The MOSEK command line tool Version 7.0 (Revision 141)

40 CHAPTER 5. THE OPTIMIZERS FOR CONTINUOUS PROBLEMS
Tolerance
ɛ p
ɛ d
ɛ g
ɛ i
Parameter name
MSK DPAR INTPNT TOL PFEAS
MSK DPAR INTPNT TOL DFEAS
MSK DPAR INTPNT TOL REL GAP
MSK DPAR INTPNT TOL INFEAS
Table 5.1: Parameters employed in termination criterion.
Finally, if
ɛ i b T y k >
‖b‖ ∞
∥
∥A T y k + s k∥ ∥ ∞
max(1, ‖c‖ ∞ )
then y k is reported as a certificate of primal infeasibility.
It is possible to adjust the tolerances ɛ p , ɛ d , ɛ g and ɛ i using parameters; see table 5.1 for details. The
default values of the termination tolerances are chosen such that for a majority of problems appearing
in practice it is not possible to achieve much better accuracy. Therefore, tightening the tolerances
usually is not worthwhile. However, an inspection of (5.5) reveals that quality of the solution is
dependent on ‖b‖ ∞ and ‖c‖ ∞ ; the smaller the norms are, the better the solution accuracy.
The interior-point method as implemented by MOSEK will converge toward optimality and primal
and dual feasibility at the same rate [5]. This means that if the optimizer is stopped prematurely then
it is very unlikely that either the primal or dual solution is feasible. Another consequence is that in
most cases all the tolerances, ɛ p , ɛ d and ɛ g , has to be relaxed together to achieve an effect.
The basis identification discussed in section 5.2.2.2 requires an optimal solution to work well; hence
basis identification should turned off if the termination criterion is relaxed.
To conclude the discussion in this section, relaxing the termination criterion is usually is not worthwhile.
5.2.2.2 Basis identification
An interior-point optimizer does not return an optimal basic solution unless the problem has a
unique primal and dual optimal solution. Therefore, the interior-point optimizer has an optional
post-processing step that computes an optimal basic solution starting from the optimal interior-point
solution. More information about the basis identification procedure may be found in [6].
Please note that a basic solution is often more accurate than an interior-point solution.
By default MOSEK performs a basis identification. However, if a basic solution is not needed, the
basis identification procedure can be turned off. The parameters
• MSK IPAR INTPNT BASIS,
• MSK IPAR BI IGNORE MAX ITER, and
• MSK IPAR BI IGNORE NUM ERROR
controls when basis identification is performed.