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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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37 votes
3 answers
16k views

How to linearize the product of two binary variables?

Suppose we have two binary variables $x$ and $y$. How can we linearize the product $xy$?
Michiel uit het Broek's user avatar
36 votes
2 answers
12k views

How to linearize the product of a binary and a non-negative continuous variable?

Suppose we have a binary variable $x$ and a non-negative continuous variable $y$. How can we linearize the product $x y$?
Michiel uit het Broek's user avatar
33 votes
8 answers
2k views

Modeling floor function exactly

Suppose we want to enforce a constraint $$ y=\lfloor{x}\rfloor $$ where $x$ is some continuous variable. One option is to use $$ x-1\leq{y}\leq{x},\quad y\in\mathbb{Z}, $$ which fails on the edge case ...
David M.'s user avatar
  • 2,107
22 votes
3 answers
2k views

How to minimize an absolute value in the objective of an LP?

I want to solve the following optimization problem $$\begin{array}{ll} \text{minimize} & | c^\top x |\\ \text{subject to} & A x \leq b\end{array}$$ Without the absolute value, this a ...
Discrete lizard's user avatar
22 votes
4 answers
3k views

Linearize or approximate a square root constraint

I encounter a nonlinear constraint that contains the square root of a sum of integer variables. Of course one could use nonlinear solvers and techniques; but I like linear programming. Are there any ...
Albert Schrotenboer's user avatar
16 votes
1 answer
10k views

How to formulate (linearize) a maximum function in a constraint?

How to formulate (linearize) a maximum function in a constraint? Suppose $C = \max \{c_1, c_2\}$, where both $c_1$ and $c_2$ are variables. If the objective function is minimizing $C$, then it can be ...
Mostafa's user avatar
  • 2,104
16 votes
1 answer
931 views

How to linearize a constraint with a maximum or minimum in the right-hand-side?

Suppose we have three variables, $x, y, z \in \mathbb R$. How can we linearize constraints with the following structure? $$z \geq \min(x, y)$$ $$z \leq \max(x, y)$$
Michiel uit het Broek's user avatar
15 votes
5 answers
8k views

How to linearize the product of two continuous variables?

Suppose we have two variables $x, y \in \mathbb R$. How can we linearize the product $xy$? If this cannot be done exactly, is there a way to get an approximate result?
Michiel uit het Broek's user avatar
14 votes
4 answers
2k views

Single reference for Mixed Integer Programming formulations to linearize, handle logical constraints and disjunctive constraints, do Big M, etc?

Is there a single crisp and accessible reference which covers how to generate Mixed Integer Programming formulations to linearize products, handle logical constraints and disjunctive constraints, do ...
Mark L. Stone's user avatar
13 votes
4 answers
739 views

The effect of choosing big M properly

I have a set of linearized constraints that are modelled using big-Ms. Now, it is, of course, common knowledge to make the value of M and small as possible in order to provide tighter LP relaxations ...
Albert Schrotenboer's user avatar
13 votes
6 answers
262 views

How to formulate: each pair of elements in $A$ has one common unit in $B$

We have two sets, $A$ and $B$. Some elements of $A$ must be connected to some elements of $B$, but no element of a given set is connected to another element of the same set. (Think of a bipartite ...
LarrySnyder610's user avatar
13 votes
2 answers
126 views

Sensible and realistic way to model truck based transport costs depending on amount

Different kinds of problems involve transporting an amount $x$ from A to B which results in a cost $c(x)$ in the objective function. Traditionally, often linearized costs are used to get an easy, ...
J Fabian Meier's user avatar
13 votes
1 answer
914 views

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 ...
Wilmer E. Henao's user avatar
13 votes
1 answer
580 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 ...
S_Scouse's user avatar
  • 803
12 votes
1 answer
563 views

How to linearize membership in a finite set

Given finite set $S$ and variable $x$, how do I linearize the set membership constraint $x\in S$?
RobPratt's user avatar
  • 32.3k
12 votes
1 answer
332 views

QA techniques for optimization problem coding

I often spend much, much, more time QAing and debugging my code than I do actually writing the optimization problem or shaping my data. Are there any tools or techniques to make it easier? I am asking ...
Zohar Strinka's user avatar
12 votes
2 answers
270 views

Linearisation techniques for MINLPs

I am wondering what kinds of linearisations people do for MINLPs outside my field of expertise. I work in global optimisation, so by "linearisation" we would typically mean one of the following: ...
Nikos Kazazakis's user avatar
11 votes
4 answers
980 views

How to linearize a constraint with a maximum of binary variables times some coefficient in the right-hand-side

I have the following constraint that I'd like to linearize: $P$ is a given set $b_p \in \{0,1\} , \forall p \in P$ a binary variable associated with each element of $P$ $c_p \in \mathbb{R}^+$, a ...
Renaud M.'s user avatar
  • 2,418
11 votes
2 answers
1k 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 ...
BarkingCat's user avatar
11 votes
4 answers
9k views

What are the benefits of linearization?

