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

Chapter 12
The optimizers for mixed-integer
problems
A problem is a mixed-integer optimization problem when one or more of the variables are constrained
to be integer valued. MOSEK contains two optimizers for mixed integer problems that is capable for
solving mixed-integer
• linear,
• quadratic and quadratically constrained, and
• conic
problems.
Readers unfamiliar with integer optimization are recommended to consult some relevant literature,
e.g. the book [16] by Wolsey.
12.1 Some concepts and facts related to mixed-integer optimization
It is important to understand that in a worst-case scenario, the time required to solve integer optimization
problems grows exponentially with the size of the problem. For instance, assume that a problem
contains n binary variables, then the time required to solve the problem in the worst case may be
proportional to 2 n . The value of 2 n is huge even for moderate values of n .
In practice this implies that the focus should be on computing a near optimal solution quickly rather
than at locating an optimal solution. Even if the problem is only solved approximately, it is important
to know how far the approximate solution is from an optimal one. In order to say something about
the goodness of an approximate solution then the concept of a relaxation is important.
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