Combinations of variables in a minimization problem

[18...@novasbe.pt]

Dear all,

I need to solve a minimization problem with 300 variables that form 1000 combinations in pairs of two. Not all possible combinations are made and sometimes the variables can combine themselves two times. Additionally: 

1- There are is also a parameter "c" that depends on the variables present in the combination, regardless of the order, 

2- And two parameters "a" and "b" that depend on the order of the variables. 

As an example:

(a*x1+b*x2 - c12 )^2 + (a*x2+b*x2 - c22 )^2 +  (a*x2+b*x1 - c21 )^2

where c21=c12 

 After reading the provided materials my main question is: how can I set up my model and data in such a way that AMPL understands that the second variable appearing corresponds to the first variable in some other combinations? That is, how can I show that a*x1 in the first combination above and b*x1 in the third combination above correspond to the same variable?

[ptiwari]

Are there any conditions on combining these variables? You could specify condition on the sum.  I will give you one example, but it may or may not work depending on your condition.

param n:= 300;

var x{1..n};

sum{i in 1...n, j in 2..n} (a*x[i]+b*x[j]-c[i,j])^2 + sum{i in 2...n, j in 2..n: i = j} (a*x[i]+b*x[j]-c[i,j]) + sum{i in 1..n,j in 1..n: j>i} (a*x[j]+b*x[i]-c[j,i])^2;

In the above formula, the first expression will have all combination of i and j. Do you have any condition on what combination should appear? The second one contains all the sum where i = j. 

Thanks,

Paras

[18...@novasbe.pt]

Thank you for your comment.

Although that is probably a way to start, it does not completely solve my question since there are no conditions on the way how the variables are combined. 

For instance, imagine I have some people that will form groups of two. Person A may combine with B and then with C while C only combines with B. And then there is D that only combines with herself. How can I organize my data and my mod files to solve this issue?

[ptiwari]

Can you express your problem using the mathematical notation? How do you write expression using summation notation ? I recommend you to forget about AMPL and write your formula in the paper using mathematical notation. After writing mathematical formula, It would be easier to translate it into AMPL syntax.