ILOG CPLEX 11.0 User's Manual

Reminder: It may help you understand the signatures of these functions to recall thefamiliar Cartesian representation of a line or segment in two dimensions, x and y:y=ax+bwhere a represents the slope of the line or segment and b represents the height at which theline theoretically crosses the y-axis at the point (0, b).Discontinuous Piecewise Linear FunctionsThus far, you have seen a piecewise linear function where the segments are continuous.Intuitively, in a continuous piecewise linear function, the endpoint of one segment has thesame coordinates as the initial point of the next segment, as in Figure 18.1.There are piecewise linear functions, however, where the endpoint of one segment and theinitial point of the next segment may have the same x coordinate but differ in the value off(x). Such a difference is known as a step in the piecewise linear function, and such afunction is known as discontinuous. Figure 18.2 shows a discontinuous piecewise linearfunction with two steps.Syntactically, a step is represented in this way:◆ The x-coordinate of the breakpoint where the step occurs is repeated in the array of thebreakpoint.◆The value of the first point of a step in the array of slopes is the height of the step.◆ The value of the second point of the step in the array of slopes is the slope of the functionafter the step.By convention, a breakpoint belongs in both segments associated with the step. For example,in Figure 18.2, at the breakpoint x=3, the points (3,1) and (3,3) are both admissible.Similarly, when x = 5, the points (5,4) and (5,5) are both admissible.However, isolated points, as explained in Isolated Points in Piecewise Linear Functions onpage 341, are not allowed, neither in continuous nor in discontinuous piecewise linearfunctions. In fact, only one step is allowed at a given point.In Concert Technology, a discontinuous piecewise linear function is represented as aninstance of the class IloPiecewiseLinear (the same class as used for continuouspiecewise linear functions). For example, the function in Figure 18.2 is declared in this way:IloPiecewiseLinear(x,IloNumArray(env, 4, 3. ,3. ,5. ,5.),IloNumArray(env, 5, 0., 2., 0.5, 1., -1.),0, 1);340 ILOG CPLEX 11.0 — USER’ S MANUAL