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72 CHAPTER 8. SENSITIVITY ANALYSIS
f( β )
f( β )
β
1 0
β
2
β
β
1
0
β
2
β
Figure 8.1: The optimal value function f l c
i
(β). Left: β = 0 is in the interior of linearity interval.
Right: β = 0 is a breakpoint.
8.4 Sensitivity analysis for linear problems
8.4.1 The optimal objective value function
Assume that we are given the problem
z(l c , u c , l x , u x , c) = minimize c T x
subject to l c ≤ Ax ≤ u c ,
l x ≤ x ≤ u x ,
and we want to know how the optimal objective value changes as li
c is perturbed. To answer this
question we define the perturbed problem for li c as follows
f l c
i
(β) = minimize c T x
subject to l c + βe i ≤ Ax ≤ u c ,
l x ≤ x ≤ u x ,
where e i is the i th column of the identity matrix. The function
(8.1)
f l c
i
(β) (8.2)
shows the optimal objective value as a function of β. Please note that a change in β corresponds to a
perturbation in l c i and hence (8.2) shows the optimal objective value as a function of lc i .
It is possible to prove that the function (8.2) is a piecewise linear and convex function, i.e. the function
may look like the illustration in Figure 8.1. Clearly, if the function f l c
i
(β) does not change much when
β is changed, then we can conclude that the optimal objective value is insensitive to changes in l c i .
Therefore, we are interested in the rate of change in f l c
i
(β) for small changes in β — specificly the
gradient
f ′ l c i (0),
which is called the shadow pricerelated to li c.
The shadow price specifies how the objective value
changes for small changes in β around zero. Moreover, we are interested in the linearity interval