Questions tagged [linearization]
For questions related to techniques for converting nonlinear expressions in optimization models into equivalent (or approximately equivalent) linear ones.
188
questions
2
votes
3
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98
views
Linearization the product of three variables (two binary & one continuous)
Consider the following binary variable $x \in \{0,1\}$ and two continuous real variables $y,p \in \mathbb{R}$.
I am trying to model the following conditional equations as constraints:
\begin{cases}
...
- 103
3
votes
1
answer
48
views
How to enforce logical implication $\sum_j a_j x_j \le b \implies \sum_j c_j x_j \le d$
Some modeling languages and solvers support indicator constraints of the form $$y=\hat{y} \implies \sum_j a_j x_j \le b,$$ where $y$ is a binary decision variable and $\hat{y}\in\{0,1\}$ is a constant....
- 27.3k
3
votes
3
answers
190
views
Equivalence between constraints in ILP
Let's have binary variables $x$ and $y$. I'd like to define a helping binary variable $z$ such that
$$ z = 1 \; \;\; \mathrm{iff} \; \; \; x + y = 2.$$
If I wanted to express the equivalence between ...
- 133
1
vote
1
answer
103
views
Linearize conditional constraint
Consider a variable c from the domain {-1,0,1}. I have the following constraint:
IF $c = 1 \Rightarrow x = 1 $ ELSE $x = 0$
How do I linearize this constraint?
- 21
2
votes
0
answers
47
views
The linearization of the logical constraints
I know the logical constraints can be linearized by either the logical representation of whose relation, (for pure binary variables e.g. CNF/DNF) or for general form by using Big-M formulation. As I ...
- 7,281
4
votes
2
answers
270
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The linearization of the (Iff-and-only-Iff) expression
I am trying to linearize the following expression without using the Big-M formulation, but I cannot convert it. I am willing to know if there exists an efficient way to do that?
$$ Iff \quad (w=1) \...
- 43
3
votes
2
answers
148
views
Reformulate bilinear binary constraint
I'm a solving a model that has the following constraint:
$$
c_{p,n} = \sum_{s\in S}\sum_{i \in \{1,2,3\} } x_{p,s,i-1} x_{n,s,i}, \forall (p,n) \in C
$$
where both the $c$ and $x$ variables are binary,...
- 183
0
votes
1
answer
53
views
Converting a piecewise function to a linear equation as a constraint
The value of one of the variable of my model (alpha_1) is given by a piecewise function. Each element of the piecewise depends on the value of some other binary decision variables (X1, x2, x3).
I'd ...
- 67
1
vote
1
answer
49
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Linearize constraints on a truncated variable
Let $K$ and $Q$ be two variables, and $Q_\min$ and $Q_\max$ be two parameters. I need a series of linear constraints to define $Q$ vis-a-vis the value of $K$ based on the following rules:
If $K \le ...
- 11
2
votes
1
answer
62
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Nonlinear fractional objective function
Could you please teach me when an optimization model with fractional terms in the objective function can be linearized or solved optimally?
I only know that if the objective function has a single ...
2
votes
2
answers
202
views
Linearizing a disjunctive expression into MILP
I want to linearize the following disjunctive form.
$$\left[\begin{gathered}w_{1}\\x \geq a\end{gathered}\right] \vee \left[\begin{gathered}w_{2}\\x \geq b\end{gathered}\right]$$
where $w_1$ and $w_2$...
- 7,281
3
votes
1
answer
199
views
How to deal this L0 norm of a vector of L2 or L1 norms in objective?
I have an optimization variable denoted as ${\bf A\in\mathbb{C}^{100\times 5}}=[{\bf a}_1\hspace{1mm} {\bf a}_2 \hspace{1mm} {\bf a}_3 \hspace{1mm} {\bf a}_4 \hspace{1mm} {\bf a}_5];$
Here, ${\bf a}_1$...
- 2,179
2
votes
0
answers
33
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Choosing upper and lower bound using big-M [duplicate]
This question is related to my previous question posted here: Piecewise constraint using big-M notation and this question posted on the math stackexchange: https://math.stackexchange.com/questions/...
- 43
3
votes
2
answers
100
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How to linearize or fix this disciplined convex programming error?
How can I linearize this constraint
$$d_{u,c}\sigma \le \|{\bf f}_{u,c}\|^2\le Td_{u,c}$$
$\sigma$ is a very small number based on scale of $f$
$T>0$, ${\bf f}_{u,c}$ is optimization variable, a ...
- 2,179
1
vote
1
answer
65
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MILP: Substituting products with additive logarithms
I would like to linearize a product, for example $a*b$. if I solve my solution in log space, I can formulate it as $a+b$ and when my final output is returned, remember to convert back to original ...
- 471
4
votes
1
answer
160
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How to solve a "nearly" linear program
Given a positive integer $n$, a constant $k=2/3$, and $7$ variables $x_1, x_2, x_3, x_{12}, x_{13}, x_{23}, x_{123}$ (non-negative reals or integers) I would like to find:
$$\min \binom{x_1}2$$
...
