ILOG CPLEX 11.0 User's Manual

Piecewise Linearity in ILOG CPLEXSome problems are most naturally represented by constraints over functions that are notpurely linear but consist of linear segments. Such functions are also known as piecewiselinear. In this chapter, a transportation example shows you various ways of stating andsolving problems that lend themselves to a piecewise linear model. Before plunging into theproblem itself, this section defines a few terms appearing in this discussion.What Is a Piecewise Linear Function?From a geometric point of view, Figure 18.1 shows a conventional piecewise linear functionf(x). This particular function consists of four segments. If you consider the function overfour separate intervals, (-∞, 4) and [4, 5) and [5, 7) and [7, ∞), you see that f(x)is linear in each of those separate intervals. For that reason, it is said to be piecewise linear.Within each of those segments, the slope of the linear function is clearly constant, though itis different between segments. The points where the slope of the function changes are knownas breakpoints. The piecewise linear function in Figure 18.1 has three breakpoints.Figure 18.15f(x)4321x012345678Figure 18.1 A piecewise linear function with breakpointsPiecewise linear functions are often used to represent or to approximate nonlinear unaryfunctions (that is, nonlinear functions of one variable). For example, piecewise linear338 ILOG CPLEX 11.0 — USER’ S MANUAL