CODEWITHSUNDEEP
constraintscplexmixed-integer-programming
user496181
user496181

Reputation: 119

binary variable depending on a continuous variable (mixed integer programming in cplex)

I'm trying to formulate a constraint for a MIP problem that involves binary variable v and continuous variable i, such as:

if i = 0, v = 0, and if i > 0, v = 1

I haven't been able to think of a solution to this and I'm not sure if there is a solution. Any suggestion is greatly appreciated. Thank you!

Upvotes: 0

Views: 1101

Answers (2)

TimChippingtonDerrick
TimChippingtonDerrick

Reputation: 2072

You can also model this using the 'traditional' Big-M formulation which is documented in many places on the internet and in many textbooks.

Usually this is done in a pair of constraints like this:

i <= M * v

This forces i to be zero if v is zero, and also if i is non-zero then v must be 1 which covers most of your requirement, but still allows i = 0 and v = 1. In many cases the objective is trying to minimise some expression including v and that may be sufficient to encourage v=0 when i=0. But don't fall into the silly error of using a really big value for M as that will adversely affect your linear relaxations and possibly overal performance.

Then you might also need to add a further constraint to force v to zero if i is zero such as:

v <= i

which would have the effect of directly forcing v to zero if i is zero.

Upvotes: 3

Alex Fleischer
Alex Fleischer

Reputation: 10059

you can rely on logical constraints. In OPL you can write

dvar boolean v;
dvar float+ i;

subject to
{
  v==!(i==0);
}

And you can do the same with all CPLEX APIs

Upvotes: 1

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