Questions tagged [linear-programming]

For questions related to problems that optimize (i.e., minimize or maximize) a linear objective subject to linear constraints.

Filter by
Sorted by
Tagged with
1
vote
1answer
56 views

LPs having a 'stable' objective value wrt changes in the constraint right-hand sides

I have a problem as: $$ \begin{align} \begin{array}{cl} \underset{x \in \mathbb{R}^n_+}{\min} & c^\top x \\ \mathrm{s.t.} & Ax \leq \mathbf{1} \cdot b , \end{array} \end{align} $$ where $A \in ...
3
votes
2answers
42 views

Write in ILP: If $x$ within range then $s=1$, else $0$

How can write the following function in LP: $$ s= \begin{cases} 1 & 1 \leq x \leq C \\ 0 & \text{otherwise} \end{cases} $$ where $x$ takes only non-negative integers and $C$ is some large ...
2
votes
1answer
98 views

Benders decompositions: Number of iterations does not remain the same

I am solving an LP (i.e 118-bus system economic dispatch for 130% loading) using Benders decomposition. The problem takes 26 iterations to converge. This means that the process adds 25 cuts to the ...
1
vote
0answers
66 views

Optimisation Multiple Constraints

I am trying to solve a linear algebra problem: an optimisation problem and I am using CVXOPT. I've split the problem into 3 components In its simplest form, The general formulation for CVXOPT is \...
4
votes
1answer
174 views

Are decomposition methods applicable on large linear programs?

Working on a very large Linear Program, we tried out some primitive implementations of decomposition techniques such as Lagrangian relaxation and column generation. However, none of these were able to ...
9
votes
2answers
153 views

Are there small LPs out there?

A fair share of academic research and software development focuses on solving ever-larger problems, particularly when it comes to LPs. I am however curious to know in what contexts and to what extent ...
2
votes
1answer
82 views

Linearly independent rows in simplex

I'm having some trouble understanding about the independent rows in a basic solution. In the book Introduction to linear optimization by B&T, the authors give the definition of a basic solution as ...
6
votes
1answer
93 views

Upper and lower bounds of a variable equal

I'm working on a MILP (Mixed-Integer Linear Programming) problem with the Java API of Cplex. In order to easily exclude some variables from my problem I thought about setting both their lower and ...
2
votes
1answer
65 views

Examples of LP problems in irrigate engineering

I need examples of linear programming problems applied to irrigation system. My idea is to give students many examples of LP applications in the irrigation engineering.
1
vote
0answers
74 views

Minimax problem with a large high dimensional feasible region

How to solve minimax mixed integer problem with a large high dimensional feasible region? \begin{aligned} \max_{\vec{x}}\min_{\vec{y}} \quad & \vec{r} \cdot \vec{x} + \vec{s} \cdot \vec{y}\\ \...
2
votes
1answer
87 views

KKT conditions analysis for binary constraints

I am wondering if boolean constraints in a linear program can be solved (after linear relaxation from $x\in\{0,1\}$ to both $x\ge0$ and $x\le1$) using KKT analysis. Most of the algorithms that I have ...
3
votes
1answer
114 views

Linear Relaxation of Boolean Constraint for Solving Integer Linear Program Using KKT

I am trying to convert a boolean LP to LP using LP relaxation by converting $x \in {0,1}$ to both $x \ge 0$ and $x \le 1$. Then to use it in my problem analysis, I am trying to build the KKT ...
5
votes
1answer
276 views

Model “if and only if” indicator constraints in Linear programming

Apologies if this question has been asked, but I haven't been able to find it. I'm modelling something with Gurobi and want to do the following: \begin{align}\text{cond} < \dfrac{1}{3} &\iff x =...
4
votes
1answer
228 views

CPLEX log always the same after hours (“Gap” and “Best Integer” always blanks)

I'm using the Java API of CPLEX 12.6.1 (with license) to solve a MILP (Mixed-Integer Linear Programming) maximization problem. The point is that, after 21 hours, CPLEX has consumed 32 GB of RAM and ...
3
votes
1answer
73 views

Need help with an appointment scheduling problem

I am currently stuck on writing a linear programming model to describe the process of appointment scheduling for an Oncological Center. I wanted to share it with you guys and see if anyone here could ...
1
vote
1answer
48 views

Employee allocation based on ranking: Mathematical Model

Suppose I have three employees and I have to assign three employees based on their ranks. If an employee has rank 1 that means he is best. Say, I have the following table ...
4
votes
0answers
191 views

Simplified risk game: writing a pratical Minimax objective for mixed integer programming

Problem To ensure fairness of the game, I am writing a bot that plays against itself. I have trouble rewriting a minimax objective to a practical maximization in mixed integer programming. The amount ...
-2
votes
1answer
71 views

MODM to create balanced groups of students to maximise diversity

I am very new to LP, Goal Seek, Decision Models etc and I have a multiple-objective decision making problem that may or may not require a mix of techniques although I am trying to solve it purely ...
4
votes
1answer
76 views

An efficient Integer programming model for the minimum spanning tree problem?

