How to write if else statement in Mixed Integer Programming?

Question

I posted a similar question earlier here

How to write the following if-else condition in Linear/MI Programming? If $a = b$ then $c = d$ else $c = e$

$a,b,c,d,e$ all are variables. How can we write a linear program without multiplying d and e with binary variables? But we can use binary variables.

real numbers. Doesn't have to be integers? – Vinay Mar 25 '20 at 01:41 — Vinay, Mar 25 '20 at 01:41
In your motivating question, $(c,d,e)=(x,1,-1)$, right? – RobPratt Mar 25 '20 at 01:49 — RobPratt, Mar 25 '20 at 01:49
You are right, but I want to generalize. – Vinay Mar 25 '20 at 01:51 — Vinay, Mar 25 '20 at 01:51

score 1 · Accepted Answer · answered Mar 25 '20 at 02:12

As in this question, you need some tolerance $\delta>0$, bounds $[L_1,U_1]$ on $a-b$, bounds $[L_2,U_2]$ on $c-d$, and bounds $[L_3,U_3]$ on $c-e$.

Introduce three binary variables $y^-$, $y$, and $y^+$ and big-M constraints: $$ y^- + y + y^+ = 1\\ L_1 y^- + 0y + \delta y^+ \le a- b \le -\delta y^- + 0y + U_1 y^+\\ L_2(1-y) \le c - d \le U_2(1-y)\\ L_3 y \le c - e \le U_3 y $$

How to write if else statement in Mixed Integer Programming?

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