Questions tagged [linearization]
For questions related to techniques for converting nonlinear expressions in optimization models into equivalent (or approximately equivalent) linear ones.
152
questions
2
votes
2
answers
139
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:
$$
a,b,...
- 65
2
votes
1
answer
143
views
Linearize function
I have a facility location problem with a non-linear objective;
There are fixed costs $S_j$ to opening facility $j$
$Y_j$ is a binary, $1$ if facility $j$ is opened, $0$ otherwise
$D_j$ is the number ...
- 31
1
vote
0
answers
82
views
Converting quadratic constrains to linear constraint [closed]
I try to convert a quadratic constraint to a linear one:
$$
w_j = \sum w_\text{j,i} \\
w_\text{j,i} = \frac{w_j}{D} \times u \\
w_j,D,u \in \mathbb{N} \\
$$
The values for $w_j$ and $D$ are constant ...
- 65
6
votes
2
answers
518
views
How to solve Rogo Puzzle with an extra constraint?
Given a n×m grid with numbered cells and forbidden cells, the objective of the Rogo puzzle is to find a loop of fixed length through the grid such that the sum of the numbers in the cells on the loop ...
1
vote
1
answer
79
views
If $x=\min\{f(\mathbf{a}),1-\epsilon\}$, how can we model and partition $x$?
I have been dealing with a problem for sometime and although tried different things and asked some questions before, I think the problem might be somewhere that we didn't look before.
Variables $0\le ...
user9659
2
votes
1
answer
61
views
Linearizing $y=\sum_{i=1}^n(z+c)\left(\frac{r_i^2}{1-r_i}\right)\phi_i$
Variables $0\le x< 1$, $y,z\ge 0$. We have a constraint
$$y=(z+c)\frac{x^2}{1-x},$$
where constant $c>0$.
We partitioned $x$ into $n$ intervals of equal length and defined a new variable $\phi_i=...
user9659
2
votes
3
answers
205
views
How to use condition in cplex?
I want to use conditions to my variable.
dvar boolean x[I][J][K][L]
dvar in h[i]
my code is
...
- 21
5
votes
1
answer
101
views
Linearizing this absolute difference objective function $\min\sum_{i=1}^{I}\sum_{j=1}^{i}|x_i-x_j|$
For $x_i>0, i=1,\ldots,I$, I tried to linearize this objective function
$$\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}$$
subject to
\begin{align}
& y_{...
user9659
6
votes
2
answers
305
views
How to measure the tightness of MILP models?
Suppose we have a MILP model. How can we say this model is tight or not?
How to make it more tight? Any advice or example?
5
votes
2
answers
749
views
Knapsack - How to optimize bonuses for each pair of items
I am trying to solve a variation of the knapsack problem where every pair of items in my knapsack has a bonus or penalty associated with it.
My knapsack can hold a dozen items
There are thousands of ...
- 197
2
votes
1
answer
63
views
How to optimize multiple linear regressions based on cost?
I have an optimization problem where I'd like to maximize revenue and I have separate variables for cost and revenue.
Building a single unit of a product takes 100 hours of labor
I have a list of ...
- 197
3
votes
3
answers
399
views
Converting if conditions to linear constraints
I have an optimization problem and I want to convert the following if conditions to linear constraints:
If $(y_1 > U_1)$ and $(m_1)$ and $(E_1)$ then $x_1=1$
If $(y_2 > U_2)$ and $(m_2)$ and $(...
- 77
8
votes
2
answers
590
views
MILP Penalty Function Only for Negative Values
This is (hopefully) an easy answer but I haven't dealt with this before.
I have a MILP which includes an unbounded, continuous decision variable. However, I generally don't want this decision ...
- 661
4
votes
1
answer
70
views
Modeling $x=1$ iff $y\leq D$ and $x=0$ otherwise (either-or-constraints)
We have decision variables $x\in\{0,1\}$ and $y>0$. We know that $x=1$ if and only if $y\leq D$ and $x=0$ iff $y>D$. $D>0$ is a model parameter.
How I modeled these constraints is
\begin{...
user9659
5
votes
1
answer
152
views
Can we linearize the division of a binary variable by a continuous variable?
I'm trying to solve an MINLP problem where the following division term is appearing in the objective:
$$z_r = \frac{x_{ry}}{\sum_r d_r x_{ry}}, \forall r, y,$$ where $x_{ry}$ is a 2D binary variable ...
2
votes
1
answer
242
views
How can I linearize this nonlinear variable relationship?
Assume a mathematical optimization problem with two positive continuous variables:
0 <= x <= 1
0 <= y <= 1000
I am seeking an efficient way to express ...
