# 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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### Linearization of two constraints: one with a conditional max and one with a sum with a variable as index

I have these two quite nasty constraints I have tried to linearize. I am trying to dynamically control if you are allowed to plan producing product p. You are allowed to do it if the product arrived (...
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### Set a limit on value change of a binary variable

I am working on an Energy Management problem. The objective is to minimize the electricity bill for the customer. I have a time-series data with 15 min. intervals spanning the course of 1 year. The ...
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### Priority based demand fulfilment in Linear Constraint

Say I have 3 sources. D1, D2, D3. their capacity is 100, 200, 400. I want to create some constraints such that First D1 is depleted then D2 and then D3. But the catch is you cant use min or max ...
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### How to model this constraint in a better way?

I have a resource allocation problem. There are $M$ users and $N$ resources (machines). One user can be assigned to multiple resources/machines. But maximum $B$ machines can be activated at a time for ...
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### How to model the constraints of min and max in cvxpy

I have a continuous variable $x_{ij}\in[0,1]$ and I need to write the following constraint: $$M_i-m_i+1\leq C_i$$ where $M_i=\max\{j: x_{ij}>0\}$ and $m_i=\min\{j: x_{ij}>0\}$
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### Linear condition between two continuous variables

There are two real variables $x$ and $y$. The conditions are such that: if $y\le 0$, then $x=0$ if $y>0$, then $x=y$ How to write linear equations or inequalities to satisfy both the conditions?
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### Reformulate constraints

I have the following constraints and am wondering whether I can formulate the whole thing more narrowly and with fewer constraints. $x_{itk}$ is binary and $u_{it}, v_{itk}\in [0,1]$. $M$ is a Big-M ...
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### Is the linearization with first-order Taylor approximation correct?

I have a QP problem as $\min \hspace{2mm} x^TQx-c^Tx$ here $x$ in binary I want to transform it into a MILP by writing the objective function as $\min \hspace{2mm} z-c^Tx$ and then adding a constraint ...
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### Converting a function composing of multipe pieces into a linear equation

I have a variable (alpha) which depends on some other binary variables, denoted as X_i. So, for some combination of other variables, alpha may take a value (Beta_j). I added some auxillary variables (...
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### How to model $\max\limits_{x\in X} \min\limits_{y\in Y} \max\limits_{z\in Z} f(z)$ as single MILP

I have the following optimization problem: \begin{align*} \max\limits_{x\in X} &\min\limits_{y\in Y} \max\limits_{z\in Z} & f(z) \\ &\text{such that} & (x, y, z)\in P \end{align*} ...
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### Add second "constraint" to model a binary variable

in my model I have the binary variable $f_{ij}$ which pushes a time-dependent $j$ integer variable $D_{ij}$ to zero if $f_{ij}$ takes the value 1 and keeps the integer number if $f_{ij}$ equals 0. Yes,...
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### Is it possible to transform MIQP into MILP without introducing new variable?

I have a QP optimization problem in the form $$\min {\bf x}^T{\bf Qx}-{\bf c}^T{\bf x}$$ here $\bf Q$ is a symmetric matrix. $\bf x$ is the optimization variable, and it is binary. Is there a way to ...
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### linearizing a constraint involving an absolute function

I would like to know what is the best way to linearize a constraint involving an absolute function. More precisely, imagine I have three binary variables and their relationships is as follows: |x-y| = ...
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I am working on a quadratic conic optimization problem, but I have discovered that it would be preferable if the quadratic constraint is linearly approximated. In other words, I need some way to make ...
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### Converting a piecewise function to linear equations

I am trying to build a MILP model. In this model, I have a dependent variable (alpha) that its value depends on the value of some other variables (or different combination of some other variables). In ...
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### 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 ...
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### Lifting a 3rd order polynomial into a higher dimensional space

An MINLP from a paper I am reading has the following expression in its constraints: $$p_{l,s}=z_lb_l\Delta\theta_{l,s}+b_l\lambda_{l,s}u_l\Delta\theta_{l,s}$$ Where from left to right: $p_{l,s}$: ...
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### How to linearize this L0 norm of a vector?

I have an QP optimization problem. $\bf x$ is the binary optimizaion variable of size $12\times 1$. One of the constraints is non-linear/non-convex. The constraint is L0 constraint. The constraint I ...
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### How to transform a binary QP into an MILP?

I have a binary quadratic problem with objective ${\bf{x}}^T{\bf{Qx}}+{\bf{c}}^T{\bf{x}}$ subject to ${\bf{A}}{\bf{x}}\le{\bf{b}}$ ${\bf{A}}_{eq}{\bf{x}}={\bf{b}}_{eq}$. here ${\bf{x}}$ is binary. ...
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### why this little constraint changes my whole program?

I'm trying to linearize a CP in ILOG CPLEX. I have the following constraint that I want to linearize (I already simplified it with the big M) : ...