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linearize
linearize, v. (ˈlɪnɪəraɪz) Also linearise. [f. linear a. + -ize.] trans. To represent in a linear form; to transform into a linear figure; to make linear.1895 Daily News 2 May 5/1 The Cretans used a symbol of a double axe-head, bipennis. They linearised this into an X with the top and bottom closed....
Oxford English Dictionary
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linearize a sum of linear and piecewise linear functions How can I linearize the following constraint: $$ c_1\max(y + |x| - d_1, 0) + c_2\max(y + |x| - d_2, 0) + e - y \leq 0 \tag{$*$} $$ where $x,y$ are scalar decisi...
That looks correct. At the end you can observe that each of your constraints boils down to $|x|\leq s$, where $s$ is some linear combination of other stuff, and that you can write as $s\geq x, s\geq -x$.
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Linearization of product of three variables Let $h=xyz$, where $x,y \in \\{0,1\\}$ and $z \in [0,T]$ with $T>0$ being a constant. Is there any method that can linearize $h=xyz$? For example, if $h=xy$ the method in he...
We first linearize $xy$ in the same way as described in the link you posted.
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Linearization of a second order differential equation > Linearize the equation $$x'' = -\alpha x-\rho x'+c \sin(t)$$ It is very easy when $c=0$ giving you a $$ x' = y $$$$ y' = -\alpha x -\rho y $$ giving you a very ...
First of all, the linearization for $c \neq 0$ is wrong. Indeed, the term $c\sin(t)$ has not been differentiated. However, it is difficult to do this since it does not depend on $x$ or $y$. How can we deal with the term $\sin(t)$? The presence of a time-varying term implies that the order of the sys...
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Linearizing a function using logs I’m trying to understand how to linearize functions using logs and I can’t quite wrap my head around this. Let’s say we have two functions: p(x) = x^2 q(x) = x(i-1)*1.5 (for x >= ...
The way you've written $q(x)$ doesn't make sense because in addition to $x$ there's a free variable $i$ that you didn't introduce. I think you probably mean $q(x)=q(x-1)\cdot1.5\text{ with } q(1)=1$, which is a recurrence relation for $q$ that can be solved in the closed form $q(x)=1.5^{x-1}$. If so...
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Linearizing a constraint containing a root square expression We are working on a combinatorial optimization problem. In order to solve it using CPLEX, we need to linearize the non-linear constraint stated in the follo...
Since you are using cplex, there is no need to linearize really, all you need is a second-order cone model.
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How to linearize (or convexify) a function that takes the maximum of binary variables? I have the following optimization problem in binary variables $x_{ij}$ and $y_i$. $$\begin{array}{ll} \text{minimize} & \displays...
Start by the rewriting the objective as $\sum y_i t_i$ subject to the constraints $\max_{j} x_{ij}c_{ij} \leq t_i$. The max constraints are modelled using standard epi-graph reformations, i.e. $x_{ij}c_{ij} \leq t_i$. The binary times continuous $y_i t_i$ terms you can do using standard big-M models...
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How to linearize a constraint of the form of a product? Is there a way to linearize a constraint of the form: $$\prod\limits_{ i=1 }^{ n }y_i\geqslant b,$$ where $y_i$ are discrete variables in the set $\\{1,2,\ldot...
Consider \begin{equation} \textrm{log} \left( \prod_{i=1}^n y_i \right) = \sum\limits_{i=1}^{n} \textrm{log}(y_i) \geq \textrm{log}(1) = 0 \end{equation} As $y_i = 2^{k_i}$ then \begin{equation} \textrm{log}(y_i) = \textrm{log}\left(2^{k_i}\right) = k_i \textrm{log}(2) \end{equation} Therefore an eq...
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Linearize non-linear constraint I have a problem which may be defined as: $$\max 5 x_{11} + 6 x_{12} + 2 x_{21} + 3 x_{22} \\\ x_{ij}\in \\{0,1\\} \\\ x_{11} + x_{12} = 1 \\\ x_{21} + x_{22} = 1 \\\ t_1,t_2 \text { i...
