Linearizing an equation for Mixed-integer linear programming (MILP)

[zod123]

Summary:: Linearizing an equation for MILP

Hi All

I have Linear programming problem where I need to linearise the following function: Y = A * log2(1 + (Bx/Cx^2)). where A, B and C are constants.

Can you please help or advise?

 

[BvU]

Hello @zod123 ,

!​

If this is homework, you need to post an attempt at solution.
I, for one, would start simplifying Bx/Cx² to (B/C) x³

Or did you mean Bx/(Cx²) and forgot the brackets ? In that case it of course simplifies to (B/C) / x

Do you know about differentation ? Taylor series ?
What is the range of x ? If x is not small and the range is big, then linearizing isn't possible

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[zod123]

Hello @zod123 ,

!​

If this is homework, you need to post an attempt at solution.
I, for one, would start simplifying Bx/Cx² to (B/C) x³

Or did you mean Bx/(Cx²) and forgot the brackets ? In that case it of course simplifies to (B/C) / x

Do you know about differentation ? Taylor series ?
What is the range of x ? If x is not small and the range is big, then linearizing isn't possible

$\$

Hello @BvU

Thank you for your response and is NOT homework.

X is large and the range is big.

 

[BvU]

X is large and the range is big.

Ok, that answers one of the four questions. And it's not homework (what is it ?).

Let me assume you meant $$Y = A \; \log_2\left (1 + {B\over C} {1\over x}\right ) = A' \;\log\left (1 + {1\over x'}\right )$$ with

$A' = A /\log 2\$ (from using $\log_2 = \log_e/\log_e 2\$ ) so we can write natural logarithms​

and $x'= Cx/B\$ (to get rid of B and C) .​

​

I drop the ' from now on (lazy me ... ) .

The standard approximation of $\displaystyle {\log\left (1 + {1\over x}\right )}\$ for large x is 1/x which doesn't linearize your Y in the sense that it gets you an expression of the form Y = ax + b.

If you want to force such a form nevertheless, you need to know about differentiation to do it analytically. The alternative is a numerical approach.

With us so far ?

$\$

 

[zod123]

Hello @BvU

I am with you so far.

RE the homework question: I am developing an app that optimises resource allocation using LP.

 

[BvU]

Clear how you can approach $1\over x$ from a minimum $x_{min}$ to a maximum $x_{max}$ by a straight line ?

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