Questions tagged [linearization]
For questions related to techniques for converting nonlinear expressions in optimization models into equivalent (or approximately equivalent) linear ones.
80
questions
3
votes
1answer
45 views
How/when can we use MINLP engines instead of linearizing MP models?
Nowadays, mathematical programming solvers have been frequently used to solve lots of practical/academic problems. Many of these might be interpreted as a MIP or MINLP to represent a specific problem (...
6
votes
1answer
141 views
Linearizing a program with multinomial logit in the objective
I have a nonlinear problem as follows: \begin{align}\min&\quad\sum_{k=1}^{K}\left|y_k - \sum_{i=1}^{N} \frac{e^{x_{k}^{i}}}{\sum_{j=1}^{K} e^{x^{i}_{j}}}\right|\\\text{s.t.}&\quad x^i_{j} \ge ...
5
votes
1answer
56 views
Maximizing a Ratio/Percent
I'm using cvxpy to model a problem. Inside a very large and complex LP, I create two continuous, affine (unconstrained) expressions: $x$ and $y$. Due to how they ...
1
vote
2answers
148 views
Switching of decision variables to be larger than or equal to a decision variable according to an indicator variable value
I would like to seek some advice on modeling the following:
I have two integer decisions variables, $x, x'$, that are either equal or greater than zero and either of them is greater than or equal to a ...
1
vote
3answers
200 views
How can I linearize this IF-THEN constraint?
Let
$P_{t,u}; t=1,2,\ldots,T, u=1,2,\ldots,U$ be known values
$\alpha$ is also a known parameter
$X_{t,u}$ an optimization variable
I have the following constraint: IF $P_{t,u}\geq\alpha$, THEN $X_{...
4
votes
3answers
195 views
How to linearize the Min function while letting the binary variable to be fixed for x1==x2 as well?
As discussed here, the min function, i.e $X = \min\{x_1,x_2\}$, can be linearized as follows:
\begin{align}
X & \le x_1 \\
X & \le x_2 \\
X & \ge x_1 - ...
4
votes
1answer
88 views
Formulating these logical constraint in an ILP
I have these two constraints :
$z \leq My$
$t \leq M'y $
where $z$ and $t$ are two integer variables $ z, t\geq 0$, $y$ is a binary variable, and $M$, $M'$ are two big numbers.
So basically these ...
2
votes
1answer
115 views
Linearize sum of continuous and boolean variable
For maximizing the objective function $\sum_i{d_i y_i}+ A x - B \cdot \mathbb{I}_{x>0}$, how can I linearize $ A x - B \cdot \mathbb{I}_{x>0}$ term where $d_i, A$ and $B$ are positive constants ...
6
votes
1answer
146 views
Strong MIP formulations for a large-scale mixed-integer nonlinear feasibility problem
I'm trying to construct a strong MIP formulation for the following integer nonlinear feasibility problem.
Informally:
We have a $m \times n$ decision matrix of binary variables
Each row of the matrix ...
3
votes
1answer
90 views
Logical constraint in ILP
I want to write the following constraint:
Let $z$ be an integer variable such that $0\le z\le M$, and $t$ be a binary variable where $M$ denotes big-M. The logical constraint is as follows:
if $z \...
3
votes
1answer
108 views
How to express this logical constraint for an ILP?
I am trying to write an ILP for a problem but I have this logical constraint and I'm stuck.
In my model I have:
two binary variables: $x$ and $y$
One Integer variable: $z$
The logical constraint I am ...
10
votes
6answers
2k views
Nonlinear integer (0/1) programming solver
I have the following optimisation problem.\begin{align}\max&\quad\sum_i\sum_j\sum_k x_{ji}y_{kj} \operatorname{cost}(i,k)\\\text{s.t.}&\quad\sum_j x_{ji}=1\quad\forall i\\&\quad\sum_k y_{...
