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lines 8-155 of file: example/abs_normal/min_nso_linear.hpp
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{xrst_begin min_nso_linear}
{xrst_spell
   dbl
   maxitr
}
Non-Smooth Optimization Using Abs-normal Linear Approximations
##############################################################

Syntax
******
| *ok* = ``min_nso_linear`` (
| |tab| *level* , *g* , *a* , *epsilon* , *maxitr* , *b_in* , *x_in* , *x_out*
| )

Prototype
*********
{xrst_literal
   // BEGIN PROTOTYPE
   // END PROTOTYPE
}

Source
******
This following is a link to the source code for this example:
:ref:`min_nso_linear.hpp-name` .

Purpose
*******
Given a current that abs-normal representation
:ref:`abs_normal_fun@g` , :ref:`abs_normal_fun@a` ,
for a function :math:`f(x)`,
this routine minimizes :math:`f(x)`.

DblVector
*********
is a :ref:`SimpleVector-name` class with elements of type ``double`` .

SizeVector
**********
is a :ref:`SimpleVector-name` class with elements of type ``size_t`` .

f
*
We use the notation *f* for the original function; see
:ref:`abs_normal_fun@f` .

n
=
We use *n* to denote the dimension of the domain for *f* ; i.e.,
*f* . ``Domain`` () .

m
=
We use *m* to denote the dimension of the range for *f* ; i.e.,
*f* . ``Range`` () .
This must be equal to one.

s
=
We use
:ref:`abs_normal_fun@f@s` to denote the number absolute terms in *f* .

level
*****
This value is less that or equal 5.
If *level*  == 0 ,
no tracing of the optimization is printed.
If *level*  >= 1 ,
a trace of each iteration of ``min_nso_linear`` is printed.
If *level*  >= 2 ,
a trace of each iteration of the ``abs_min_linear`` sub-problem is printed.
If *level*  >= 3 ,
a trace of the :ref:`lp_box-name` sub-problem is printed.
If *level*  >= 4 ,
a trace of the objective and primal variables :math:`x` are printed
at each :ref:`simplex_method-name` iteration.
If *level*  == 5 ,
the simplex tableau is printed at each simplex iteration.

g
*
This is the function :ref:`abs_normal_fun@g`
in the abs-normal representation of *f* .

a
*
This is the function :ref:`abs_normal_fun@a`
in the abs-normal representation of *f* .

epsilon
*******
This is a vector with size 2.
The value *epsilon* [0] is convergence criteria in terms
of the infinity norm of the difference of *x_out*
between iterations.
The value *epsilon* [1] is convergence criteria in terms
of the derivative of :math:`f(x)`.
This derivative is actually the average of the directional derivative
in the direction of the sub-problem minimizer.

maxitr
******
This is a vector with size 3.
The value *maxitr* [0] is the maximum number of
``min_nso_linear`` iterations to try before giving up on convergence.
The value *maxitr* [1] is the maximum number of iterations in the
``abs_min_linear`` sub-problem.
The value *maxitr* [2] is the maximum number of iterations in
the :ref:`simplex_method<simplex_method@maxitr>` sub-problems.

b_in
****
This the initial bound on the trust region size.
To be specific, if :math:`b` is the current trust region size,
at each iteration affine approximation is minimized with respect to
:math:`\Delta x` and subject to

.. math::

   -b \leq \Delta x_j \leq b

for *j* = 0 , ..., *n* ``-1`` .
It must hold that *b_in* > *epsilon* [0] .

x_in
****
This vector *x_out* has size *n* .
It is the starting point for the optimization procedure; i.e.,
the ``min_nso_linear`` iterations.

x_out
*****
This vector *x_out* has size *n* .
The input value of its elements does not matter.
Upon return,
it is the approximate minimizer
of the abs-normal approximation for :math:`f(x)` over the trust region
is :math:`x = \hat{x} + \Delta x`.
{xrst_toc_hidden
   example/abs_normal/min_nso_linear.cpp
   example/abs_normal/min_nso_linear.xrst
}
Example
*******
The file :ref:`min_nso_linear.cpp-name` contains an example and test of
``min_nso_linear`` .

{xrst_end min_nso_linear}