Questions tagged [linearization]

For questions related to techniques for converting nonlinear expressions in optimization models into equivalent (or approximately equivalent) linear ones.

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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,...
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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 ...
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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 ...
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  • 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 ...
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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 ...
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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=...
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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 ...
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  • 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_{...
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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?
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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 ...
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  • 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 ...
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  • 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 $(...
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  • 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 ...
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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{...
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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 ...
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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 ...
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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 $x$. ...
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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 ...
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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\...
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  • 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{...
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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,...
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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 ...
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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 ...
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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 ...
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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 ...
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  • 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 ...
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  • 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 ...
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  • 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 ...
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  • 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 $x \cdot y$?
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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 $B_i \in \{0,1\}$ ...
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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,&\...
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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 ...
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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 ...
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  • 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 ...
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  • 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?
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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 $...
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  • 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-...
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  • 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 ...
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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, ...
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  • 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 &\...
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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+\...
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  • 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 $x\in S$?
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  • 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 =...
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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 ...
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  • 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 ...
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  • 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 ...
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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 ...
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  • 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 ...
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  • 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_{...
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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 ...
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