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Questions tagged [linearization]

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

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Linearized formulas and constraints

I am researching how to linearize the maximum and minimum functions in optimization problems and have encountered some confusion. I am linearizing an equation, and ...
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partial feedback linearization with observation

I am considering a partial feedback linearization system with relative degree 2. I have to stabilize a state $x_1$, during estimating $x_2$. Can I describe it below? and recommendation for book and ...
Younghwa Park's user avatar
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Which nonlinear PDEs can be converted into linear PDEs?

In Section 4.4 of Partial Differential Equations by Evans, the author describes several techniques for converting certain nonlinear equations into linear equations. First, the author introduces the ...
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4 votes
3 answers
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What is the proper way of interpreting "linearizing a differential equation"? [closed]

Suppose we have the following differential equation: $$\ddot{\theta} + \cos{\theta} = 0 \quad,$$ where $\theta$ is an unknown function of a real variable. It isn't uncommon to say we will "...
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0-1 Linear programming and non-optimal multidimensional knapsack

I would like to create a set of constraints forcing a set of knapsacks to be filled. The knapsacks should be filled, so that no further element of a set of elements fits into it. It is not a classical ...
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Can anyone see a way to linearize this function for linear regression?

I have the following function: $$f(x) = \dfrac{a_1}{(x+b_1)^2+c_1} + \dfrac{a_2}{(x+b_2)^2+c_2}.$$ From multiple measurements of $f$ at known $x$ values I would like find the values of $a_1,a_2,b_1,...
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Percentage From Linearized Exponential Decay

I have approximated a function to resemble some data that I took: $$ f(x) = Ae^{B(x - C)} + D $$ with values of: A = 248 B = -0.0025 C = 20 D = 105 I have attempted to linearize it and got the ...
user98055's user avatar
2 votes
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Is this problem linear?

$$\begin{array}{lcl} &\max\min\{3x+4y,3x\}&\\ s.t. & \frac{x+y}{3x-1}\le8&\\ &x,y\le0& \end{array}$$ I seem to have difficulty understanding this problem, which I guess is some ...
raffaello.sanzio's user avatar
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Differential Equation to Linear State-Space Model

I derived a differential equation for a system that I am studying that takes the following form: $$\dot{x}-au_1(x-x_0)=bu_2+cu_1$$ This was derived from expanding a non-linear differential equation as ...
b7031719's user avatar
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Approximate the function cos(x) by a polynomial of degree N using linear optimization

I want to find an approximation to the cos(x). I formulated the problem as a linear optimization problem as follows: $$ \min \sum_{i=1}^{M} e_i $$ subject to: $-(a_0 + a_1x_i + \dots + a_nx_i^{N})-...
Felipe Oliveira's user avatar
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Tractable formulation of a mixed integer program

Given constant matrices $A_1\in\mathbb{R}^{1\times l}$ and $A_2\in\mathbb{R}^{1\times l}$, and constants $b_i$, $i=1,\dots,n$. Consider the following mixed integer program (MIP) with decision ...
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SQP and first order Taylor approximation

I am trying to use the SQP solver with a nonlinear constraint. The solver requires a linear constraint so I am trying to approximate the constraint with the first-order Taylor approximation. Is this ...
zenzu's user avatar
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How to linearize time-variant ordinary differential equation (ODE), in particular, if there is a convolution term?

Letting $(\ * \ )$ denote convolution, I have an ODE of the form \begin{align} \dot{r} &= -7.4r -1.6f - 8.8 (f(t)*e^{-t}) - 10.4(f(t)*(te^{-t})) \\ \dot{f} &= 0.25r \end{align} with initial ...
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A linearization in optimization problem

The step $\boldsymbol{\beta}^{t+1}$ in solving an optimization problem is given below. $$\boldsymbol{\beta}^{t+1}=\mathrm{argmin}_{\boldsymbol{\beta}}\frac{1}{2}\left\Vert \mathbf{u}^{t}-\boldsymbol{\...
mert's user avatar
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How to "linearize" a term of the form $\frac{\partial}{\partial x} \left( m \frac{\partial c}{\partial x} \right)$ in a nonlinear PDE?

I am currently studying the following nonlinear PDE model for cells migrating under the influence of diffusion and chemotaxis: \begin{align} \hspace{2cm} \frac{\partial m}{\partial t} & = \...
Leonidas's user avatar
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What considerations should I have into account when linearizing a non-linear model for linear regressions?

