Linked Questions
17 questions linked to/from How to linearize the product of a binary and a non-negative continuous variable?
6
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2
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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.&\...
- 1,589
10
votes
1
answer
603
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MIP: If integer variable $>0$ it should be equal to other integer variables $>0$
I have an MIP problem where $n$ different types of cars are delivering packages. Sometimes multiple types of cars are required to go to a single location. For example if car $1$ makes two deliveries ...
- 205
12
votes
1
answer
749
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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 ...
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2
votes
2
answers
458
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Mixed Integer Programming with product of a binary variable and multiple continuous variables
Suppose we have a binary variable $x$ and two non-negative continuous variables $y_1\ge 0$ and $y_2 \ge 0$. How can we linearize $xy_1 y_2$ ?
FYI, this is a follow up question to this:
How to ...
- 31
12
votes
2
answers
161
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Pricing of blends/mixtures across multiple timesteps
I have a simple blending problem, where each final product is a blend or mixture of several raw materials, and want to calculate the price per unit of weight for each of the products. So for a given ...
5
votes
1
answer
613
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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.
$$\min_{\mathbf{d}...
- 1,239
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votes
1
answer
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How to linearize the multiplication of an integer and a binary integer variable?
I have the following constraints
\begin{align}\sum_{i=1}^{N}{x_it_i}&= M\\\sum_{i=1}^{N}{t_i}&\le S\end{align}
where $x_i\ge 0$ is an integer variable, $t_i\in\{0,1\}$ is a binary variable ...
- 2,211
5
votes
1
answer
387
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Can we linearize the division of a binary variable by a continuous variable?
I'm trying to solve an MINLP problem where the following division term is appearing in the objective:
$$z_r = \frac{x_{ry}}{\sum_r d_r x_{ry}}, \forall r, y,$$ where $x_{ry}$ is a 2D binary variable ...
4
votes
1
answer
304
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How to express this constraint?
I have the constraint \begin{align}\max&\quad\gamma\\\text{s.t.}&\quad a\ge\gamma b\\&\quad\gamma\le 1\end{align} where $\gamma$ is an optimization variable and $a$ is a function of some ...
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3
votes
1
answer
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Linearizing division of two variables
For all $i \in I,j\in J$ and $k\in K$, define variables $x_{ij}, z_{ijk}\in\{0,1\}$, $y_{ij}\geq 0$ and constants $c_j, e_{ijk}, d_j, f_j >0$.
We have the following constraint
$$\sum_{j\in J_1}c_j\...
- 103
4
votes
1
answer
239
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Can we simplify (perhaps linearize) this constraint?
We are dealing with a stochastic model and one of the constraints is
\begin{align}
y_j=\frac{\sum_{i \in I}\sum_{k \in K}\mathbb{E}\left[X_{ik}^2\right]x^k_{ij}}{\sum_{i \in I} \sum_{k \in K} \mathbb{...
user9659
2
votes
1
answer
110
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Correct way to set a quadratic constraint Xpress
I'm implementing on Xpress a problem with different solution proposed on a paper. The idea is to decompose a matrix $X$ into a convex sum $\sum_{t}\lambda_t M^{(t)}$, where each $M^{(t)}$ has only ...
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3
votes
1
answer
102
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using milp for a linear complementarity problem
I have to minimize $c^Tx$
subject to $Ax = b$, $x_iw_i = 0$ for all $i$, with $x$ non negative continuous and $w$ binary.
What model should I use to solve this problem?
2
votes
2
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89
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How to linearize the product of a binary and a negative continuous variable?
Suppose we have a binary variable $x$ and a negative continuous variable $y$. How can we linearize the product $u=xy$?
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2
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1
answer
74
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Linearizing $y=\sum_{i=1}^n(z+c)\left(\frac{r_i^2}{1-r_i}\right)\phi_i$
Variables $0\le x< 1$, $y,z\ge 0$. We have a constraint
$$y=(z+c)\frac{x^2}{1-x},$$
where constant $c>0$.
We partitioned $x$ into $n$ intervals of equal length and defined a new variable $\phi_i=...
user9659