Questions tagged [linearization]
For questions related to techniques for converting nonlinear expressions in optimization models into equivalent (or approximately equivalent) linear ones.
56
questions
3
votes
1answer
71 views
How to fomulate the following conditional constraint in MILP?
How can I formulate the following conditional constraint to a linear constraint using indicator variables? Please note that all variables are continuous and $c \ge 0$
$\text{1: if} \ c=0 \ \& \ ...
5
votes
1answer
65 views
Linearize a product of an integer variable (not just binary) and a continuous variable?
I have a constraint in my formulation that contains multiplication of an integer variable $y$ and a continuous variable $x$, which is $xy=q$ where $y$ is the number of units in which $q$ gets equally ...
5
votes
3answers
188 views
Problem with binary decision variable constraints in VRP
I would like to create non-linear violation costs in my VRP. I already created my whole VRP with time windows in which I have these decision variable:
...
2
votes
1answer
38 views
Inequality Constraint Linearization of a product of an integer and a binary variable
I have thought I had found the answer here: How to linearize the multiplication of an integer and a binary integer variable?
But the answers to that questions didn't help me find a solution for my ...
3
votes
1answer
71 views
How to linearize a weighted maximum coverage problem?
Is it possible that the binary variables below be modeled as continuous variables?
\begin{alignat}2\max&\quad\sum _{{e\in E}}w(e_{j})\cdot y_{j}\\\text{s.t.}&\quad\sum {x_{i}}\leq k,\quad&...
5
votes
2answers
102 views
Minimizing $x_1/x_2$ over a simplex in the positive orthant
I need to solve the following problem
\begin{align}\min&\quad x_1/x_2\\\text{s.t.}&\quad Ax \leq b\\&\quad x > 0\end{align} where $A$ is a positive matrix.
The best thing I can think ...
3
votes
2answers
189 views
Mocking up conditional statements in LP
I would like to know how if condition statements in linear programming can be reformulated using indicator constraints, and hence solved as a mixed integer linear program. Specifically:
1. Is it ...
5
votes
2answers
90 views
Linearizing objective function with variables inside an indicator function
I am working on a problem in which I am trying to maximize the average of a variable only for the data that meet a certain condition with a constraint on the number of data that meet this condition. I ...
4
votes
1answer
96 views
Transforming a Quadratic constraint to SOCP
I have a problem similar to Markowitz portfolio optimization that I would like to transform into second-order cone programming. I have a linear objective function with a quadratic constraint (assuming ...
8
votes
1answer
259 views
if-else condition for the objective variable using big M notation
Let $0\leq \beta\leq 1$ be an objective variable. The size of $\beta$ is $N\!\times\!N$.
Now, how can I impose the following?
if $\beta_{i,j}>0$ then $\beta_{j,i}=0$
Big M notation can be ...
5
votes
1answer
498 views
How can I deal with a possibly undefined constraint?
I have a minimization problem minimizing $d_k \geq 0$ and some other variables with all strictly positive coefficients. I leave my objective function below to better convey my goal.
$$\min_{\mathbf{d}...
6
votes
1answer
100 views
Linearizing the square root of binary summations
My question is similar to this one and almost identical with this. I am so confused due to indexing and could not make sure if I could apply the solution in here to this indexed version as shown below....
6
votes
1answer
49 views
Linearizing the square root of two binary summations
My question is similar to this one though a bit more complicated. Though my question also includes indices, I am removing them to ease readability.
Let binary variables $x,y\in\{0,1\}$, non-negative ...
10
votes
2answers
958 views
Linear programming: objective function with “buckets”
I had a linear programming problem with the following objective function
$$f(x) = \sum_{j}x_jq_jp_j - \sum_{i}\left(\sum_{j}x_jq_jC_{ij} \right) c_i$$
Where $q, p, C, c$ are known.
This problem was ...
3
votes
1answer
38 views
defining Mixed integer linear inequalities for a set of variables
The problem is described as follows:
considering $n$ variables which are continuous and bounded such that
$$L_i \le x_i \le U_i\quad \forall i=1,2,\dots,n.$$
How can i define a set of mixed integer ...
4
votes
2answers
148 views
How to model If $A \le B$ then $Y = 1$, otherwise $Y = 0$
Somehow I don't get it right.
I would like to model the following:
If $a\le B$ then $Y=1$ otherwise $Y=0$ where $A, B$ are reals and $Y$ is binary.
I can model as follows:
$Y \cdot A \le B$ and ...
5
votes
2answers
167 views
Trade off between number of constraints and bounds of a variable
I am not familiar with the inner working of the solvers. I mostly use the python pulp or IBM CPLEX solver.
For fast execution ...
