{ijt}^k$ that is $1$ iff job $i$ is assigned to machine $j$ at time $t$ using processor $k$. I would like to express the following constraints: If job $i$ is assigned to ...
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1answer
58 views

Conditional milp formulation

I have two binaries, $\alpha_{ts,it}$ and $\alpha_{ts,gshp} \in \{0,1\} $, and two reals $T_{it}$ and $T_{ts}$ which have upper and lower bounds. How can I model $\alpha_{ts,it}=1$ if the following ...
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0answers
57 views

Can somebody suggest what is wrong with these constraint? [closed]

I have written two constraints for Mixed integer linear problem. I am working on the scheduling problem i.e., Scheduling of hybrid appliances. For example, the washing machine is appliance indicated ...
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2answers
45 views

How to create constraints for Mixed integer linear problem?

i am a beginner to Discrete optimization domain. I am working on the real world problem, i.e., Scheduling of hybrid appliances. I have hybrid appliances which can ...
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1answer
115 views

How does cycling happen in the simplex method?

I'm reading Schrijver's Theory of Linear and Integer Programming, and I have a problem understanding cycling happens in the simplex method. The simplex is described as below: Solving $\max\{cx\mid x \...
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0answers
29 views

Does floor and ceiling in LP implies more than $P=NP$?

We know ability to take floor and ceiling in Linear Programming (LP) implies $P=NP$ (just apply floor and ceiling to variable in $(0,2)$ to get binary variable and from this it follows $0/1$ ...
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1answer
37 views

Can we split a vector into positive and negative parts in LP?

Say you have a vector $v$ with $n$ length $v=\begin{bmatrix}v_1&\dots&v_{n}\end{bmatrix}$ can we write as $v=v_+-v_-$ where $v_+$ agrees with $v$ on non-negative components and is $0$ ...
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1answer
35 views

Find coverage for plane from half-planes

The document (see pic. below) states that it is possible to find a cover of the plane by a subset of 3 half-planes. It proposes to use linear programming for this. How to formulate such a program? ...
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0answers
56 views

How to solve this optimization problem with logarithmic objective function?

I have this optimization problem I have no idea how to solve it. Is Lagrange suitable for this problem? Thank you
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0answers
34 views

mLP to mAGTSP formulation

In the paper Scheduling Twin Yard Cranes in a Container Block authors provide a mILP to solve scheduling twin cranes to execute requests in a block at a seaport to minimize makespan of the cranes. ...
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0answers
62 views

Is $P\neq NP\iff 3SAT$ not reducible to LP?

If $3SAT$ in $n$ variable and $m$ clauses reduces to LP with $O((nm)^c)$ variables and $O((nm)^c)$ constraints at a fixed $c$ then $P=NP$. Conversely if $P=NP$ then does $3SAT$ in $n$ variable and $m$...
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1answer
34 views

Fractional vertex cover number may not be feasible? Very confusing!

For my project, I try to use minimum fractional vertex cover number (MFVC). (Please find below definition details) MFVC can be formulated as optimal solution of a linear program relaxation. However I ...
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1answer
63 views

How do you proceed if your milp is not solvable

We are currently developing an ilp/milp model to fit the best routes with given resources (people) in a given timeframe and given visits and costs to travel from one visit to another (asymetrical). ...
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0answers
40 views

On a condition in real linear programming

For me $a,x,y\in(0,1]$ and I want to select $b=x$ if $a\leq0.5$ or else I want $b=y$. Is it possible to set this condition in real linear program?
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1answer
68 views

Can we use ILP here?

Is it possible to encode $y=0\implies G=0$ else $G=x$ by Integer Linear Programming where $x,y,G$ are integer variables? The answer mentioned below gets to the point of taking absolute value of ...
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1answer
237 views

On if then condition in linear programming?

I have variables $a,b\in\mathbb R$ and if $a>1$ I want $b=1$ or else $b=0$. Can this be encoded by linear programming (no integer variables)? Even $b<0.5$ and $b>0.5$ is ok.