Questions tagged [linearization]

On the many different ways to turn non-linear systems of equations into linear ones.

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Linearization of a binary decision variable [closed]

I tried finding a replacement for the decision variable $X_{ij}$ using linear functions. $X_{ij}$ is the variable and $y_i$ and $y_j$ are integer parameters between $1$ and $8$. $X_{ij} = 1$ if $y_i \...
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How to reformulate or linearize the phrase "become redundant" or "not needed"?

I am an electrical engineer and currently I have to deal with an optimization problem with a very specific requirement: $\begin{array}{*{20}{c}} {\mathop {Min}\limits_x }&{f\left( x \right)}\\ {{...
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Linear optimization with max function in constraint

I have a linear optimization: $$ \min_{x_i} -\sum_i x_i $$ $$ \mathrm{s.t.} Ax \le b$$ Where $x$ is a variable vector. I would like to introduce into the linear problem an additional constraint that ...
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How to linearize If-then constraint in linear programming?

I have the following decision variables: $a_i, x_i^t$ and $x_i^0$ are binary variables. I want to realize the following four conditions: if $a_i = 1, \sum_{t=0}^n x_i^t = 0$, then $x_i^0 = 0$; if $...
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How to linearize or formulate optimization constraints that are stated in terms of "if-then" sentence?

I am a engineer who is working on an optimization problem and my constraints are in the form of this statement "if $x_1=1$ then $d_1=1T$" where $T$ is just a given time period. Scenario 1 ...
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Linearization of Multiplicative model

I've read a paper about Huff model, and I have a question for linearization technique of Multiplicative model. How does following linearization work? $U_{ij} = X_{1j}^\alpha X_{2j}^\beta X_{1j}^{\...
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Maximize sum of absolute values over a box set

I am interested in the following linear problem: $$ \begin{array}{cl} \max & |a_{11} x_1 + a_{12} x_2| + |a_{21} x_1 + a_{22} x_2| \\ \mathrm{s.t.} & 0 \leq x_1 \leq b_1 \\ & 0 \leq x_2 \...
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Meaning of complex eigenvalues for 2D matrix realtive to dynamical systems

I am studying non-linear dynamical systems with the linearization method around an equilibrium point, but I don't get the geometrical meaning of complex eigenvalues. (Let's focus on a 2D case) For ...
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Same sign constraint in linear optimization problem

I'm trying to find a way to force a group of variables to take the same sign. Either positive or negative. Clearly, the fact that an OR statement is required implies that the problem becomes non-...
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linearized operator for ODE system

Consider a system of reaction-diffusion equations where we write as $$ \begin{cases} u_t=\Delta u+f(u,v),\\ v_t=\Delta v+g(u,v) \end{cases} $$ In vector form, we also have $U_t=F(U)$ where $U=\begin{...
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Help on fractional programming

I need help transforming this Mathematical Programming model into a solvable model $ \min \delta$ $\frac{190(E_j-\sum_ix_{ij})}{P_j-\sum_iQ_ix_{ij}}\leq\delta \ \ \ \ \forall j$ $\sum_j x_{ij} \leq ...
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Linear programming-piecewise linear minimization

I am reading Bertsimas and Tsitsiklis which says that minimization of piecewise linear functions(maximum of linear functions) can be reduced to linear programing. For example, $$\min |x|+|y|$$ is ...
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The linearized equation associated to a given equation.

In the paper titled "On the existence and stability properties of positive solution for some p-Laplacian Dirichlet problems", World Journal of Modelling and Simulation, Vol. 3 (2007) No. 1, ...
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Can I linearize this piecewise function so it can be used in an objective function for my LP optimization model?

Thanks for taking the time to read this. I am looking for methods to linearize this piecewise function so that it can be added to an optimization function of a linear programming problem. I figured ...
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Why do we linearize optimization problems?

I am currently doing research on the calibration of the robots' geometry, which is a standard and well-studied topic. In fact, it can be formulated as a nonlinear non-convex optimization problem: ...
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Linearize and then substitute vs. Substitute and then linearize

If I have some nonlinear function $f(x,u)$, I can linearize it as $$ f(x,u) \approx f(x_{ss},u_{ss})+\left.\frac{\partial f(x,u)}{\partial x}\right|_{x_{ss},u_{ss}}(x - x_{ss}) + \left.\frac{\partial ...
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Proper linearization of ODEs of the form $\dot{x}(t) + f(x(t)) + \sigma(t) = 0$?