3
votes
1answer
79 views
Linearizing constraint with continuous and boolean variables
I have two continuous variables $A$, $B$ and two binary variables $x$, $y$.
Condition: if $A = B \wedge x = 1 \wedge y=1$ then $z = 1$ else $z = 0$ from
In an integer program, how I can force a ...
5
votes
1answer
58 views
How to formulate case distinctions in AMPLs objective function?
This is my first real optimisation problem I formulated and now trying to solve by using AMPL.
The following objective function is from a linear 0-1 LP means all variables $x_i^b\in\{0,1\}$, with $i\...
3
votes
0answers
33 views
Extract binary value from continuous variable [duplicate]
I have a continuous variable $c$ which has value in between $[-R, +R]$.
I want to create a boolean variable $x$ and,
$x = 1$ when $c = 1.0$ otherwise $x = 0$
In more general form:
$x = 1$ when $c \...
1
vote
0answers
60 views
Converting Nonlinear Program into an LP
I have a problem with a nonlinear objective function which is
\begin{align}\min&\quad Z_j\cdot(N_j)^{0.5}\end{align} where $j$ is the index.
I want to know how can I turn it into a linear ...
6
votes
1answer
145 views
How to propagate time using linear inequalities?
I have an adjacency matrix $G_{i,j}$ that tells the distance between $i$ to $j$ (between 0 to 1) if there is no edge between $i$ to $j$ I am putting a large integer $100$.
This is my previous ...
6
votes
1answer
68 views
Linearizing objective function with absolute differences
I want to turn this objective function
$$\max \sum_{i=1}^{N-1} \sum_{j=i+1}^N |TX_i^T - TX_j^T|$$
where $T$ is just a vector with increasing integers (e.g $[1 \ 2]$) and $X_i$ is a vector ...
11
votes
1answer
218 views
k-means/k-medoids Clustering Implementation in CPLEX Java
I am trying to model a grouping algorithm as k-means clustering problem, by referring to the general definition as mentioned in Wikipedia.
In my system, I have $N$ nodes that I want to group in $m$ ...
8
votes
2answers
248 views
How can I transform this MILP into an LP problem?
I have a MILP problem with one of the constraints is given below. Sometimes, even for a small-sized problem, the solver takes a very long time to find a solution. What could be an efficient ...
10
votes
1answer
162 views
Linearization of the product of two real valued variables - Binary expansion approach
I want to minimize the following objective function:
\begin{align}\min &\quad x\cdot y\\\text{s.t.}&\quad2 \le x \le 5\\&\quad5 \le y \le 10.\end{align}
Since the objective function is ...
10
votes
2answers
147 views
Linearization $\max(c_1 x_2, c_2 x_2, \ldots, c_nx_n) \geq q$ constraint
I have a MIP minimization problem where I have a maximization in the constraints:
$$\max(c_1x_2,\, c_2x_2,\, \ldots,\, c_nx_n) \geq q$$
Where:
$c_n$ constants
$x_n$ binary variables
$q$ constant
$...
11
votes
4answers
624 views
Integer programming problem with simple quadratic objective function in Python
I have $n$ objects that need to be divided among $k$ groups. Each group must receive at least $5$ objects. In addition, the percentage of objects in group $i$ should be as close as possible to $p_i$ ...
13
votes
6answers
198 views
How to formulate: each pair of elements in $A$ has one common unit in $B$
We have two sets, $A$ and $B$. Some elements of $A$ must be connected to some elements of $B$, but no element of a given set is connected to another element of the same set. (Think of a bipartite ...
10
votes
3answers
1k views
Is there a heuristic approach to the MILP problem?
I have the following optimization problem which is a MILP. I can solve it with a MILP solver.
\begin{align}\min_t&\quad t\\\text{s.t.}&\quad d_{c}-t\le \sum_{n=1}^{N} B_{n,c}x_{n}\le d_{c}+t,...
7
votes
2answers
421 views
Why does a Max constraint work, but this non-negativity constraint does not?
Suppose I have the following constraint: \begin{align}x_{t} &= x_{t-1} + y_{t-1} - z_{t-1}\\x_{t} &\ge 0\end{align}
From my limited experience in coding my own problem, I have found that my ...
8
votes
1answer
197 views
How to linearize the multiplication of an integer and a binary integer variable?
I have the following constraints
\begin{align}\sum_{i=1}^{N}{x_it_i}&= M\\\sum_{i=1}^{N}{t_i}&\le S\end{align}
where $x_i\ge 0$ is an integer variable, $t_i\in\{0,1\}$ is a binary variable ...
11
votes
4answers
1k views
What are the benefits of linearization?
So I am new to OR (not my field, but I have found myself working in it for a thesis project). My problem is a non-linear problem by design and unfortunately I cannot linearize everything, however, ...
