The MOSEK Python optimizer API manual Version 7.0 (Revision 141)

188 CHAPTER 15. SENSITIVITY ANALYSIS
In summary, the basis type sensitivity analysis is computationally cheap but does not provide complete
information. Hence, the results of the basis type sensitivity analysis should be used with care.
15.4.3 The optimal partition type sensitivity analysis
Another method for computing the complete linearity interval is called the optimal partition type
sensitivity analysis. The main drawback of the optimal partition type sensitivity analysis is that it
is computationally expensive compared to the basis type analysts. This type of sensitivity analysis is
currently provided as an experimental feature in MOSEK.
Given the optimal primal and dual solutions to (15.1), i.e. x ∗ and ((s c l )∗ , (s c u) ∗ , (s x l )∗ , (s x u) ∗ ) the optimal
objective value is given by
The left and right shadow prices σ 1 and σ 2 for l c i
z ∗ := c T x ∗ .
are given by this pair of optimization problems:
and
σ 1 = minimize e T i s c l
subject to A T (s c l − s c u) + s x l − s x u = c,
(l c ) T (s c l ) − (u c ) T (s c u) + (l x ) T (s x l ) − (u x ) T (s x u) = z ∗ ,
s c l , s c u, s c l , s x u ≥ 0
σ 2 = maximize e T i s c l
subject to A T (s c l − s c u) + s x l − s x u = c,
(l c ) T (s c l ) − (u c ) T (s c u) + (l x ) T (s x l ) − (u x ) T (s x u) = z ∗ ,
s c l , s c u, s c l , s x u ≥ 0.
These two optimization problems make it easy to interpret the shadow price. Indeed, if ((s c l )∗ , (s c u) ∗ , (s x l )∗ , (s x u) ∗ )
is an arbitrary optimal solution then
Next, the linearity interval [β 1 , β 2 ] for l c i
(s c l ) ∗ i ∈ [σ 1 , σ 2 ].
is computed by solving the two optimization problems
and
β 1 = minimize β
subject to l c + βe i ≤ Ax ≤ u c ,
c T x − σ 1 β = z ∗ ,
l x ≤ x ≤ u x ,
β 2 = maximize β
subject to l c + βe i ≤ Ax ≤ u c ,
c T x − σ 2 β = z ∗ ,
l x ≤ x ≤ u x .