# 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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### How to model If $A \le B$ then $Y = 1$, otherwise $Y = 0$

Somehow I don't get it right. I would like to model the following conditional: If $A\le B$ then $Y=1$ otherwise $Y=0$ where $A, B$ are reals and $Y$ is binary. I can model as follows: $Y \cdot A \le B$...
398 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: ...
207 views

### How to linearize a quadratic constraint to add it then via a callback function

Suppose we have a positive continuous variables $0 \le x \le UB$ where $UB$ is a known upper bound. How can we linearize the term $x^2$? Detailled problem: Suppose that via a callback we compute a ...
187 views

### Linearizing a constraint with square root of a variable

I am trying to linearize the constraint set (2) in the following simplified program. The parameters: $A,C,D,T\in\mathbb{R}^+$. The set $\mathcal{J}$ is polynomially-sized. \begin{alignat}2\min &\...
139 views

### Linearizing power term in objective function

I would like to linearize $x^2$ term in my objective function. I understand this can be solved using quadratic programming solver; however, for my use case linearization is necessary to convert it to ...
134 views

### Linearize sum of product in objective function

Notation: I have an optimisation problem with objective function: \begin{align}\max&\quad\sum_n Q_n\\\text{s.t.}&\quad Q_n=x_{ij}^{nk}(y_i^{nk}-c_n), \forall n \in N, k \in K, (i,j) \in P.\end{...
820 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 ...
81 views

### Linearization of the shifted copy of a function

Suppose in a model I have the expression $y_{1}(x) = 10 + 5 x^2$ where $x \in [0,20]$ is a continuous variable. In order to be able to use an MILP solver, I piecewise linearise $z_{1} = x^2$, by ...
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143 views

### Linearizing the square root of binary summations

My question is similar to this one and almost identical with this. I am so confused due to indexing and could not make sure if I could apply the solution in here to this indexed version as shown below....
66 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 ...
45 views

### defining Mixed integer linear inequalities for a set of variables

The problem is described as follows: considering $n$ variables which are continuous and bounded such that $$L_i \le x_i \le U_i\quad \forall i=1,2,\dots,n.$$ How can i define a set of mixed integer ...
257 views

### Trade off between number of constraints and bounds of a variable

I am not familiar with the inner working of the solvers. I mostly use the python pulp or IBM CPLEX solver. For fast execution ...
144 views

### Linearizing constraint with continuous and boolean variables

I have two continuous variables $A$, $B$ and two binary variables $x$, $y$. Condition: if $A = B \wedge x = 1 \wedge y=1$ then $z = 1$ else $z = 0$ from In an integer program, how I can force a ...
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### How to propagate time using linear inequalities?

I have an adjacency matrix $G_{i,j}$ that tells the distance between $i$ to $j$ (between 0 to 1) if there is no edge between $i$ to $j$ I am putting a large integer $100$. This is my previous ...
414 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 $m$ ...
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### Converting Nonlinear Program into an LP

I have a problem with a nonlinear objective function which is \begin{align}\min&\quad Z_j\cdot(N_j)^{0.5}\end{align} where $j$ is the index. I want to know how can I turn it into a linear ...
1k 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 ...
86 views

### Linearizing objective function with absolute differences

I want to turn this objective function $$\max \sum_{i=1}^{N-1} \sum_{j=i+1}^N |TX_i^T - TX_j^T|$$ where $T$ is just a vector with increasing integers (e.g $[1 \ 2]$) and $X_i$ is a vector ...
443 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 ...
364 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 ...
306 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 ...
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 $p_i$ ...