- 29
3
votes
2
answers
208
views
Piece-wise linear approximation of a constraint
We have a decision variable $0<y<1$ and the following constraint
$$z=\frac{y^2-y+1}{y(1-y)},\tag{1}$$
We also have another constraint
$$y=f(x),\tag{2}$$
where $f(x)$ is a linear function of $x$.
...
user9659
5
votes
2
answers
155
views
Linearizing $x^2/(1-x)$ by partitioning the interval $0<x\le X$
We have two decision variables
\begin{align}
& 0<x\le X,\\
& 0<y\le Y,
\end{align}
where both $X$ and $Y$ are two sensible upper bounds on our decision variables.
We also have a ...
user9659
3
votes
1
answer
199
views
Linearizing division of two variables
For all $i \in I,j\in J$ and $k\in K$, define variables $x_{ij}, z_{ijk}\in\{0,1\}$, $y_{ij}\geq 0$ and constants $c_j, e_{ijk}, d_j, f_j >0$.
We have the following constraint
$$\sum_{j\in J_1}c_j\...
- 103
4
votes
1
answer
217
views
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
3
votes
0
answers
60
views
Function approximation of a complex objective function
I would like to approximate the following objective function using a simpler function that can use be defined in gurobi.
\begin{equation}
\min_{I_{i,v}} \ \sum^{N_v}_{v}\sum^{TT_v}_{i} \ C_{loss,...
- 327
2
votes
2
answers
86
views
Change the objective function formula change the complexity of a linear program?
I have a linear program, where I can use it with the same constraint to minimize objective 1 or minimize objective 2. I noted that when I use the formula of objective 2 the problem can be solved with ...
- 335
4
votes
1
answer
145
views
Can you calculate the mean of some MIP variables using linear constraints?
got a lingering question from a graduate course in integer programming that's been bugging me ever since.
Is it possible to find the mean of some variables in a MIP without resorting to quadratic ...
- 138
4
votes
0
answers
232
views
Linearize a highly non-linear objective function
[EDIT] : The formula below is updated to remove the radical, 0.5 in the term $(I_{i,v} \cdot \Delta t)$ and constant temperature $T$ replces temperature as function of current.
[EDIT] :The values of ...
- 327
3
votes
1
answer
194
views
Range limits on terms in the objective function of an LP
I have a linear maximization problem with an objective as follows: $$\sum c_i\cdot\text{exp}_i$$ where $c_i$ are constants (positive or negative) and $\text{exp}_i$ are linear expressions of the free ...
- 542
3
votes
1
answer
199
views
If variable falls below a certain value, include difference to set value in objective
I think its easiest to describe my goal first and continue with my implementation and the resulting problems!
My goal:
Using Pyomo as interface and Gurobi as solver, if a variable $x_{i,t}$ falls ...
- 33
6
votes
1
answer
348
views
Optimize for bonuses within a group (knapsack)
I am trying to create an LP problem which is like the knapsack problem but with groups of items. Let's say there are 10 groups of items each with up to 5 items. Each group has one special item and you ...
- 197
5
votes
1
answer
310
views
How to optimize on a fixed-cost based on number of results?
I am trying to create an LP problem which is like the knapsack problem but where there is a fixed bonus/penalty based on the number of items chosen.
There are 20 items to choose from with some weight ...
- 197
2
votes
1
answer
373
views
How to linearize the product of two integer variables?
Given two integer variables $L_x \leq x \leq U_x$ and $L_y \leq y \leq U_y$, how can we linearize the product $x \cdot y$?
- 1,192
2
votes
1
answer
86
views
Linearize product of $x\cdot y \text{ with } x,y \in \{-1,0,1\}$ for MILP
I have a problem where I have many products between variables drawn out of $\{-1,0,1\}$. Could you suggest a linearization in terms of variables in $\{-1,0,1\}$ or $B_1 - B_2$ where $B_i \in \{0,1\}$ ...
- 3,477
3
votes
1
answer
108
views
Linearization of problem with affine linear functions
Problem: Write the following task as a linear program: $\min f(x),x\in[-2,5]$ where
\begin{align}f(x) := \begin{cases}
-2x+2,&\quad-2\le x<-1\\
-x+3,&\quad-1\le x < 1\\
2,&\...
- 41
7
votes
2
answers
231
views
Product of weighted binary variables equivalent to sum of weighted binary variables?
I'm working on an optimization problem with a non-linear objective function of the form $$\max\prod_{i=1}^{n}(1-a_{i}x_{i}).$$
The objective function represents the combined probability of success for ...
- 248
5
votes
1
answer
307
views
If continuous variable < constant then same variable = 0
When I come across with a situation needs an if-then constraints I visit Larry's post. I am a bit confused with the titled constraint this time because I am not trying to set $y$ based on $x$ but ...