For a scalar ODE of the form $$\dot{x}(t) + f\left(x(t)\right) = 0 \label{1}\tag{1}$$ where $f \colon \mathbb R \to \mathbb R$ is some smooth function admitting a unique root $x^*$ such that $f(x^*) = ...
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Write Newton-Raphson for Lagrangian Navier-Stokes

I am trying to write the finite element formulation of the Cauchy momentum equation (Navier Stokes) in Lagrangian formalism: $\begin{aligned} &\nabla\cdot\sigma+\rho\mathbf{g}=\rho\dfrac{\partial\...
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Help linearizing simple nonlinear PDE [closed]

I'm new to PDEs and need help linearizing this system around some equilibrium $(w_0, u_0)$: $$\partial_t w = \partial_x u + w u$$ $${\partial_t}u = w({\partial_x}w) + b$$ where $u = u(x,t)$, $w = w(x,...
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Why the linearization $(1+u\left(x\right))^k \approx 1+ku\left(x\right)$ is eligible to use when $u\left(x\right)\approx0$?

I know, that applying standard linear approximation to $(1+x)^k$ at $0$ for any real k results in $1+kx$. However while reading Thomas' Calculus 14th Ed., in section 3.9 I encountered many examples ...
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CNF form of the logical $atleast(b) x_{i,k}$

Suppose there exists the following logical expression: $$(\sum_{i=1}^I x_{(i,k)} \leq b) \implies (z_{(j,k)}=1) \quad \forall j \in J, k \in K \tag{1}$$ where all of variables, $x_{(i,k)}$ and $z_{(j,...
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Linearising a polynomial using logs

I have been a bit rusty with my natural logs. I want to linearize this polynomial using logs. Does this look correct? y = (v-x)/x^2 ...
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Eigenvalues of Jacobian around non-stationary points

This is a two-part question Part 1 We have often done stability analysis by calculating the eigenvalues of the Jacobian of a nonlinear function around a stationary solution. I accidentally repeated ...
Paddy's user avatar
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Linearization of constraint in linear programming

I have the following decision variables: $x$, which is binary, and $a, b, c > 0$, which are continuous. I would like to express in linear form for a linear programming model the following ...
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In linear-fractional programming why can't we require denominator to be equal to one without introducing a new variable?

Usually, linear-fractional programming problems are solved by transforming them into equivalent linear programming problems. That is done using Charnes-Cooper method which involves introducing a new ...
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How to linearize a weighted average using Pyomo?

I'm learning about linear optimization problems using Pyomo. At this time I'm looking to practice and ended up getting stuck in a constraint that contains the weighted average formula. Using this ...
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Linearizing equations of motion around a circular orbit

I am given the equations of motion: $$\dot{r} = f(r)p_r,$$ $$\dot{p_r} = -\frac{V'(r)}{2f(r)}-\frac{f'(r){p_r}^2}{2}+\frac{V(r)-E^2}{2f^2(r)}f'(r)$$ Along with the conditions that $$E^2-V(r_0)=V'(r_0)=...
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Description linearized tanks

Description linearized tanks I have a task to complete with the following content: Given a system of two conical tanks connected in a cascade, H1=15m , D1=5m, H2=10m, D2=7m and the free outflow is ...
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Graph of $XY$ against $Y$

Consider this equation: $$xy = 4y + 9$$ When we plot a graph of $xy$ against $y$, is the vertical intercept a defined value? My question is because I feel that the vertical intercept should be ...
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Is it correct to linearize a part of a differential equaztion?

if I'm not wrong, if I have a differential equation like this: $Dl=a_0k_1 + k_1a_1x + k_1a_2x^2 - k_1l$ and I want to linearize it, I can linearize only the quadratic part (ergo, $k_1a_2x^2$ ). Right? ...
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Using linearisation to determine stability of equilibria.

I have been given the system: $$\begin{cases}\dot{x}=-x+y^2\\\\ \dot{y}=x^2-y\end{cases}$$ (Note: the left hand sides should be x and y with a dot on top however I can't quite find how to write that, ...
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Integrality Gap and Linear Relaxation vs. Binary Problem

For the following problem, can we say that its linear relaxation is equivalent to the binary problem? Problem 1 ($y_j$ and $u_j$ are $0-1$ parameters.): Given that $u_j=0$ the problem becomes (as $z_{...
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Is it necessary to consider $\epsilon$ when it come to strict inequality linearization?

My decisions variables are all binary. One the constraints I'd like to include into a model is in the from of $\sum_{(i,j) \in E} \delta_{i,j}^t \ge 1 \implies \theta^t =1 \qquad \forall t$ I wanted ...
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Linearization of if-then when having both binary and non-negative [closed]

How to linearize the if-then statements when the condition is a binary and the result is equality between two non-negative? Three variables: $\gamma_{i}^{t,s} \in \{0,1\}$ and $x_{i}^{t} \in \mathbb{...
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Smooth action $G\curvearrowright M$ induces smooth action $TG \curvearrowright TM$

Setup: Let $G$ be a Lie group acting smoothly on a manifold $M$ by a map $$ A\colon G\times M \to M.$$ The derivative of the mutliplication $\mu\colon G\times G \to G$ defines a group structure on $TG$...
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Can I linearize a System of SDE without losing the constant terms?

