### Question & Answer

## Question

What is the difference between using indicator constraints and a big-M formulation?

## Answer

CPLEX 10 introduced indicator constraints as an alternative to big-M formulations.

Both of these constructions typically use a binary variable to turn on or turn off the enforcement of a constraint, or to relate a binary variable to a continuous variable or expression.

For example, to express this idea:

if z = 0, then x = 0 where z binary, and x >= 0

we could use the big-M formulation:

x - M * z <= 0

or the indicator constraint formulation:

z = 0 -> x = 0

Big-M formulations are relatively straightforward, but the value of the M term needs to be chosen carefully. If M is smaller than the upper bound of x, this situation may cut off valid solutions. If M is too large, the model may become numerically difficult or exhibit trickle flow.

Indicator constraints have the advantage of avoiding these types of problems, as they do not rely on a separate constant value. However, they tend to have weaker relaxations during the MIP optimization, a condition which may lead to longer solve times in a model.

Consider using the big-M form instead of indicators:

- When the big-M factor is not much larger than other coefficients in the model.
- If the big-M factor is eliminated in presolve. You can write out the presolved model to check this condition.
- If the model does not show any side effects from a big-M formulation.
- If neither CPLEX 10.0 nor 10.1 can efficiently solve the model formulated with indicator constraints.

Consider using indicator constraints instead of big-M:

- When the big-M factor remains very large, relative to other coefficients in the model.
- When the big-M formulation is difficult to express, such as an if-then constraint on complex expressions.

In all cases, defining upper bound information on the continuous variable will generally yield a much tighter formulation and nearly always helps with performance.

With Concert and CPLEX 10, the default formulation of linearized operators,
like `IloIfThen`

, `IloAbs`

, and `IloOr`

,
involve indicator constraints.
However, you can always write your own big-M formulation, if you wish.

For more information about indicator constraints, see the "Using Indicator Constraints" chapter of the CPLEX user's manual.

## Historical Number

cplex/FAQ/167

### Document Information

**Modified date:**

16 June 2018

## UID

swg21400084