You can linearize it as follows: $$ t_2-t_1\le M(2-x_{11}-x_{21}) $$ where $M$ is a large constant.
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Linearization without operating point assuming a nonlinear dynamical system like $$ \begin{split} \dot{\mathbb{x}} &= \mathbb{f}(\mathbb{x}, \mathbb{u}) \\\ \mathbb{y} &= \mathbb{h}(\mathbb{x}, \mathbb{u}) \end{split...
Linearization is basically just a first order Taylor polynomial of that function. But usually you perform a linearization around an equilibrium point, since you usually diverge rather quickly from non equilibrium points, so the linearization would become a bad approximation rather quickly as well. S...
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How to linearize the following LP I want to minimize $|d_1-d_2|+e1+e2+e3$ where $d_1,d_2,e_1,e_2,e_3>=0$ and $|.|$ denotes the absolute value, for some linear constraints. Is there any way I can linearize the objectiv...
Is this homework? If you have the problem min $|d_1-d_2|+e_1+e_2+e_3$ with the constraints $d_1,d_2,e_1,e_2,e_3\geq0$ then by introducing a new variable $\lambda$ with $\lambda\geq d_1-d_2\geq -\lambda$ we have the following LP whos optimum is just the same as the above min $\lambda+e_1+e_2+e_3$ wit...
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How to linearize this in equilibrium? Consider the pde system $$ u_t=-uu_x-vu_y+fv-g\cdot (H+c)_x\\\ v_t=-uv_x-vv_y-fu-g\cdot (H+c)_y\\\ H_t=-(uH)_x-(vH)_y $$ where $u=u(x,y,t), v=v(x,y,t), H=H(x,y,t)$ and $f\in\math...
We linearize by dividing $u$, $v$, and $H$ into the following:
$$u(x,y,t)=u_0(x,y)+\epsilon u_1(x,y,t)$$ $$v(x,y,t)=v_0(x,y)+\epsilon v_1(x,y,t)$$ $$H
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How can I linearize this IF-THEN constraint? Let, $m=1,2,\cdots,M$ $b_{m}$ is a binary variable $c_{m}$ is a continuous variable I have an IF-THEN constraint like this IF $a_{m}=1$, THEN $f_{m}> 0$ IF $a_{m}=0$, ...
A similar approach as in this answer can be used. Let $K$ be a sufficiently large constant. Then add constraints: $$\begin{align} f_{m,n} & \le Ka_{m,n} \\\ f_{m,n} & \ge 0.2\sum_{l=1}^M f_{l,n} - K(1-a_{m,n}) \end{align}$$ (Note that I changed the summation index to $l$ since $m$ is already used in...
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How to linearize this IF-THEN Constraint? I have a nonlinear constraint as below If $x_k=0$, Then $||{\bf w}_k||==0$ If $x_k=1$, Then $||{\bf w}_k||>0$ Here, $x_k\in\\{0,1\\}$ is a binary variable and $||\bf x||$ ...
The generic big-M model would be $$ -Mx_k \leq w_k \leq Mx_k, ~||w_k||\geq \epsilon -M(1-x_k) $$ However, this is a nasty model as you have a non-convex constraint on the norm. Hence, you will have to work on an elementwise level to force at least one element to be non-zero. To do that, introduce a ...
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How to linearize a min-max problem including a variable and parameters? How to linearize the following model: $\min\limits_{x,y}$ $\max\limits_{i \in \\{1, ..., m\\}}(|x-a_i| +b_i)$ s.t. x $\geq$ 0, y $\geq$ 0 wher...
You are not done with the linearization, you have something quadratic. Besides, you miss some constraints on $z$ which would tell you it is equal to $1$ iff $x-a_i$ is positive. But there is a simpler way to go here... You can solve the equivalent problem $\min\limits_{x,y,t} t $ where the variable ...
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