I have the following system: $dx/dt = con_1 + a_1k_1y - k_1x$ $dy/dt = con_2 + b_1k_2x + b_2k_2x^2- k_2y$ that I want to linearize. However, I'm not able to do it properly, because my aim is to ...
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Linearizing a nonlinear differential equation

So I have a nonlinear equation which is $m\ddot{x}+(b\dot{x})^3+kx-u=0$ that I want to linearize. Can anyone help me do this? Thanks!
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Linearization of a system of equation at the identity jet

Consider the following first order system $$X_y=X_u=0,\quad Y=y,\quad Y_y=1,\quad UX_x=u, \quad U_uX_x=1,$$ where $(x,y,u)$ are independent variables and $(X,Y,U)$ are dependent variables. If ‎ $$\...
Mostafa's user avatar
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Feedback linearization of a controllability form system

Given the system: $$\begin{cases}\dot{x_1}= x_2\\\dot{x_2} = -10x_1+1.8{x_1^2}-0.25x_2 +u,\end{cases}$$ where $$u=-1.8{x_1^2}+v,$$ I get the system: $$\begin{cases}\dot{x_1}= x_2,\\\dot{x_2} = -10x_1-...
mdeli's user avatar
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Is type of variables can help in linearization?

I was wondering if the type of variables, whether binary or non-negative can affect the linearization? For example assuming that $x_{i,j,t}$ and $y_{i,t,s}$ are two binary variables, then when we want ...
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different control input on linearized and nonlinear system

I have a nonlinear sytem, i need to design the controller for the linearized system and then apply the same controller to the nonlinear system it self to see how it performs, i've done so, but i get ...
mdeli's user avatar
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How to linearize an if-then statement?

I want to write constraints for finding a path for some rider, driver on a directed network problem. Let $x_{i,j}^d$ indicate whether driver travels from $i$ to $j$ and $z_{i,j}^{r,d}$ indicates ...
Hemfri's user avatar
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The concept of the solution space and its possible combination in the logical (Binary) modeling

There is a following logical optimization problem. $$\text{Max} \text{ } \text{OF} = z$$ $$\text{S.t:} \quad (x+y=1) \implies (z \leq 20)$$ $$x,y \in \text{{0,1}} ,z \in \mathbb{R}^+$$ I know, there ...
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Log-linearization of an integral

I'm working on Neo-Keynesian model and I want to log-linearize this simple equation: $$P_t = \int_0^1 P_t(i)di$$ In the steady state: $$P = \int_0^1 P(i) di$$ Let me denote $$e^{\hat{P}_t(i)} = \frac{...
wiktorze's user avatar
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How to model this constraint linearly in binary integer linear programming?

I have a directed acyclic graph, and two binary decision variables: $a_{ij}$, which is equal to one when the corresponding edge between the nodes $i$ and $j$ of the graph is selected, and zero ...
E-O's user avatar
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Two approaches to change the absolute value in Linear Programming

Usually a MP with absolute value $|x|$ can be linearlize by using the transformation of $$|x|=x^++x^- \ \ and \ \ x=x^+-x^-. (A)$$ But I find someone also use $$y\geq x \ \ and \ \ y\geq -x. (B)$$...
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Linearize tricky constraint to MILP, LP

It's hard to describe this constraints. Please check this: It is a selection problem. Item Name Brand Value Price item 1 A 0.1 0.2 item 2 B 0.2 0.3 item 3 A 0.1 0.3 We want to select several ...
W Lewis's user avatar
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Linearizing product of binary variables

How would I linearize the following expression $$ z = (1-x)y $$ where $x,y \in \{0,1\}$? Ideally, I would want to formulate this as a system of linear inequalities.
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Algorithm to solve optimization problem with step function on cost function

I have an optimization problem where I have two sets containing $n$ items, which I must place inside $m$ arrays with the same capacity $c$. The cost of each array is equal to the cost of the most ...
Samuel's user avatar
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How to define linearization of a dynamical system on a manifold with affine connection?

In Euclidean space, if I have a smooth dynamical system $\dot{x}=F(x)$, it's linearization about a solution $x(t)$ is $\dot{v}(t) = DF(x(t))v(t)$, where $DF(x)$ is the Jacobian matrix of $F$ at $x$. I ...
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