Let $T=(V, E')$ be a spanning tree of a graph $G=(V, E)$. Rather than verifying for any subset of vertices $S\subseteq V$ that $|E'(S)|=|S|-1$, is there an efficient way to satisfy the spanning tree ...
3
votes
1answer
114 views

Best way to add dummy to transportation problem? Zero cost will be always chosen first?

I know that an unbalanced transportation problem could be made a balanced transportation problem by adding a dummy node which equals the difference between demand and supply. In literature, dummy ...
5
votes
1answer
240 views

What is a good way to penalise LP relaxation?

I have a binary integer program. It is of a large size and the solver is taking longer time. I am thinking of relaxing the binary integer variable and making it a continuous variable. How can I ...
2
votes
0answers
52 views

How is the dual revised simplex method equivalent to running the RSM on the dual problem?

I've seen the claim (in the title) several places, but can't quite understand why it's true. From what I understand so far, the revised simplex method solves an LP in standard computational form, $$\...
2
votes
2answers
91 views

Transformations that leave the linear program unchanged

A typical linear program is written as $$L_0:\min_{x \geq 0; A^\top x \leq b}c^\top x.$$ Here, $x \in \mathbb{R}^n$, $c \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. ...
0
votes
1answer
85 views

Objective function with exponential coefficients

I have a linear programming problem, with $n$ variables and $a\leq x_{i} \leq b$ for each variable $x_{i}$, where the objective function is $\min \sum\limits_{i=1}^{n}{2^{i} x_{i}}$ Is it true that, ...
10
votes
1answer
612 views

Linear Optimization Library for C++ with GPU Support

Does anyone know any linear optimization libraries for C++ supporting GPUs for parallelization? If multiple, which do you recommend? The GPU support is important to me since I am dealing with large ...
-4
votes
1answer
95 views

Multiple If else constraints in Mixed integer programming

How to formulate the following as constraints in MILP? a[0][0] = y, if x[0]= 0, a[0][0] = 0, if x[0] != 0, . . . . a[i][j] = b[i][j-1] + y, if x[j]=i, a[i][j] = a[i][j-1], if x[j] != i, ... . . . b[0]...
1
vote
0answers
46 views

Specializing Iterations of Dantzig-Wolfe Decomposition with an Oracle

This arises from an engineering problem I am working on. Let $\mathbf{c}_i,\mathbf{a}_i\in \mathbb{R}^{d}$ be a given set (collection) of vectors where $i\in\{1,\dots,n\}$. Define the bounded ...
5
votes
2answers
159 views

Asymmetric time-constrained capacitated vehicle routing problem

I am trying to add some more constraints to the flow-based ADVRP model in Almoustafa et al. (2013)1 (pp.4). The mentioned model caps the travel distance, while I cap the travel time. Let $U$ represent ...
4
votes
1answer
113 views

Extreme points of a simple polyhedron

Consider the polyhedron given by the set of inequalities \begin{align} \mathbf{b}^T\mathbf{x} ~&\leq~ c \\ \mathbf{e}^T\mathbf{x} - 1 ~&\leq~0 \\ \mathbf{x}~&\geq~0 \end{align} where $\...
4
votes
2answers
143 views

Any Solution for $k$-means with minimum and maximum cluster size constraint?

I am looking for an efficient approach to $k$-means clustering with minimum cluster size constraints. The clusters are non overlapping, so, one point can belong to only one cluster. $N$ be the number ...
2
votes
1answer
66 views

How to linearize inequalities having max or min?

I'm modeling an LP problem in which I have to maximize an objective function. Two of the constraints are the following, where $k_i$ are constants and $x_i$ decision variables (continuous). Could ...
5
votes
4answers
430 views

How to visit a subset of network nodes in a single trip?

I have a connected network where I want to visit a set of destinations which may require visiting intermediate nodes as well because there may be no direct edge between source and destination nodes. I ...
2
votes
2answers
95 views

How to deal with a decision variable in the objective function that depends on if-else conditions involving other decision variables?

I'm modeling an optimization problem in which a decision variable $x_1$ in the objective function depends on if-else conditions involving decision variables $x_2$ and $x_3$, as the following equation, ...
4
votes
1answer
311 views

Is there any automatic way to spot contradictory constraints in linear programming?

