Semicontinuous Variables

This question pops up from time to time on various forums.  In a math programming model, how does one handle a variable that can either be zero or in some range not containing zero?  The mathematical description of the variable would be something like $$x\in {0} \cup [a,b]$$where $a>0$.  A couple of common applications in supply chain models include quantity discounts (a vendor offers a discounted price if you buy in bulk; $x$ is the amount purchased at the discounted price and $a$ is the minimum amount to qualify for the discount) and minimum lot sizes (where $x$ is the amount of a custom-made product obtained from a vendor who requires a minimum order of $a$ to justify the setup cost of the order).

The variable $x$ is known in math programming jargon as semicontinuous.  For whatever reason, the term is not well publicized; I taught a graduate course on integer programming for years without coming across it.  Some solvers can model these variables directly.  For instance, the C++ and Java APIs for CPLEX contain a class named IloSemiContVar.  I'm not sure which other solvers directly support semicontinuous variables. For solvers that do not recognize semicontinuous variables, there is a generic formulation.  You introduce a binary variable $y$ that will be 1 if $x$ is nonzero and 0 if $x$ is zero, and you add the contraints $x\ge ay$ and $x\le by$.

AMPL allows you to define a semicontinuous variable without explicitly adding constraints. Our example could be written in AMPL as -->

param a > 0;
param b > a;
var x in {0} union interval[a, b];

Currently, AMPL converts that into the pair of constraints mentioned above, even if the chosen solver is CPLEX.  Whether that will change in the future I do not know.

One potential virtue of directly modelling semicontinuous variables is that it may lead to a more parsimonious formulation.  If the solver understands semicontinuous variables, it can modify the branching logic to separate nodes based on $x=0$ versus $x\ge a$ without having to add the binary variable $y$ and the extra constraints.