- 1,229
2
votes
1
answer
110
views
Linearization of constraints in a ILP
I have been working on a Graph Theory problem for my thesis and got stuck about the linearization of some constraints. I am hiding everything, constraints, variables and so on, of my problem not ...
- 533
4
votes
1
answer
68
views
How to know if a combinatorial optimization problem is linear or not?
I want to know if a combinatorial problem like the knapsack problem is linear or not. And how do we know if a given problem is convex or not?
- 335
4
votes
1
answer
243
views
How to linearize a non-convex optimization objective function?
The non-convex multi-objective optimization problem in my case is defined below:
Objective 1: Minimize $f_1(X_1,X_2)=C_0+C_1(1/X_1)+C_2(X_2/X_1)+C_3X_1+C_4X_2+C_5(X_2^2/X_1)$
Objective 2: Minimize $...
- 159
4
votes
1
answer
145
views
Alternate formulation for modeling inventory constraints
I'm working on a inventory optimization problem where inventory used at a time-period is computed based on price-bucket that is selected for an item. Problem contains multiple items (around 10K), 15-...
- 959
3
votes
2
answers
452
views
How to linearize a constraint with a maximum of a linear function
I want to linearize the following statement into a MILP: $\forall x\in \mathbb{R}^{m}$ satisfying $Cx \le d$, $\exists i\in \{1,\cdots,m\}$ such that $a_i^Tx \ge b_i$, where $a_i$ and $b_i$ are given ...
- 109
6
votes
2
answers
363
views
How to model this expression?
Suppose $0\le x \le 1$ is a decision variable and $\gamma(x)$ is defined as follows:
$$
\gamma(x)= \begin{cases}
\theta & x>0\\
0 & x=0
\end{cases}
$$
where $0\le \theta\le 1$.
In my model, ...
- 2,073
6
votes
2
answers
135
views
Linearise $\max\{ y_{t-1} + a_t - z_t ,0\}$
I'm trying to linearise these constraints, but I am not able to do correctly do it.
$$y_t = \max\{ y_{t-1} + a_t - z_t, 0 \} $$
My attempt was this:
\begin{align}y'_t &\geq a_t - z_t\\y'_t &\...
user4387
5
votes
2
answers
331
views
How to linearize specific range constraints?
I would like to know about the linearization of the $(If, Then)$ constraints as follows:
$$\begin{array}{l}
\text { If: } \\
15 \leqslant x \leqslant 25 \\
\text { then: } \quad y=\color{blue}{a} x+\...
- 5,833
10
votes
1
answer
337
views
How to linearize membership in a finite set
Given finite set $S$ and variable $x$, how do I linearize the set membership constraint $x\in S$?
- 22.2k
5
votes
1
answer
373
views
Model "if and only if" indicator constraints in Linear programming
Apologies if this question has been asked, but I haven't been able to find it. I'm modelling something with Gurobi and want to do the following:
\begin{align}\text{cond} < \dfrac{1}{3} &\iff x =...
- 315
3
votes
0
answers
65
views
Linearization of a quadratic model, what is the difference while using gurobi?
I have a quadratic model of parking $N$ cars in $S$ separate lanes as follows. Each car has an arrival time and a departure time. Departure follow the last in first out principle. The objective is to ...
- 379
4
votes
1
answer
170
views
Problems modeling a constraint in network design problem
I'm working on a network design problem where the objective is to minimise the network design cost. Given a graph G = (V, E) and a set of point-to-point demands K, the task is to route the demands and ...
- 143
5
votes
1
answer
337
views
What is a good way to penalise LP relaxation?
I have a binary integer program. It is of a large size and the solver is taking longer time.
I am thinking of relaxing the binary integer variable and making it a continuous variable.
How can I ...
- 2,181
6
votes
2
answers
246
views
Forbid transformation of max(x,y) into MILP
The function $\max(x,y)$ can be linearized by making use of additional binary variables. I suppose global optimisers are implemented to perform this transformation automatically.
Is there a global ...
- 1,936
2
votes
2
answers
225
views
Linearize a product of binary variables
I have a function to minimize which has the following term $$\sum_{i\in I}\sum_{j\in J}\sum_{k\in K}x_{ijk}N_{ij}a_{ijk},$$ where the variables are $x_{ijk}\in\{0,1\}$, $a_{ijk}$ are given as input ...
- 381
2
votes
1
answer
178
views
Which linearisation technique is correct?
I have the objective function (Maximally Diverse Grouping Problem) as
$$\max\sum_{g=1}^G\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}d_{ij}x_{ig}x_{jg}$$
Here, $d_{ij}$ are known parameters, and $x_{ig}$ and $x_{...
- 2,181
2
votes
1
answer
103
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 ...
- 59