If there are so many constraints like that you should use integer linear programming or nonlinear programming. But, if there is only one such constraint you can solve the problem by solving 4 linear programming.

Use constraints like these where two of them caused nonlinearity of the model:

x(3) + s1 - s2 = k

y(5) + t1 - t2 = m

t1 , t2 , s1 , s2 >=0

s2

Thanks Majid.

It looks like a nice trick to avoid MINLP. I also came up with a different idea to solve the same problem. How about this:

Assuming that B is the binary variable, I can replace the 'if' condition with the following two constraints:

(1 - B) * K >= X(3) - K

-B * K = B * m

How does that sound to you?

I think that the modeling you represented has a problem.

For representing P -> Q relation in a mathematical program, we can use ~P or Q relation and then by using binary variables we can transform these OR constraints to AND constraints. Hence, you can use the following constrains too, where U is a large enough number like U=1000000:

x(3)

Majid Zohrehbandian: Many thanks. Although I think your latest recommendation looks more efficient than solving 4 linear programming in terms of model complexity and run time, I am not sure what you proposed as a pair of new constraints works correctly.

If B=0 the pair of constraints will give x(3) = m. That results in a different outcome to the 'if...then...' type rule I am looking for. Am I missing something here or there is a mistake in those constraints?

You're welcome.

I think that my first recommendation is more suitable for you.

You can solve an NLP problem or transform it to some LP problem (if there are a small number of if ... then ... relations).

But about the last recommendation, I said that in this manner we can model the relation P -> Q as ~P or Q. These relations are false if and only if P=True and Q=False. Hence, in solution space we have not any solution with the result of P=True and Q=False.

In other words, there is not any feasible solution in the model that x(3)=m, since as you said above, for one of them we should have B=0 and for the other B=1.

However, as I said before I think that you don't want to model P -> Q relation. What you want is that x(3)>=k has an outcome of y(5)>=m. Then I proposed the first NLP model. But, when you proposed using binary variables, I said that by using binary variables you can model the if P then Q relations as not P or Q relations.

Good luck.