The MOSEK command line tool Version 7.0 (Revision 141)

42 CHAPTER 5. THE OPTIMIZERS FOR CONTINUOUS PROBLEMS
5.2.3 The simplex based optimizer
An alternative to the interior-point optimizer is the simplex optimizer.
The simplex optimizer uses a different method that allows exploiting an initial guess for the optimal
solution to reduce the solution time. Depending on the problem it may be faster or slower to use an
initial guess; see section 5.2.4 for a discussion.
MOSEK provides both a primal and a dual variant of the simplex optimizer — we will return to this
later.
5.2.3.1 Simplex termination criterion
The simplex optimizer terminates when it finds an optimal basic solution or an infeasibility certificate.
A basic solution is optimal when it is primal and dual feasible; see (4.1) and (4.2) for a definition
of the primal and dual problem. Due the fact that to computations are performed in finite precision
MOSEK allows violation of primal and dual feasibility within certain tolerances. The user
can control the allowed primal and dual infeasibility with the parameters MSK DPAR BASIS TOL X and
MSK DPAR BASIS TOL S.
5.2.3.2 Starting from an existing solution
When using the simplex optimizer it may be possible to reuse an existing solution and thereby reduce
the solution time significantly. When a simplex optimizer starts from an existing solution it is said to
perform a hot-start. If the user is solving a sequence of optimization problems by solving the problem,
making modifications, and solving again, MOSEK will hot-start automatically.
Setting the parameter MSK IPAR OPTIMIZER to MSK OPTIMIZER FREE SIMPLEX instructs MOSEK to
select automatically between the primal and the dual simplex optimizers. Hence, MOSEK tries to
choose the best optimizer for the given problem and the available solution.
By default MOSEK uses presolve when performing a hot-start. If the optimizer only needs very few
iterations to find the optimal solution it may be better to turn off the presolve.
5.2.3.3 Numerical difficulties in the simplex optimizers
Though MOSEK is designed to minimize numerical instability, completely avoiding it is impossible
when working in finite precision. MOSEK counts a ”numerical unexpected behavior” event inside the
optimizer as a set-back. The user can define how many set-backs the optimizer accepts; if that number
is exceeded, the optimization will be aborted. Set-backs are implemented to avoid long sequences
where the optimizer tries to recover from an unstable situation.
Set-backs are, for example, repeated singularities when factorizing the basis matrix, repeated loss
of feasibility, degeneracy problems (no progress in objective) and other events indicating numerical
difficulties. If the simplex optimizer encounters a lot of set-backs the problem is usually badly scaled;
in such a situation try to reformulate into a better scaled problem. Then, if a lot of set-backs still
occur, trying one or more of the following suggestions may be worthwhile: