I have the following optimization problem:
Model I: $$f(x,y) \\ s.t., \\ y\leq x+M(1-V)\\ y \leq MV \\ x \geq 0, y \geq 0$$
where x and y are continuous variables whereas V is a binary variable. M is a sufficiently big number (not too big to make computation difficult).
Model 2: $$f(x,y) \\ s.t., \\ y\leq x \\ x \geq 0, y \geq 0$$
It seems to me that Models 1 and 2 are equivalent and should give me the same result. However, my computational results on small instances on Gurobi showed that Models 1 and 2 don't give me the same results; the optimality gap for both models are 0. Can anyone let me know if Models 1 and 2 are not equivalent and why?
Thank you!
Assuming $f(x,y)$ is the objective and does not depend on $V$, I would guess the two models give the same objective. But not necessarily the same solution values. You probably should give the two different solutions and tell us what you find surprising about them.