Questions tagged [linearization]

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

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How to linearize a max function about a small change??

My question is the following: Given a vector field $\mathbf{v}$, we have the following functional: $f\{\mathbf{v}\}=1/|\mathbf{v}|_{\text{max}}$, where $|\mathbf{v}|_\text{max}$ is the maximum of the ...
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Linear Instability vs. Nonlinear Stability

I have an N-body system that I can simulate directly. For certain initial conditions, this system is unstable to a buckling behavior (exponential growth in an angle), and for others it's stable (...
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Can these quadratic constraints be linearized?

I have the following optimization problem: $\max s$ $\displaystyle \left (\sum_{n=1}^NA_nx^i_n\right )^2 + (1-y_i)M\geq B\sum_{n=1}^N(x^i_n)^2,\forall i=1,2,\ldots S$ $\displaystyle \sum_{i=1}^Sx^i_n=...
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How to linearize following if-else statement with inequality and two decision variables?

How can I linearize following equation: If $a \neq b$, then $d = 1$, else, $d = 0$. It is also important to know that b and d are decision variables and only a is known. The value of $b$ is already ...
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How to linearize z <-> x == y?

I have a constraint that reads $z \iff x = y$ where $z$ is a 0-1 variable and $x,y$ are non-zero, positive integer variables. I'm managed to formulate equivalence going in the right direction, but not ...
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linearization around operating point

I׳m trying to understand what is the importance of linearization around operating point of phase of a transfer function. (in that matter the operating point is the resonant frequency) thank you.
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Linearization of Gradient Flow

As someone who has only "theoretical" knowledge in Riemannian geometry, I have a hard time trying to wrap my head around how to actually compute the so called "linearization" of a ...
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how can I linearize a constraint of the form $\sum (\min(x(i),y(i)))$ for a linear optimisation problem? [closed]

I have an linear optimization problem, and I'd like to impose a constraint of the following form: $∑_{i=0}^N \min⁡(x_i,y_i)≥C$ where $x_i,y_i$ are rational numbers greater or equal to 0. How can I ...
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Log-linearization of Euler consumption Equation

I've been using a resource from Sims (2011) to log-linearize everything. However I'm struggling with the following equation: $$C_{t}^{-\gamma}=\beta E_{t}[(\alpha \frac{Y_{t+1}}{K_{t+1}}+1-\delta) C_{...
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How to linearize a state space equation with higher order $>2$?

Let us consider the following nonlinear polynomial system $$\dot{x} = f(x,u),$$ where $x=[x_1, ... , x_n]$. A Taylor expansion about $(x_0,u_0)$ gives $$f(x,u) = f(x_0,u_0) + \frac{\partial f}{\...
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Is there a procedure to locally linearize nonlinear operators from $\Bbb R^n$ to $\Bbb R^n$?

If I have a nonlinear operator $T: \mathbb R^n \to \mathbb R^n$ such that $\| T(x) \| = \| x \|$ for each $x$ in $\mathbb R^n$, is there a procedure to locally approximate it with a linear orthogonal ...
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center manifold and bifurcation: 2D Bifurcation system reduction

I have this system to study $$ \left\{ \begin{aligned} \frac{dx}{dt} &= y-x - x^2 \\[5pt] \frac{dy}{dt} &= \mu x - y - y^2 \end{aligned} \right. $$ I have derived the Jacobian around fixed ...
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How can I convert non-linear constraint to linear one?

Problem: Suppose I have $n$ finished products and each product has its own completion time, such as C$_i$ (C$_i$=completion time of product $i$, where $i=\{1,2,...,n\}$). These products will be ...
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State error propagation of ODE with uncertain parameters

I have an idea on how to integrate uncertainty of the parameters of an ODE in the state error propagation. But I am unsure if my idea is correct. I have a non-linear ODE of the form: $$ \frac{d}{dt} \...
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Minimize sum of absolute value with linear constraint

Consider a minimization problem: $$ \begin{aligned} & \min \sum_{i=1}^n |x_i|,\\ & A x = b, \end{aligned} $$ where $A$ is an $m\times n$ matrix of rank $m$. I know that the minimum points ...
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Build an explicit polyhedral representation of $\operatorname{Epi}(f)$

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $f(x) = \max\limits_{1\leq i < j \leq n} \{ |x_i| + |x_j| \}$. Furthermore, let $\operatorname{Epi}(f) = \{ [x; t] \; : \; f(x)\leq t\}$. ...
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How to build an explicit polyhedral representation of $P_n$

I am having difficulty with the following question. Let $P_n = \{x\in\mathbb{R}^n \; : \; |x_i| \leq 1, i\leq n, \sum_i |x_i| \leq 2 \}$. Build an explicit polyhedral representation of $P_n$ that is ...
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Rewriting an equation as a set of inequalities

I have a set $\\{x_1,x_2 \dots , x_d \in \mathbb{R} : |x_1| + |x_2| + \dots + |x_d| \leq 1 \\}$ and i would like to rewrite it as $\\{ u \in \mathbb{R}^d : Au \leq \textbf{1} \\}$ , where $\textbf{1}$ ...
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How can I convert a non-linear constraint to a linear constraint for the mixed integer programming?

