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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!

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  • $\begingroup$ Well, is the solution for one model feasible for the other and vice versa? $\endgroup$ Commented Dec 4, 2015 at 16:25

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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.

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  • $\begingroup$ Thanks for comment Erwin. In fact, these two models can potentially give different results. When V=1, both provide the same objective value. When V=0, in the former model y has to be zero whereas in the latter y stays less than or equal to x. Therefore, the latter formulation has a larger feasible space. $\endgroup$ Commented Dec 5, 2015 at 23:49
  • $\begingroup$ But it would not pick V=0 if it had a detrimental impact on the objective. $\endgroup$ Commented Dec 5, 2015 at 23:53
  • $\begingroup$ Good point. With this reasoning, both should have the same objective value. I double check my code and dig into the solution then. Thanks! $\endgroup$ Commented Dec 6, 2015 at 0:22

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