In addition to allowing the solver to work more efficiently, SOS sets also help to reduce the number of variables and constraints in your model. Running, the model, as predicted, we see that for an X value of 8.5, total cost is 25.5: In particular, each weight W(i) is a member of the Type SOS2 set titled SOS2_SET.įor this particular example, we have chosen to pick an x-value and then let LINGO compute the corresponding y-value, or cost. A linear spline with knots at \xik with k 1, \dots, K is a piecewise linear polynomial continuous at each knot.
As you recall, this is the definition of an SOS2 set, which we create at the end of the model with the expression: Just like the global polynomials and the piecewise constant functions, we can represent this as a linear expansion in basis functions that are transformations of the variables. In the case where we lie exactly on a breakpoint, then only one of the W(i) will be nonzero and equal to 1.įor this strategy to work correctly, only two, at most, of the W(i) may be nonzero, and they must be adjacent. For example, if W(2)=W(3)=0.5, then we are exactly halfway between the second and third breakpoints : (5,10) and (12,41), i.e., at point (8.5,25.5). Answer: In terms of a number of discrete intervals, sets, or pieces, these are piecewise continuous functions. We defined an attribute, W, whose members act as weights, placing us on an particular segment of the curve. Weights can be B( I): 'SOS2_SET', W( I))) ! the weights are SOS2: at most two adjacent Variable will automatically be set to the ! set x to any value in interval-the cost
! See "Optimization Modeling with Lingo", The following sample model, SOSPIECE.LG4, uses a Type 2 SOS set to model this piecewise-linear function using what is referred to as the lambda method: For example, suppose we want to model the following cost function, where cost is a piecewise-linear function of volume, X: Many cost curves exhibit the piecewise-linear property. As we mentioned above, SOS2 sets are particularly useful for implementing piecewise-linear functions.
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