Linearing absolute

Min x+y

a-b=x-y

x,y=>0

a,b=free

Reza Lotfi Thank you Reza,

Do you mean just minimizing the summation of the new positive variables a & b?

Regards,

It is applied for minimizing the summation, but this formula is for Min |a-b|

Hello Monsef. I had an optimization problem I was formulating that had to address the same issue. I have attached the resulting paper. I hope it helps you.

Thank you Alejandro Hernandez for sharing your paper.

Introduce one additional non-negative variable x >= 0. Then

min x

s.t. a-b

Armin Fügenschuh How is this different from what Reza Lotfi suggested? Is your variable x integer here?.

One more question Sir, what if I want to minimize the term |V-1| where V is a positive continuous variable in this case. Is the solution:

Min x

s.t

V-1=x : x positive variable with upper/lower limits

X>= 0.

Thank you,

To minimise |V-1|, where V is a positive continuous variable, just minimise x subject to x >= V-1 and x >= 1- V.

Reading the answers provided for the same question can also be useful!

https://www.google.com/amp/s/www.researchgate.net/post/How_can_I_make_linear_the_Absolute_Value_function_x_in_optimization_problems/amp

Adam N. Letchford Thank you your contribution

Is the term |a-b] minimized as:

introduce free positive varaible x

then Min x

s.t

a-b

Yes, this is a folklore trick in linear programming.

Dear Monsef Tahir ,

Mathematically, if one needs to minimize

f(x)=|a(x)-b(x)|

It is equivalent to minimize F(x)=(a(x)-b(x))²

assuming that a(x) and b(x) are given differentiable functions

that provide integres when x runs over integers.

Next we solve (a(x)-b(x))(da(x)/dx - db(x)/dx) = 0

implies that a(x)-b(x)=0 or da(x)/dx - db(x)/dx = 0

seeking integer solutions if any. Otherwise, we consider the approximation

of the solutions ( as the floor or ceiling functions) this will solve the problem.

The main point is that, what are the forms of a(x) and b(x)?

Also, one may ask, are there any constraints?

PS. we may consider x as multidimensional that based on

the definition of a and b.

Best regards

Anyone here can help me with my post too? I am stuck in a similar issue that i need some help with. :-)

https://www.researchgate.net/post/How_to_linearize_the_absolute_difference_of_all_elements_in_an_array_in_MILP_problem

to minimize an absolute value

min |X1-X2|

s.t. (any constraints)

just add new complementary variable U1=|X1-X2| with two constraints as follows

min U1

s.t. (the same original problem constraints)

(X1-X2)

https://www.coursera.org/lecture/operations-research-modeling/4-6-linearizing-an-absolute-value-function-Hicdh