Forbidden range for a linear programming variable

I would like to express a linear program having a variable that can only be greater or equal than a constant $c$ or equal to $0$. The range $]0; c[$ being unallowed.

Do you know a way to express this constraint in a linear program, in a way that it can be solved using an unmodified simplex implementation ?

For example this constraint : $x_1 \geq 4$ or $x_1 = 0$.

Typical relation between all constraints in a linear program is AND. Here this is an OR between two constraints.

Note : I need to solve problems having multiple variables like this in a computationally efficient way

Thanks

• This can't be done with strict linear programming. If you're willing to employ binary variables and branch and bound, you can do this. – Michael Grant Jun 27 '14 at 14:53
• I were hopping I could use a home made linear simplex solver with some tricks :(. – infiniteLoop Jun 27 '14 at 17:25
• I'm afraid not. It's not linear. There is no way to avoid some sort of branching approach where you decide which "side" of the disjunction to try at each stage. – Michael Grant Jun 27 '14 at 17:31
• Thanks. That's bad news. I wish there was some tricks, like the ones used to compute absolute value or min between variables – infiniteLoop Jun 27 '14 at 17:36

Here's the bad news: you can't do this with a straight-up linear program.

Here's the good news: you can do this with an integer linear program.

Introduce an additional binary decision variable $z$. Let $z=0$ whenever $x=0$ and $z=1$ whenever $x\ge 4$. Furthermore, pick an arbitrarily large number, call it $M$, such that $M$ can not bound your $x$ variable too soon(e.g. if your problem data is on the order of $10^2$, pick $M=10^5$ or something). Now add the following constraints to your problem:

$$x \ge 4z \\ x \le Mz$$

If $z=0$, the constraints force $x=0$. If $z=1$ the constraints force $x \ge 4$ (since $M$ is large enough by definition).

In general, the modeling issue is capturing a situation like this: $$x = 0 \lor x\in[a,b], \quad0<a<b<\infty$$ $x$ is called a semicontinuous variable, and the trick that I've shown you above extends naturally to the following pair of constraints: $$x \ge az \\ x \le bz$$

Unless you are coding the algorithm yourself, be aware that most commercial solver packages can handle semicontinuous variables internally (by doing the constraint modeling internally and branching on $z$). Read the appropriate documentation for the syntax.

• Thanks for your answer and the term "semicontinuous" that describe my variable. Unfortunately I have a home made implementation of the two-phase simplex and I were hopping I could use it with some tricks :(. – infiniteLoop Jun 27 '14 at 17:24
• @trinita Nope, you will have to also implement a home-brew version of a branch-and-bound scheme in order to accommodate semi-continuity. Honestly, you're better off examining the source code of some open-source projects such as GLPK or COIN-OR. – baudolino Jun 27 '14 at 17:43
• @trinita BTW, welcome to Math.SE. If you like this answer and feel it adequately captures your question, please upvote and accept. – baudolino Jun 27 '14 at 17:47
• Ok I will look into branch-and-bound method – infiniteLoop Jun 27 '14 at 17:53

Another possible constraint:

$y_1(x_1-4-z_1)+(1-y_1)\cdot x_1=0$

$y_1 \in \{0,1 \}, z_1 \geq 0$

• This is not linear programming, cause there is a product between variables and also binary variables. – infiniteLoop Jun 27 '14 at 17:29
• You are right. As baudolina has said already, you can´t mangage the problem with a straight-up linear program. It was only another (simple) suggestion. – callculus Jun 27 '14 at 17:37
• Ok thanks for the suggestion – infiniteLoop Jun 27 '14 at 17:45
• You are welcome. If you are looking for a optimization program try Lingo. The "coding language" is quiet simple. And it is for free for a certain maximum number of variables. – callculus Jun 27 '14 at 18:52