The MOSEK command line tool Version 7.0 (Revision 141)

5.5. NONLINEAR CONVEX OPTIMIZATION 45
5.5 Nonlinear convex optimization
5.5.1 The interior-point optimizer
For quadratic, quadratically constrained, and general convex optimization problems an interior-point
type optimizer is available. The interior-point optimizer is an implementation of the homogeneous and
self-dual algorithm. For a detailed description of the algorithm, please see [9], [10].
5.5.1.1 The convexity requirement
Continuous nonlinear problems are required to be convex. For quadratic problems MOSEK test this
requirement before optimizing. Specifying a non-convex problem results in an error message.
The following parameters are available to control the convexity check:
• MSK IPAR CHECK CONVEXITY: Turn convexity check on/off.
• MSK DPAR CHECK CONVEXITY REL TOL: Tolerance for convexity check.
• MSK IPAR LOG CHECK CONVEXITY: Turn on more log information for debugging.
5.5.1.2 The differentiabilty requirement
The nonlinear optimizer in MOSEK requires both first order and second order derivatives. This of
course implies care should be taken when solving problems involving non-differentiable functions.
For instance, the function
is differentiable everywhere whereas the function
f(x) = x 2
f(x) = √ x
is only differentiable for x > 0 . In order to make sure that MOSEK evaluates the functions at points
where they are differentiable, the function domains must be defined by setting appropriate variable
bounds.
In general, if a variable is not ranged MOSEK will only evaluate that variable at points strictly within
the bounds. Hence, imposing the bound
x ≥ 0
in the case of √ x is sufficient to guarantee that the function will only be evaluated in points where it
is differentiable.
However, if a function is differentiable on closed a range, specifying the variable bounds is not sufficient.
Consider the function