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

156 CHAPTER 12. THE OPTIMIZERS FOR MIXED-INTEGER PROBLEMS
Name Run-to-run deterministic Parallelized Strength Cost
Mixed-integer conic Yes Yes Conic Free add-on
Mixed-integer No Partial Linear Payed add-on
Table 12.1: Mixed-integer optimizers.
The mixed-integer optimization problem
z ∗ = minimize c T x
subject to Ax = b,
x ≥ 0
x j ∈ Z, ∀j ∈ J ,
(12.1)
has the continuous relaxation
z = minimize c T x
subject to Ax = b,
x ≥ 0
(12.2)
The continuos relaxation is identical to the mixed-integer problem with the restriction that some
variables must be integer removed.
There are two important observations about the continuous relaxation. Firstly, the continuous relaxation
is usually much faster to optimize than the mixed-integer problem. Secondly if ˆx is any feasible
solution to (12.1) and
then
¯z := c T ˆx
z ≤ z ∗ ≤ ¯z.
This is an important observation since if it is only possible to find a near optimal solution within a
reasonable time frame then the quality of the solution can nevertheless be evaluated. The value z is
a lower bound on the optimal objective value. This implies that the obtained solution is no further
away from the optimum than ¯z − z in terms of the objective value.
Whenever a mixed-integer problem is solved MOSEK rapports this lower bound so that the quality of
the reported solution can be evaluated.
12.2 The mixed-integer optimizers
MOSEK includes two mixed-integer optimizer which is compared in Table 12.1. Both optimizers can
handle problems with linear, quadratic objective and constraints and conic constraints. However, a
problem must not contain both quadratic objective and constraints and conic constraints.