Re: [Help-glpk] Compare two matrices

[Andrew Makhorin]

>> I have the following problem: I have a parameter A with (3 indices)
>> and a decision variable X also with 3 indices (same matrix form).
>> The idea is as follows: Parameter A is the old matrix, X is the new
>> matrix. Now I need to be sure that at most 10 changes are made. So I
>> would like to have at most 10 different X[a,b,c] to A[a,b,c] for all
>> a,b,c.

>> Now I was used to programs that could work with absolute values
>> (SUM[a,b,c]: abs(A[a,b,c]-X[a,b,c]) <= 10), however GLPK is not able
>> to work with absolute values of decision variables.

>> Does anyone have an idea how to solve this?

>> I tried introducing a Dummy: D[a,b,c] = 1 for A[a,b,c]<=X[a,b,c], 0
>> else but that didn't work out....

> Big M formulation:

> "if z1 then x <= a - eps else x <= +M" can be modeled as

>    x <= (a - eps) * z1 + M * (1 - z1)

"if z2 then x >>= a + eps else x >= -M" can be modeled as

>    x >= (a + eps) * z2 - M * (1 - z2)

> "if z then x <= a - eps or x >= a + eps else a - eps < x < a + eps"
> is equivalent to "z = z1 or z2" and can be modeled as

>    0 <= 2 * z - z1 - z2 <= 1

> where z1, z2, z are binary variables.

Since z1 and z2 cannot be 1 at the same time, i.e. z1 + z2 <= 1,
the latter constraint can be simplified as

   z = z1 + z2

that allows eliminating z.