- 393
2
votes
1
answer
85
views
Multiple absolute values with multiple variables in an LP
Assume that we have a LP with the constraint
$$ \sum_{j} \left(c_j x_j + |c_j x_j - \alpha_j + \beta_j|\right) \leq y $$
and
$$\alpha_j + \beta_j \leq \lambda_j $$
for all $j$, where the (positive) ...
- 145
5
votes
3
answers
183
views
MIP constraint with sum of decision variables having certain value : $\sum_{i=1}^nx_i = 2 \implies \delta = 1$
I want to formulate a MIP constraint such that :
$$\sum_{i=1}^nx_i = 2 \implies \delta = 1$$
$x_i, \delta \in \{0, 1\}$.
My problem is that delta should be one when this sum is exactly 2 and not ...
- 235
3
votes
1
answer
62
views
Outer approximation approach for MINLP
Does anybody know why in the outer approximation approach for MINLP it is not necessarily/needed to solve MILP to optimality? What is the rationale or explanation behind it?
2
votes
1
answer
57
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Expressing inner product of binary variables in MIP
I have a $m$ by $n$ matrix $X$ of binary variables in my MIP which represents a list of $m$ items each belonging to one of $n$ categories. $m$ is usually around $1,000$ while $n$ is much lower at ...
- 123
3
votes
0
answers
80
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Linearize objective function with non-linear terms
I have a problem with linear constraints but in the objective function I want to have some linear terms along with a $x^2$ term. So it is like the following:
$$\min \sum \limits _i \sum \limits _j (a[...
2
votes
1
answer
364
views
Optimization problem with if condition as constraint
I am trying to solve an optimization problem where the constraint contains absolute values and I am not sure how I can express this in a 'Pyomo-friendly' way.
Consider the following optimization ...
- 175
3
votes
1
answer
212
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Linearize objective function in MILP
I have an objective function that I want to linearize but want to confirm that I'm doing it correctly. There are some constraints that are linear in $x$ but they're unimportant for the problem. The ...
- 45
3
votes
2
answers
196
views
How to model logic constraint: $y=1$ if $a\le x\le b$ and $y=0$ otherwise?
I am trying to formulate indicator-type of constraints. $y$ is binary $0$ or $1$ and $x$ is a continuous variable.
$$ y =
\begin{cases}
1, & \text{ if } a \leq x \leq b \\
0, & \...
- 53
2
votes
1
answer
120
views
Linearize a higher order polynomial objective function?
My question up front with context below:
Is there a generalized linearization possible for a higher order polynomial (max degree 6 in my case) involving a mix of binary and real variables? If not, ...
- 23
2
votes
1
answer
140
views
Why MiniZinc do not do convert to linear constraint a quadratic constraint?
I would like to know which are the advantage to do not convert quadratic expressions into linear expression in MiniZinc. For example let be the following simple MiniZinc code
...
- 179
2
votes
1
answer
106
views
Linearize a product of binary variables with 2 indexes
I have the following inequality that I would want to linearize.
Consider that $r_{ij}, x_{ij}, y_{ij}$ are binary variables defined for every pair of nodes $(i,j) \in A$. Also, I have a set of nodes $...
5
votes
1
answer
206
views
Linearize minimum and maximum constraint with variable and constant
Let's say I want to linearize the restrictions: $ \min(0, y) \leq x \leq \max(0, y) $
Then I can define $y_{\max}$ and $y_{\min}$ such that:
$$ y_{\max} \geq 0 \\ y_{\max} \geq y \\ y_{\min} \leq 0 \\ ...
- 153
5
votes
1
answer
141
views
Absolute value in an equality constraint
What is the best way to model or represent an equality constraint which includes an absolute term in the expression:
$$ x = |y| $$
$x \in \mathbb{R^+}$ and $y \in \mathbb{R}$
- 103
6
votes
2
answers
163
views
When should we avoid linearizing a quadratic term?
Some solvers like Gurobi can handle mixed-integer quadratically-constrained quadratic models regardless of their nonconvexity. I have some experience that Gurobi can handle instances of the max $k$-...
- 169
2
votes
1
answer
152
views
How to write constraint with sum of absolutes in Integer Programming?
I found a solution for just one term here
How can we formulate constraints of the form
$$ \sum_{i=1}^n |x_i -a_i| \ge K $$
in Mixed Integer Linear Programming ?
- 203
2
votes
2
answers
339
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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
2
votes
1
answer
201
views
linearize bilinear quadratic objective terms
I need to model a problem as a linear program. However my working solution contains a (bilinear) quadratic objective term:
$$
\sum x_i * y_i \\
x \in \{0,1\} \\
y \in \mathbb{R}^+
$$
The value of $y$ ...
- 147
3
votes
1
answer
515
views
How to minimize the sum of absolute values
How can I solve a problem such as the following:
$$
\text{minimize}~~~ \sum_{i=1}^n |x_i|
\\
\text{subject to}~~~ A x \geq b
$$
?