Let's have the following trivial linear program: \begin{align}\max&\quad z=20A+30B\\\text{s.t.}&\quad A\le60\\&\quad B\le50\\&\quad A+2B\ge220\\&\quad A,B\ge0\end{align} It's easy ...
4
votes
1answer
60 views

Name for subclass of ILP without any inequality constraints (including constraints on x)

In "Myths and Counterexamples of Mathematical Programming" myth "IP Myth 21" says: The problem of finding $x\in \mathbb{Z}$ such that $Ax=b$, where $A\in\mathbb{Z}^{m\times n}$ ...
1
vote
0answers
64 views

LP instead of IP formulation of assignment problem

In the example files of GLPK, the assignment problem is written as a linear program. I don't understand why this isn't an integer programming problem. The problem formulation: ...
4
votes
3answers
108 views

warmstarting simplex algorithm- how much can problems differ from each other?

I'm working on an implementation of the simplex algorithm. I want to solve problems in real time every 30 minutes. They could be interpreted as a classic transportation problem. I couldn't really say ...
1
vote
1answer
137 views

How to linearize the product of a binary and a continuous variable? [duplicate]

Suppose we have a binary variable $b \in \{0, 1\}$ and a continuous (possibly negative) variable $y \in \mathbb{R}$. How can we linearize the product $b \cdot y$?
4
votes
2answers
156 views

Scheduling optimisation constraint on consecutive shifts & consecutive night shifts (`python`)

I am trying to write a program to schedule a team of 8 individuals into shifts. I want to know how to model that every individual must get at least one night shift break, and must not work two ...
2
votes
3answers
122 views

Linearizing a Max Function in the constraint - not working

I have a minimization function which is in its simplest form looks like below. I am including the index of the variables. ...
0
votes
1answer
256 views

How to mathematically formulate the optimization problem?

I have a system with $S$ service points. There are also $U$ users in the system. We have $$U>S>G$$ One group can have maximum $M$ service points, but there is no restrictions on the number of ...
0
votes
1answer
90 views

Mixed Integer Programming - How to model the dependency of two variables in an objective function

I have two variables $a$ and $b$, in which $a$ is the amount of goods and $b$ is the amount of boxes of the given sizes. So $b$ (box size + number) is dependent on a (goods quantity). If $a$ is ...
3
votes
0answers
76 views

How to linearize a max min objective function?

Let us suppose that I have a $\max \min$ objective function that only depends on one set of variables: $\underset{x}\max \underset{y}\min dy$ Associated with the linear set of constraints and right ...
1
vote
1answer
78 views

Job Shop Scheduling Problem: jobs are scheduled on the same machine at the same time

I want to solve a job shop scheduling problem. I got $n$ Jobs that have to be scheduled on $k$ Machines. A Job $i$ has 2 or 3 Tasks $j$, and there is a known sequence of the Tasks of a Job. One ...
3
votes
0answers
104 views

0 1 solution of linear programming problem with only equality constraints

I have a linear programming problem $LP$ where all the variables $x_{i}$ take value in $\left[0, 1\right]$ (that is $0\leq x_{i} \leq 1$). All the constraints are as follow: $a_{1}+a_{2}+a_{3}=1$ that ...
0
votes
0answers
37 views

linear equation with conditions in Java

I have the following problem: I want to determine a linear equation for a data table with $x$ and $y$ value. Point 1 should be the point where the first time $y > 0$ and point 2 should be the point ...
1
vote
1answer
93 views

complexity order of the interior point method

I was wondering why the complexity order of the interior point method is O()^3 or O()^3.5? Much appreciate your time and consideration.
0
votes
1answer
79 views

How to linearise this nonlinear constraint?

I have a constraint in the form $\sum_{n=1}^{N}x_{m,n}\omega_{m,n}\ge (t_u-1)\beta_u, \forall u, u=1,2,\cdots, U$ where $x_{m,n}$ is binary variable $t_u$ and $\beta_u$ are continuous optimization ...
1
vote
1answer
104 views

How to transform this problem with logarithmic objective function into an approximated convex optimization problem?

I have an objective function as follows $\underset{x_{m,n}}{\max}\hspace{1mm}\hspace{1mm}\sum_{m=1}^{M}\log_2\left(\frac{\sum_{n=1}^{N}(1-x_{m,n})\omega_{m,n}+z}{\sum_{n=1}^{N}x_{m,n}\omega_{m,n}}\...
0
votes
0answers
40 views

Linear equation from pairs of values with conditions in Gurobi (Java)

How can I create a linear equation in Gurobi (Java) from values (x-y value pairs) that also has the following properties: $\forall x \leq 0 \Longrightarrow y = 0$ The linear line/equation should have ...

1
2 3 4 5
8