I'm looking for some bibliography about what I should/must have into account when I have a model and experimental data that can be expressed in a way such that I can use a linear regression method to ...
CosmeticMichu's user avatar
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Linearization of nested absolute value objective $|a-b-|c||$

I am trying to define an optimization problem that minimizes the distance between $a(x)$ and $b(x)$, where I need to adjust $b(x)$ downwards using the cost function $c(x)$ (hence, the cost must always ...
Jean-Paul's user avatar
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Linearizing a physics equation

I'm working on a physics experiment that involves a parachute made from a plastic garbage bag, and I'm trying to understand the deployment time in relation to the parachute's apex vent radius. I have ...
Miles Jarra Gloekler's user avatar
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How to linearize or reformulate an implication constraint that implies that a decision variable belong to an interval?

I am an electrical engineer who is working in computer network and I need to model my delay with respect to a binary variable $x$ as folow $\left\{ {\begin{array}{*{20}{c}} {x = 1 \Rightarrow \left( {...
Tuong Nguyen Minh's user avatar
-1 votes
1 answer
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How to linearize a max function in a constraint? [closed]

I have linear program that has constraint as follows: $ \max(x,y) \geq 0 $ where $x$ and $y$ are variables. How to linearize this inequality? How to write this constraints in google or tools?
edhi wiyoto's user avatar
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Reference Request: LP representation of problems

I am preparing for an exam on Linear optimisation and came across different problems where some apparently non linear problems can be modelled as LP (For example here, here and here). I was wandering ...
abc's user avatar
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Pseudo if statements in LP programming

When designing LPs for exams I often run across problems where I would like to input an "if-statement". For example: $5\leq x_b$ if $p_a\geq 10$ I've tried dividing by itself and using floor ...
Aron Fredriksson's user avatar
1 vote
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Writing a nonconvex program as a linear program

I want to write the following non-convex program $\texttt{P}_1$ as a linear program (LP) $$ \begin{align} \min_x \sum_i \frac{a_i^Tx+b_i}{c^Tx+d} \\\\ s.t. \ Ax \geq b \end{align} \tag{$\texttt{P}_1$}$...
abc's user avatar
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I'm a bit confused on how to linearize an Exponential Graph

I've seen other people answer questions on how to do this, I tried it and it didn't work. As of right now, I'm using the following desmos graphs. Desmos, Exponential Graph Desmos, Linearized ...
Yashwak's user avatar
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Minimize $\max (x_1, y_1, x_2, y_2) + \min (x_2, y_2) + z_1 + z_2$

Minimize $\max (x_1, y_1, x_2, y_2) + \min (x_2, y_2) + z_1 + z_2$ where $x_1+x_2+y_1+y_2+z_1+z_2=1$ and $0 \leq x_1, x_2, y_1, y_2, z_1, z_2 \leq 1$ The only thing I can think to do is the standard ...
H_R's user avatar
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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)}\\ {{...
Tuong Nguyen Minh's user avatar
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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 ...
Bur Nor's user avatar
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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 $...
Long's user avatar
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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 ...
Tuong Nguyen Minh's user avatar
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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}^{\...
최용훈's user avatar
2 votes
3 answers
64 views

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 \...
Eason Mao's user avatar
1 vote
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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 ...
ohhConti's user avatar
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1 answer
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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-...
Jean-Paul's user avatar
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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{...
79999's user avatar
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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 ...
Ria Migo's user avatar
1 vote
1 answer
142 views

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 ...
stochs's user avatar
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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, ...
Salah Khafagy's user avatar
1 vote
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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 ...
pennydreams's user avatar
4 votes
2 answers
395 views

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: ...
Ali Baradaran Birjandi's user avatar
1 vote
1 answer
96 views

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 ...
bsprenger's user avatar
6 votes
1 answer
208 views

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^*) = ...
Fei Cao's user avatar
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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\...
Kyraz's user avatar
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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,...
twist_lsk's user avatar
1 vote
2 answers
90 views

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 ...
Nijat Hamidov's user avatar
1 vote
2 answers
77 views

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,...
A.Omidi's user avatar
  • 137
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1 answer
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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 ...
PythonDabble's user avatar
1 vote
1 answer
355 views

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 ...
E-O's user avatar
  • 99
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1 answer
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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 ...
g00dds's user avatar
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1 answer
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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 ...
Yuri Santos's user avatar
1 vote
0 answers
106 views

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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