The MOSEK command line tool Version 7.0 (Revision 141)

8.4. SENSITIVITY ANALYSIS FOR LINEAR PROBLEMS 73
for which
β ∈ [β 1 , β 2 ]
f ′ l c i (β) = f ′ l c i (0).
Since f l c
i
is not a smooth function f
l ′ may not be defined at 0, as illustrated by the right example in
i
c
figure 8.1. In this case we can define a left and a right shadow price and a left and a right linearity
interval.
The function f l c
i
considered only changes in li c . We can define similar functions for the remaining
parameters of the z defined in (8.1) as well:
f u c
i
(β) = z(l c , u c + βe i , l x , u x , c), i = 1, . . . , m,
f l x
j
(β) = z(l c , u c , l x + βe j , u x , c), j = 1, . . . , n,
f u x
j
(β) = z(l c , u c , l x , u x + βe j , c), j = 1, . . . , n,
f cj (β) = z(l c , u c , l x , u x , c + βe j ), j = 1, . . . , n.
Given these definitions it should be clear how linearity intervals and shadow prices are defined for the
parameters u c i etc.
8.4.1.1 Equality constraints
In MOSEK a constraint can be specified as either an equality constraint or a ranged constraint. If
constraint i is an equality constraint, we define the optimal value function for this as
f e c
i
(β) = z(l c + βe i , u c + βe i , l x , u x , c)
Thus for an equality constraint the upper and the lower bounds (which are equal) are perturbed
simultaneously. Therefore, MOSEK will handle sensitivity analysis differently for a ranged constraint
with li c = uc i and for an equality constraint.
8.4.2 The basis type sensitivity analysis
The classical sensitivity analysis discussed in most textbooks about linear optimization, e.g. [12], is
based on an optimal basic solution or, equivalently, on an optimal basis. This method may produce
misleading results [13] but is computationally cheap. Therefore, and for historical reasons this
method is available in MOSEK We will now briefly discuss the basis type sensitivity analysis. Given
an optimal basic solution which provides a partition of variables into basic and non-basic variables, the
basis type sensitivity analysis computes the linearity interval [β 1 , β 2 ] so that the basis remains optimal
for the perturbed problem. A shadow price associated with the linearity interval is also computed.
However, it is well-known that an optimal basic solution may not be unique and therefore the result
depends on the optimal basic solution employed in the sensitivity analysis. This implies that the
computed interval is only a subset of the largest interval for which the shadow price is constant.
Furthermore, the optimal objective value function might have a breakpoint for β = 0. In this case the
basis type sensitivity method will only provide a subset of either the left or the right linearity interval.