I have a nonlinear constraint: $\sum\limits_{i\in N}\sum\limits_{j\in J} A_{ijt}\times Z_{ijt}\geq \sum\limits_{i\in N}\sum\limits_{j\in J} D_{ij} \hspace{0.5cm} \forall{t}$ Here, $Z_{ijt}$={0,1}; $...
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Rewriting the one norm into a set of linear inequalities.

I am trying to solve this exercise in linear programming: Express $\|Ax - b\|_1$ As a linear program of the form: $$\text{minimize } v^T y$$ Subject to $$Bx \leq b_1$$ $$Cx = b_2$$ $$l \leq y \leq u$$...
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How to linearize the function?

I have a function $$F(x)=\left[ a\ln\frac{y(x)}{b}-c\ln\frac{y(x)}{d} \right]^{-2},$$ where $a$, $b$, $c$, $d$ are constant. If the function $y(x)$ can be written as it mean value plus a disturbance, $...
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A question while proofing the equation of linearization error proof $g(a)=\frac{1}{2}{f}''(c)(x-a)^2$

I've got a question the linearization error proof part from the Book The calculus Lifesaver, we want to proof the error $g(a)=\frac{1}{2}{f}''(c)(x-a)^2$. Original derivation progress: The ...
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Transfer a nonlieanear function to a linear function

I'm using Java to solve a maximization problem in Cplex. My objective function is quite complex. In a nutshell, there are two parts, A and B. Both of them contain variables. My goal is to maximize A/B,...
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Log linearize with a sum

I have the following equation for a variable $s$: $$ s = -\beta\frac{\alpha(1-w)^{\zeta-1}-(1-\alpha)w^{\zeta-1}}{\alpha(1-w)^{\zeta}+(1-\alpha)w^{\zeta}} $$ where $\zeta$, $\alpha$, and $\beta$ are ...
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Version of non-linear functions with linearly increasing arc-length (proof)

I am trying to find a function "f" such that: \begin{align}{ \int_{a}^{a+b} \sqrt{(\frac{d}{dt}(x(f(t))))^2+(\frac{d}{dt}(y(f(t))))^2 } \,dt}=B \hspace{3cm}a,b,B\in{\mathbb{R}}\end{align} ...
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Linearization of a Matrix function

I have the following non linear affine system $$x_{k+1} = f(x_k)+G(x_k)u_k$$ where $f(x_k)$ is a non linear function in terms of the state vector $x_k$, $G(x_k)$ is a square matrix in terms of $x_k$ ...
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Linearize optimization problem with absolute value

Is there any method to linearize the following optimization problem? \begin{align} \min_{x,y} &~~ c~[x; y] \\ \text{s.t.} &~~ \sum x\leq \alpha_1 \\ &~~ \sum |y|\leq \alpha_2 \\ &~~ \...
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Converting this non-linear minimization problem to linear

My Problem: I would like to convert the following non-linear minimization problem into a linear programming problem, to solve it with the simplex method. The non-linear function could be of any shape ...
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Linearization around the equilibrium point for a system of differential equations

Suppose we have a system of differential equations $$ \frac{dN}{dt} = \mu N \left(1-\frac{N}{K}\right) + N \int_{0}^{\infty}p(a,t)da $$ $$ \frac{\partial p}{\partial t} + \frac{\partial p}{\partial a}...
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Linearisation of a Quasilinear Elliptic PDE

I have noticed that the word 'linearisation' can have different meanings in different places in the literature. For example, if one has a second-order quasilinear elliptic PDE, what would be the ...
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How to convert non-linear equation to linear equation [closed]

I have a problem with X machines, each one with a specific production. All the production needs to be sent to an specific place via different routes which may or may be not cheaper. I need to minimize ...
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How to linearize and get the state-space representation of this system of nonlinear ODE's?

