# Questions tagged [linear-programming]

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

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### 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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### 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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### 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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### (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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Finding integrality gap for maximum weight independent set

One of the exercises I was given was to formulate Integer Linear Program (ILP) and relaxed version of it (LP) to solve the maximum weight independent set, and I need to find an integrality gap of my ...
The feasible region of a Linear Program (LP) is $\{x \in {\bf R}^n: Ax \le b, x\ge 0 \}$. This is an intersection of halfspaces, a polyhedron. If the LP is bounded and feasible, its optimal value will ...