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

11.2. LINEAR OPTIMIZATION 139
11.2.2 The interior-point optimizer
The purpose of this section is to provide information about the algorithm employed in MOSEK interiorpoint
optimizer.
In order to keep the discussion simple it is assumed that MOSEK solves linear optimization problems
on standard form
minimize c T x
subject to Ax = b,
x ≥ 0.
(11.1)
This is in fact what happens inside MOSEK; for efficiency reasons MOSEK converts the problem to
standard form before solving, then convert it back to the input form when reporting the solution.
Since it is not known beforehand whether problem (11.1) has an optimal solution, is primal infeasible
or is dual infeasible, the optimization algorithm must deal with all three situations. This is the reason
that MOSEK solves the so-called homogeneous model
Ax − bτ = 0,
A T y + s − cτ = 0,
− c T x + b T y − κ = 0,
x, s, τ, κ ≥ 0,
(11.2)
where y and s correspond to the dual variables in (11.1), and τ and κ are two additional scalar variables.
Note that the homogeneous model (11.2) always has solution since
is a solution, although not a very interesting one.
Any solution
(x, y, s, τ, κ) = (0, 0, 0, 0, 0)
to the homogeneous model (11.2) satisfies
(x ∗ , y ∗ , s ∗ , τ ∗ , κ ∗ )
x ∗ j s ∗ j = 0 and τ ∗ κ ∗ = 0.
Moreover, there is always a solution that has the property
First, assume that τ ∗ > 0 . It follows that
τ ∗ + κ ∗ > 0.