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

88 CHAPTER 7. ADVANCED API TUTORIAL
7.2 Solving linear systems involving the basis matrix
A linear optimization problem always has an optimal solution which is also a basic solution. In an
optimal basic solution there are exactly m basic variables where m is the number of rows in the
constraint matrix A. Define
B ∈ R m×m
as a matrix consisting of the columns of A corresponding to the basic variables.
The basis matrix B is always non-singular, i.e.
det(B) ≠ 0
or equivalently that B −1 exists. This implies that the linear systems
and
B¯x = w (7.1)
each has a unique solution for all w .
B T ¯x = w (7.2)
MOSEK provides functions for solving the linear systems (7.1) and (7.2) for an arbitrary w-¿.
7.2.1 Identifying the basis
To use the solutions to (7.1) and (7.2) it is important to know how the basis matrix B is constructed.
Internally MOSEK employs the linear optimization problem
where
maximize
c T x
subject to Ax − x c = 0,
l x ≤ x ≤ u x ,
l c ≤ x c ≤ u c .
(7.3)
x c ∈ R m and x ∈ R n .
The basis matrix is constructed of m columns taken from
[ A − I ].
If variable x j is a basis variable, then the j ’th column of A denoted a :,j will appear in B. Similarly,
if x c i is a basis variable, then the i ’th column of −I will appear in the basis. The ordering of the