3
votes
1answer
51 views
Linearizing separable functions: SOS2 sets or binary variables
When linearizing a separable nonlinear function is there an advantage/disadvantage in using SOS2 sets in comparison to using binary variables?
5
votes
1answer
95 views
Minimize binary variable's distance with respect to the index values
For a given binary decision variable $x[i,j,k]$ my goal is to get as dense results in terms of k for successive values of j. Distance of k value to be kept as close as possible throughout j values:
$d ...
8
votes
2answers
316 views
knapsack problem with non-linear constraint
I have a basic knapsack problem where I need to fit the most weight possible in a bin:
...
5
votes
2answers
142 views
How to linearize a quadratic constraint to add it then via a callback function
Suppose we have a positive continuous variables $0 \le x \le UB$ where $UB$ is a known upper bound.
How can we linearize the term $x^2$?
Detailled problem:
Suppose that via a callback we compute a ...
5
votes
1answer
109 views
Linearizing a constraint with square root of a variable
I am trying to linearize the constraint set (2) in the following simplified program. The parameters: $A,C,D,T\in\mathbb{R}^+$. The set $\mathcal{J}$ is polynomially-sized.
\begin{alignat}2\min &\...
1
vote
0answers
45 views
linearisation of if-then constraint [duplicate]
Is it possible to translate this if-then constraint in linear manner to be solved with a linear programming solver?
If not, how can it be translated into a integer programming manner to be solved by ...
3
votes
1answer
77 views
Linearizing power term in objective function
I would like to linearize $x^2$ term in my objective function. I understand this can be solved using quadratic programming solver; however, for my use case linearization is necessary to convert it to ...
2
votes
1answer
85 views
Linearize sum of product in objective function
Notation:
I have an optimisation problem with objective function: \begin{align}\max&\quad\sum_n Q_n\\\text{s.t.}&\quad Q_n=x_{ij}^{nk}(y_i^{nk}-c_n), \forall n \in N, k \in K, (i,j) \in P.\end{...
3
votes
0answers
72 views
Linearization of the shifted copy of a function
Suppose in a model I have the expression $y_{1}(x) = 10 + 5 x^2$ where $x \in [0,20]$ is a continuous variable. In order to be able to use an MILP solver, I piecewise linearise $z_{1} = x^2$, by ...
6
votes
1answer
133 views
Convert summation of min functions into linear constraints for optimization
I have the following optimization problem:
$$
\mbox{maximize } j^{*} \mbox{ subject to:} \sum_{j^{*}\leq j\leq J} \min({\bf A}_j,{\bf B}_j) \geq \lambda, \lambda \in \mathbb{R} \mbox{ and } {\bf A}_j,{...
1
vote
0answers
38 views
Linearize max function in a constraint [duplicate]
I have a constraint as follows:
$ \sum_i {r_i} \geq \max \{g_j, B_j\} $
where, $r_i$, $g_j$ are variables and $B_j$ is a parameter.
How do I linearize the constraint (I suppose using big-M method)?...
3
votes
0answers
46 views
Linearisation using SOS2
I am trying to linearise the following expresssion.
$C(k) = B(k) e^{-d(k)}, B(k) \ge 0 , d(k) \ge 0 $
I am trying to do this by using SOS2 sets.
I set $X(k) = e^{-d(k)}$ and I get $C(k) = B(k) X(...
3
votes
1answer
116 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
83 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
198 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
63 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
74 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
117 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
232 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
143 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
172 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
285 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
513 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
118 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
58 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 ...
11
votes
2answers
990 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
42 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 ...
5
votes
2answers
250 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 conditional:
If $A\le B$ then $Y=1$ otherwise $Y=0$
where $A, B$ are reals and $Y$ is binary.
I can model as follows:
$Y \cdot A \le B$...
5
votes
2answers
187 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
102 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
61 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
36 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
64 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
71 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
259 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
274 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 ...
11
votes
1answer
231 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 ...
