Questions tagged [linearization]
On the many different ways to turn non-linear systems of equations into linear ones.
338
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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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2
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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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1
answer
40
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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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2
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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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1
answer
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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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0
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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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3
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Maximize sum of absolute values over a box set
I am interested in the following linear problem:
$$
\begin{array}{cl}
\max & |a_{11} x_1 + a_{12} x_2| + |a_{21} x_1 + a_{22} x_2| \\
\mathrm{s.t.} & 0 \leq x_1 \leq b_1 \\
& 0 \leq x_2 \...
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Meaning of complex eigenvalues for 2D matrix realtive to dynamical systems
I am studying non-linear dynamical systems with the linearization method around an equilibrium point, but I don't get the geometrical meaning of complex eigenvalues.
(Let's focus on a 2D case)
For ...
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1
answer
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Same sign constraint in linear optimization problem
I'm trying to find a way to force a group of variables to take the same sign. Either positive or negative. Clearly, the fact that an OR statement is required implies that the problem becomes non-...
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1
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linearized operator for ODE system
Consider a system of reaction-diffusion equations where we write as
$$
\begin{cases}
u_t=\Delta u+f(u,v),\\
v_t=\Delta v+g(u,v)
\end{cases}
$$
In vector form, we also have $U_t=F(U)$ where $U=\begin{...
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1
answer
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Help on fractional programming
I need help transforming this Mathematical Programming model into a solvable model
$ \min \delta$
$\frac{190(E_j-\sum_ix_{ij})}{P_j-\sum_iQ_ix_{ij}}\leq\delta \ \ \ \ \forall j$
$\sum_j x_{ij} \leq ...
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1
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Linear programming-piecewise linear minimization
I am reading Bertsimas and Tsitsiklis which says that minimization of piecewise linear functions(maximum of linear functions) can be reduced to linear programing. For example,
$$\min |x|+|y|$$ is ...
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The linearized equation associated to a given equation.
In the paper titled "On the existence and stability properties
of positive solution for some p-Laplacian Dirichlet problems", World Journal of Modelling and Simulation, Vol. 3 (2007) No. 1, ...
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2
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Can I linearize this piecewise function so it can be used in an objective function for my LP optimization model?
Thanks for taking the time to read this.
I am looking for methods to linearize this piecewise function so that it can be added to an optimization function of a linear programming problem. I figured ...
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Why do we linearize optimization problems?
I am currently doing research on the calibration of the robots' geometry, which is a standard and well-studied topic. In fact, it can be formulated as a nonlinear non-convex optimization problem:
...
1
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1
answer
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Linearize and then substitute vs. Substitute and then linearize
If I have some nonlinear function $f(x,u)$, I can linearize it as
$$
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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1
answer
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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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1
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1
answer
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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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2
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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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2
answers
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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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0
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21
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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 ...
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1
vote
1
answer
136
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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 ...
- 75
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votes
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 ...
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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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0
answers
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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$$
When we plot a graph of $xy$ against $y$, is the vertical intercept a defined value?
My question is because I feel that the vertical intercept should be ...
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Is it correct to linearize a part of a differential equaztion?
if I'm not wrong, if I have a differential equation like this:
$Dl=a_0k_1 + k_1a_1x + k_1a_2x^2 - k_1l$
and I want to linearize it, I can linearize only the quadratic part (ergo, $k_1a_2x^2$ ). Right?
...
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answer
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Using linearisation to determine stability of equilibria.
I have been given the system:
$$\begin{cases}\dot{x}=-x+y^2\\\\
\dot{y}=x^2-y\end{cases}$$
(Note: the left hand sides should be x and y with a dot on top however I can't quite find how to write that, ...
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1
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Integrality Gap and Linear Relaxation vs. Binary Problem
For the following problem, can we say that its linear relaxation is equivalent to the binary problem?
Problem 1 ($y_j$ and $u_j$ are $0-1$ parameters.):
Given that $u_j=0$ the problem becomes (as $z_{...
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2
answers
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Is it necessary to consider $\epsilon$ when it come to strict inequality linearization?
My decisions variables are all binary. One the constraints I'd like to include into a model is in the from of $\sum_{(i,j) \in E} \delta_{i,j}^t \ge 1 \implies \theta^t =1 \qquad \forall t$
I wanted ...
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1
answer
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Linearization of if-then when having both binary and non-negative [closed]
How to linearize the if-then statements when the condition is a binary and the result is equality between two non-negative?
Three variables:
$\gamma_{i}^{t,s} \in \{0,1\}$ and $x_{i}^{t} \in \mathbb{...
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vote
0
answers
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Smooth action $G\curvearrowright M$ induces smooth action $TG \curvearrowright TM$
Setup: Let $G$ be a Lie group acting smoothly on a manifold $M$ by a map
$$ 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,$$
where $(x,y,u)$ are independent variables and $(X,Y,U)$ are dependent variables.
If
$$\...
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votes
1
answer
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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:
$$\begin{cases}\dot{x_1}= x_2,\\\dot{x_2} = -10x_1-...
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Is type of variables can help in linearization?
I was wondering if the type of variables, whether binary or non-negative can affect the linearization?
For example assuming that $x_{i,j,t}$ and $y_{i,t,s}$ are two binary variables, then when we want ...
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1
answer
65
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different control input on linearized and nonlinear system
I have a nonlinear sytem, i need to design the controller for the linearized system and then apply the same controller to the nonlinear system it self to see how it performs, i've done so, but i get ...
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1
vote
1
answer
87
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How to linearize an if-then statement?
I want to write constraints for finding a path for some rider, driver on a directed network problem.
Let $x_{i,j}^d$ indicate whether driver travels from $i$ to $j$ and $z_{i,j}^{r,d}$ indicates ...
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The concept of the solution space and its possible combination in the logical (Binary) modeling
There is a following logical optimization problem.
$$\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}^+$$
I know, there ...
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0
answers
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Log-linearization of an integral
I'm working on Neo-Keynesian model and I want to log-linearize this simple equation:
$$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{...
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1
vote
3
answers
75
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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 ...
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votes
1
answer
93
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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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1
vote
1
answer
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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 ...
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votes
1
answer
72
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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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votes
1
answer
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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 ...
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votes
0
answers
53
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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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