So I am new to OR (not my field, but I have found myself working in it for a thesis project). My problem is a non-linear problem by design and unfortunately I cannot linearize everything, however, ...
GrayLiterature's user avatar
11 votes
1 answer
671 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 $m$ ...
Betty's user avatar
  • 544
10 votes
6 answers
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_{...
Rajya's user avatar
  • 109
10 votes
4 answers
2k views

Integer programming problem with simple quadratic objective function in Python

I have $n$ objects that need to be divided among $k$ groups. Each group must receive at least $5$ objects. In addition, the percentage of objects in group $i$ should be as close as possible to $p_i$ ...
Rohit Pandey's user avatar
10 votes
3 answers
1k views

Is there a heuristic approach to the MILP problem?

I have the following optimization problem which is a MILP. I can solve it with a MILP solver. \begin{align}\min_t&\quad t\\\text{s.t.}&\quad d_{c}-t\le \sum_{n=1}^{N} B_{n,c}x_{n}\le d_{c}+t,...
KGM's user avatar
  • 2,297
10 votes
2 answers
407 views

Linearization $\max(c_1 x_2, c_2 x_2, \ldots, c_nx_n) \geq q$ constraint

I have a MIP minimization problem where I have a maximization in the constraints: $$\max(c_1x_2,\, c_2x_2,\, \ldots,\, c_nx_n) \geq q$$ Where: $c_n$ constants $x_n$ binary variables $q$ constant $...
Tim's user avatar
  • 205
9 votes
2 answers
966 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 ...
KGM's user avatar
  • 2,297
8 votes
2 answers
980 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 ...
Ralph Asher's user avatar
8 votes
3 answers
473 views

Linearization of a scheduling objective function

I am trying to maximize the workload per employee. An example: $e$ the index of an employee $j$ the index of a project decision variable: $x_{e,j} \in \mathbb{Z}$ and $0 \leq x_{e,j} \leq 100$ ...
Georgios's user avatar
  • 1,193
8 votes
2 answers
756 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: ...
Cesar Canassa's user avatar
8 votes
2 answers
3k views

How to linearize a constraint with max

I would like to linearize a constraint with max. I have the following constraint: $$\max_{pcj}X_{pwcj}\leqslant L_{wk}.$$ With this constraint, I would like to ensure that for $\forall w \in W$, no ...
campioni's user avatar
  • 1,133
8 votes
1 answer
891 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 ...
user199's user avatar
  • 83
8 votes
1 answer
2k views

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 ...
KGM's user avatar
  • 2,297
7 votes
2 answers
882 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 $...
Vogtster's user avatar
  • 205
7 votes
2 answers
2k views

Mixed-integer optimization with bilinear constraint

So I have an optimization problem of the following form: \begin{aligned} \max_{x,y} \quad & \sum_i x_i \\ \text{s.t.} \quad & \sum_i x_iy_i \leq a \\ \quad & x_{\min} \leq x \leq x_{\max} ...
Johnny's user avatar
  • 293
7 votes
2 answers
500 views

Why does a Max constraint work, but this non-negativity constraint does not?

Suppose I have the following constraint: \begin{align}x_{t} &= x_{t-1} + y_{t-1} - z_{t-1}\\x_{t} &\ge 0\end{align} From my limited experience in coding my own problem, I have found that my ...
GrayLiterature's user avatar
7 votes
2 answers
496 views

How can I linearize or convexify this binary quadratic optimization problem?

I have an optimization problem as below. I am having a hard time with the last constraint. $\max \eta$ subject to ${\bf U}(:,m)^T{\bf A}{\bf U}(:,m)=0,m=1,2,\cdots,M$ here $\bf{A}$ is a Binary ...
KGM's user avatar
  • 2,297
7 votes
1 answer
10k views

How to linearize min function as a constraint?

I'm trying to solve an optimization problem including following constraint, and I need to linearize it in a maximization nonlinear programming model. Please help me to reformulate it with mixed ...
Vida's user avatar
  • 87
7 votes
2 answers
250 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 ...
Solver Max's user avatar
7 votes
1 answer
289 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 ...
ProAmateur's user avatar
7 votes
1 answer
414 views

How to reformulate (linearize/convexify) a budgeted assignment problem?

I have a scheduling problem at hand. In my system, there is a service station with $M$ service outlets, therefore, the service station can serve $M$ users at a time. But, there are $N$ users $N>M$ ...
KGM's user avatar
  • 2,297
7 votes
1 answer
279 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 ...
Alex's user avatar
  • 173
7 votes
1 answer
99 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 ...
tcokyasar's user avatar
  • 1,249
7 votes
1 answer
380 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 ...
Adi Shavit's user avatar
6 votes
2 answers
878 views

How to transform this logical if-then constraint?

Consider the binary variables $x, y, z \in \{0,1\}$. I'd like to formulate the two if-then constraints: $$ x + y \geq 2 \implies z = 0, \tag{1} $$ $$ x + y \leq 1 \implies z = 1. \tag{2} $$ Constraint ...
Ronaldinho's user avatar
6 votes
2 answers
2k views

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.&\...
ooo's user avatar
  • 1,589
6 votes
2 answers
377 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, ...
Amin's user avatar
  • 2,160
6 votes
2 answers
593 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?
Optimization team's user avatar
6 votes
2 answers
333 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 ...
Clement's user avatar
  • 2,252
6 votes
1 answer
781 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 ...
optimizationguy's user avatar
6 votes
2 answers
558 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 ...
Optimization team's user avatar

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