So I recently had to derive equations of motion of a seesaw system seen in the image simplified seesaw system. The equations of motion are as follows: $$(J+m_B*b^2+m_C*c^2+m_C*s^2)\phi'' - m_C*c*s'' +...
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Linearising an exponential system for linear programming

I have a system of n constraint equations in the form of $\prod C_i/C_j \leq b_i$ where all the $b_i$ are known constants and C are unknowns. All the values of C are in the range $10^{-6}$ to $10^{-2}$...
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Unstability test for non-autonomous nonlinear systems

Consider the non-autonomous nonlinear dynamical system $$ \dot{x}=f(t,x) $$ It is a known result (Theorem 4.13 in Khalil (Nonlinear Systems)) that if the linearization $$ \dot{x}=A(t)x $$ where $A(t)=\...
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Linearize a highly non-linear objective function

I would like to minimize the following objective function. \begin{equation} \min_{I_{i,v}} \ \sum^{N_v}_{v}\sum^{TT_v}_{i} \ C_{\text{loss, cyc}} \end{equation} $$C_{loss,cyc} = \left[\left( a \cdot \...
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Linearization of system of Nonlinear reaction diffusion equation

I came across many problems in my course and I solved them but the forth one, it seems the hardest for me. I will show the problem I want to solve at first, after that I will show my solution for ...
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How to linearize the following if-then constraint with Or constraint involved?

I have the following constraint set that contain an if-then condition and or condition at the same time: $$ \eta_i = 1 \...
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Diffusion-Reaction problem $u_t = \int_{-\infty}^{\infty} K(x-y) u(y) dy . u + u^3$.

I have bee stuck in this problem since more than a week. During my study I kinda understand how to find the adjoint operator for the linearization. But I have no Idea how to find the linearization to ...
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Linearisation of a structured-population model with a nonlocal nonlinearity

I would like to linearise the following differential operator (which is the McKendrik-von Forster equation combined with a nonlocal nonlinearity): \begin{align} \dfrac{\partial n(t, x)}{\partial t} + \...
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Turning a piecewise affine optimization problem into an equivalent linear program

$$\begin{array}{ll} \underset{x \in \mathbb{R}^4}{\text{minimize}} & x_1 + 6 x_2 - \min\{10x_3, 5x_4\} + \left| \displaystyle\sum_{i=1}^{4}x_i \right| \\ \text{subject to} & \displaystyle\sum_{...
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Is this how this question supposed to be solved? (Writing a system of constraints that represents this connection between variables).

Given that $x,y,z,w,v\in \{0,1\}$. The connection between the variable is given by: $\max\{\min\{x,y\},z,v\}=w$. Write a system of linear constraints that represents this connection. My Work: Let $...
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Help in expressing in McCormick Envelope

The statement which I want to express in McCormick envelope is $\sum_ix_iM_{ij}\leq F_j$ for all $j$ The initial McCormick envelope I wrote where $w_{ij}=x_iM_{ij}$, is: $\sum_iw_{ij}\leq F_j$ for all ...
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Taylor series for linearization

I need to apply Taylor series to linearize following equation $$f_{dr}= Au'+ Bu + C u^2 +D$$ Here, $A$, $B$, $C$, $D$ are constants and $$u'=\frac{{\rm d}u}{{\rm d}t}, u=\frac{{\rm d}x}{{\rm d}t}$$ ...
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Linearization of a Second Order Nonlinear Differential Equation

I am in need of help with the linearization of this equation around the initial condition u=pi My main confusion lies in the middle term .5xdotx. I cannot for the life of me figure out what to do with ...
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Find the linearization of the following curves at a suitably chosen integer near a [closed]

Find the linearization of the following curves at a suitably chosen integer near a $f(x) = x^2 + 2x$ $a=0.1$ in this i used the formula $L(x)=f(a) + f'(a)(x-a)$ and solved and got $L(x)=2.2x-0.01$ is ...
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Can Linearization from Calculus I be used as a Multivariable function?

In my math class yesterday we learned about linearization of a function to approximate values in a small range of x values (followed by Newton's method of approximating the zeros of functions). We ...
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Linearising an inverse exponential function $A\exp(-C*x) + B$

I am doing a math-intensive Physics IA experiment, which is looking at the relationship between pressure and bounce height of a ball. With all the data, I came across a best-fit line, which is in form ...
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Linear problem with min of hyperbolic functions as the objective [closed]

I am trying to convert the problem below to linear programming problem and solve it with simplex algorithm. I am aware that converting max and min in goal function usually means adding proper ...
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3 answers
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Linearization of product of a continuous and a discrete variable

From my previous questions, I have a variable : $Q$, which is function of a discrete known vector, $P$ and a binary variable $x$ : $Q=f(P,x)$. I know, we can linearize the products of (a) two / ...
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I have two questions about linearizing the specific nonlinear, non-autonomous system of equations given below.

I have the following nonlinear and non-autonomous system to solve ($a,b,c$ are specific constants): $$\begin{align} x^\prime &= -a\,\omega\,y+\frac{b}{\cosh t}\,\left(z-\frac{1}{z}\right),\\ ...
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