8
votes
3answers
384 views
Linearization of a scheduling objective function
I am trying to maximize the workload per employee.
An example:
$e$ the index of an employee
$j$ the index of a project
decision variable: $x_{e,j} \in \mathbb{Z}$ and $0 \leq x_{e,j} \leq 100$ ...
6
votes
1answer
189 views
Network flow model - How can I turn this diagram into a matrix that when converted to RREF solves for max flow?
I have the following network flow model diagram and I have already calculated maximum flow using the R package igraph to be 28. However, what I would like to know ...
6
votes
1answer
146 views
How To Linearize $X = \max\{x_1,x_2\}$
I am new to thinking about math programming and I have a particular constraint I am hoping to reformulate, I just don't know the proper mathematical translation for what I am hoping to do. Enforcing ...
8
votes
2answers
588 views
How to linearize a constraint with max
I would like to linearize a constraint with max. I have the following constraint:
$$\max_{pcj}X_{pwcj}\leqslant L_{wk}.$$
With this constraint, I would like to ensure that for $\forall w \in W$, no ...
12
votes
2answers
156 views
Linearisation techniques for MINLPs
I am wondering what kinds of linearisations people do for MINLPs outside my field of expertise.
I work in global optimisation, so by "linearisation" we would typically mean one of the following:
...
6
votes
1answer
588 views
How to linearize min function as a constraint?
I'm trying to solve an optimization problem including following constraint, and I need to linearize it in a maximization nonlinear programming model. Please help me to reformulate it with mixed ...
6
votes
2answers
483 views
Linearization of objective function
Notation
$\text{src}_{h,s},\text{dst}_{h,s},\text{ch}_{h,s}$ are constants.
$a_{h,s},x_{i,j,s}$ are binary variables.
$\text{wt}_{h,s}$ are continuous variables.
Problem
\begin{align}\min.&\...
20
votes
4answers
1k views
Linearize or approximate a square root constraint
I encounter a nonlinear constraint that contains the square root of a sum of integer variables. Of course one could use nonlinear solvers and techniques; but I like linear programming. Are there any ...
7
votes
1answer
242 views
How to reformulate (linearize/convexify) a budgeted assignment problem?
I have a scheduling problem at hand. In my system, there is a service station with $M$ service outlets, therefore, the service station can serve $M$ users at a time. But, there are $N$ users $N>M$ ...
7
votes
2answers
326 views
How can I linearize or convexify this binary quadratic optimization problem?
I have an optimization problem as below. I am having a hard time with the last constraint.
$\max \eta$
subject to
${\bf U}(:,m)^T{\bf A}{\bf U}(:,m)=0,m=1,2,\cdots,M$
here
$\bf{A}$ is a Binary ...
12
votes
1answer
164 views
QA techniques for optimization problem coding
I often spend much, much, more time QAing and debugging my code than I do actually writing the optimization problem or shaping my data. Are there any tools or techniques to make it easier? I am asking ...
15
votes
1answer
759 views
How to formulate (linearize) a maximum function in a constraint?
How to formulate (linearize) a maximum function in a constraint? Suppose $C = \max \{c_1, c_2\}$, where both $c_1$ and $c_2$ are variables. If the objective function is minimizing $C$, then it can be ...
10
votes
4answers
260 views
How to linearize a constraint with a maximum of binary variables times some coefficient in the right-hand-side
I have the following constraint that I'd like to linearize:
$P$ is a given set
$b_p \in \{0,1\} , \forall p \in P$ a binary variable associated with each element of $P$
$c_p \in \mathbb{R}^+$, a ...
31
votes
8answers
705 views
Modeling floor function exactly
Suppose we want to enforce a constraint
$$
y=\lfloor{x}\rfloor
$$
where $x$ is some continuous variable. One option is to use
$$
x-1\leq{y}\leq{x},\quad y\in\mathbb{Z},
$$
which fails on the edge case ...
12
votes
2answers
81 views
Sensible and realistic way to model truck based transport costs depending on amount
Different kinds of problems involve transporting an amount $x$ from A to B which results in a cost $c(x)$ in the objective function.
Traditionally, often linearized costs are used to get an easy, ...
12
votes
1answer
163 views
McCormick envelopes and nonlinear constraints
I have a problem with a nonlinear constraint. The non-linearity stems from a term of the form $xb$, where $x \in \mathbb{R}^+$, $x < M$ and $b \in \{0, 1\}$. I am able to remove this non-linearity ...
14
votes
5answers
691 views
How to linearize the product of two continuous variables?
Suppose we have two variables $x, y \in \mathbb R$. How can we linearize the product $xy$?
If this cannot be done exactly, is there a way to get an approximate result?
