How can I implement constraints of x1 not equal to x2 in intlinprog ?

Considering that mixed integer linear programming has a discontinuous search space this should be possible. my current implementation tries to use something like abs(x1-x2)>=E but I end up with a no feasible region error.

[RJ]

to note: Splitting the abs function into two separate inequalities does not lead to a feasible space and that is the exact problem I am trying to solve here. thank you

Answer 1

[Alan Weiss]

I'm not sure that this is the best approach, but I think that it is a correct approach. Assume that x1 and x2 are each bounded by M, so |x1 - x2| <= 2 M. Define two binary (Boolean) variables y1 and y2 as control variables. Tie them to the optimization as follows:

x1 - x2 - 2*M*y1 <= 0; % ensures y1 = 1 when x1 - x2 >= 1
x1 - x2 - 2*M*y1 >= -2*M + 1 % ensures y1 = 0 when x1 - x2 <= 0
x2 - x1 - 2*M*y2 <= 0;
x2 - x1 - 2*M*y2 >= -2*M + 1
y1 + y2 >= 1;

The first constraint ensures that y1 = 1 whenever x1 > x2. The second constraint ensures that y1 = 0 whenever x1 <= x2, so these two constraints together ensure that y1 is the indicator function that x1 > x2. The third and fourth constraints are the parallel to the first, and ensure that y2 is the indicator function of x2 > x1. The last constraint ensures that at least one of y1 and y2 is 1.

I think that this is correct, but I am not sure that it is the most efficient formulation. Might as well give it a try and see.

Alan Weiss

MATLAB mathematical toolbox documentation

Answer 2

[Stephan]

Hi,

im highly interested in that question and i hope anybody has a better answer then me. I found that in the forum so far:

https://de.mathworks.com/matlabcentral/answers/52821-all-different-and-is-integer-constraint-in-optimization

Maybe this will help you.

Best Regards

Stephan