If-then constraints in MIP programming

Question

For continuous variables $x$ and $y$, the constraints are:

if x >= 0 and x < 1 then y <= 10 and
if x >= 1 and x < 2 then y <= 5 and

(up to the ten lines of inequalities)

if x >= 2 then y <= 2

The problem is on modeling nonlinear behaviour of gas storage pumping efficiency, which decreases for high load in case of very large gas storages (decrease is seen for storages with capacity above 5 million cubic meters). I have to write these constrains using MIP (or LP) formulation. I can use SOS1/SOS2 variables in GAMS with cplex/gurobi solvers.

Answer 1

Let's just consider one constraint, since they all have the same form:

if x >= 0 and x < 1 then y <= 10 and

First, you really can't test for $x<1$, with a strict inequality. The best you can do is something like $x \le 1-\delta$, for small $\delta$. So I'll assume you're taking that approach.

Introduce a new binary variable $z$ that equals 1 if the "if" condition ($x \ge 0$ and $x\le \delta$) holds. To do this, you can use two more binary variables: $w_1=1$ if $x\ge 0$ and $w_2 = 1$ if $x \le \delta$. See this Q&A for a discussion of how to set those variables: In an integer program, how I can force a binary variable to equal 1 if some condition holds?

Then you can set $z$ using:

$$\begin{align*} z & \le w_1 \\ z & \le w_2 \\ z & \ge w_1 + w_2 - 1 \end{align*}$$

Finally, you can enforce the implication "if $z=1$ then $y\le 10$" using the discussion here: In an integer program, how can I “activate” a constraint only if a decision variable has a certain value?

Other related posts:

• Expressing a chain of boolean ORs using ILP involving different variables
• Expressing an implication as ILP where each implication term comprises a chain of boolean ORs