Without the absolute values on the variables, it is a simple linear ...
7
votes
2
answers
841
views
Is there a better way of defining a constraint on positive integer variables such that no two variables are the same and are uniquely assigned a value
So suppose I have integer variables $x_1,x_2,\dots,x_N$ and I enforce that the integer variables are bounded i.e $1 \leq x_i \leq N$
I was interested in posing a constraint so that in the collection $...
- 205
4
votes
2
answers
415
views
Transform nonlinear cost function to get LP or MILP
I'm trying to schedule power of multiple prosumers in a microgrid.
The problem includes a cost function with min and max ...
- 143
2
votes
2
answers
180
views
Difference between constraint formulation and performance
I am wondering about the characteristics and performance of some constraints with only binary variables. I assume that solving (integer) linear programs is faster than quadratic ones.
At first:
$$
a,b,...
- 147
2
votes
1
answer
178
views
Linearize function
I have a facility location problem with a non-linear objective;
There are fixed costs $S_j$ to opening facility $j$
$Y_j$ is a binary, $1$ if facility $j$ is opened, $0$ otherwise
$D_j$ is the number ...
- 99
1
vote
0
answers
154
views
Converting quadratic constrains to linear constraint [closed]
I try to convert a quadratic constraint to a linear one:
$$
w_j = \sum w_\text{j,i} \\
w_\text{j,i} = \frac{w_j}{D} \times u \\
w_j,D,u \in \mathbb{N} \\
$$
The values for $w_j$ and $D$ are constant ...
- 147
6
votes
2
answers
539
views
How to solve Rogo Puzzle with an extra constraint?
Given a n×m grid with numbered cells and forbidden cells, the objective of the Rogo puzzle is to find a loop of fixed length through the grid such that the sum of the numbers in the cells on the loop ...
- 1,076
1
vote
1
answer
84
views
If $x=\min\{f(\mathbf{a}),1-\epsilon\}$, how can we model and partition $x$?
I have been dealing with a problem for sometime and although tried different things and asked some questions before, I think the problem might be somewhere that we didn't look before.
Variables $0\le ...
user9659
2
votes
1
answer
70
views
Linearizing $y=\sum_{i=1}^n(z+c)\left(\frac{r_i^2}{1-r_i}\right)\phi_i$
Variables $0\le x< 1$, $y,z\ge 0$. We have a constraint
$$y=(z+c)\frac{x^2}{1-x},$$
where constant $c>0$.
We partitioned $x$ into $n$ intervals of equal length and defined a new variable $\phi_i=...
user9659
2
votes
3
answers
733
views
How to use condition in cplex?
I want to use conditions to my variable.
dvar boolean x[I][J][K][L]
dvar in h[i]
my code is
...
- 21
5
votes
1
answer
112
views
Linearizing this absolute difference objective function $\min\sum_{i=1}^{I}\sum_{j=1}^{i}|x_i-x_j|$
For $x_i>0, i=1,\ldots,I$, I tried to linearize this objective function
$$\min\sum_{i=1}^{I}\sum_{j=1}^{i}|x_i-x_j|$$
as
$$\min\sum_{i=1}^{I}\sum_{j=1}^{i}y_{ij}$$
subject to
\begin{align}
& y_{...
user9659
6
votes
2
answers
404
views
How to measure the tightness of MILP models?
Suppose we have a MILP model. How can we say this model is tight or not?
How to make it more tight? Any advice or example?
- 1,076
5
votes
2
answers
792
views
Knapsack - How to optimize bonuses for each pair of items
I am trying to solve a variation of the knapsack problem where every pair of items in my knapsack has a bonus or penalty associated with it.
My knapsack can hold a dozen items
There are thousands of ...
- 197
2
votes
1
answer
84
views
How to optimize multiple linear regressions based on cost?
I have an optimization problem where I'd like to maximize revenue and I have separate variables for cost and revenue.
Building a single unit of a product takes 100 hours of labor
I have a list of ...
- 197
3
votes
3
answers
692
views
Converting if conditions to linear constraints
I have an optimization problem and I want to convert the following if conditions to linear constraints:
If $(y_1 > U_1)$ and $(m_1)$ and $(E_1)$ then $x_1=1$
If $(y_2 > U_2)$ and $(m_2)$ and $(...
- 77
8
votes
2
answers
712
views
MILP Penalty Function Only for Negative Values
This is (hopefully) an easy answer but I haven't dealt with this before.
I have a MILP which includes an unbounded, continuous decision variable. However, I generally don't want this decision ...
- 701
4
votes
1
answer
89
views
Modeling $x=1$ iff $y\leq D$ and $x=0$ otherwise (either-or-constraints)
We have decision variables $x\in\{0,1\}$ and $y>0$. We know that $x=1$ if and only if $y\leq D$ and $x=0$ iff $y>D$. $D>0$ is a model parameter.
How I modeled these constraints is
\begin{...
user9659