--------------------------------------------------
lines 8-198 of file: example/abs_normal/qp_box.hpp
--------------------------------------------------

{xrst_begin qp_box}
{xrst_spell
   maxitr
   rl
   xin
   xout
}
abs_normal: Solve a Quadratic Program With Box Constraints
##########################################################

Syntax
******
| *ok* = ``qp_box`` (
| |tab| *level* , *a* , *b* , *c* , *C* , *g* , *G* , *epsilon* , *maxitr* , *xin* , *xout*
| )

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

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

Purpose
*******
This routine could be used to create a version of :ref:`abs_min_linear-name`
that solved quadratic programs (instead of linear programs).

Problem
*******
We are given
:math:`a \in \B{R}^n`,
:math:`b \in \B{R}^n`,
:math:`c \in \B{R}^m`,
:math:`C \in \B{R}^{m \times n}`,
:math:`g \in \B{R}^n`,
:math:`G \in \B{R}^{n \times n}`,
where :math:`G` is positive semi-definite.
This routine solves the problem

.. math::

   \begin{array}{rl}
   \R{minimize} &
   \frac{1}{2} x^T G x + g^T x \; \R{w.r.t} \; x \in \B{R}^n
   \\
   \R{subject \; to} & C x + c \leq 0 \; \R{and} \; a \leq x \leq b
   \end{array}

The matrix :math:`G + C^T C` must be positive definite on components
of the vector :math:`x` where the lower limit minus infinity
and the upper limit is plus infinity; see *a* and *b* below.

Vector
******
The type *Vector* is a
simple vector with elements of type ``double`` .

level
*****
This value is less that or equal two.
If *level*  == 0 ,
no tracing is printed.
If *level*  >= 1 ,
a trace of the ``qp_box`` operations is printed.
If *level*  == 2 ,
a trace of the :ref:`qp_interior-name` sub-problem is printed.

a
*
This is the vector of lower limits for :math:`x` in the problem.
If *a* [ *j* ] is minus infinity, there is no lower limit
for :math:`x_j`.

b
*
This is the vector of upper limits for :math:`x` in the problem.
If *a* [ *j* ] is plus infinity, there is no upper limit
for :math:`x_j`.

Lower c
*******
This is the value of the inequality constraint function at :math:`x = 0`.

Upper C
*******
This is a :ref:`row-major<glossary@Row-major Representation>` representation
of thee the inequality constraint matrix :math:`C`.

Lower g
*******
This is the gradient of the objective function.

Upper G
*******
This is a row-major representation of the Hessian of the objective function.
For :math:`j = 0 , \ldots , n-1`,
:math:`- \infty < a_j` or
:math:`b_j < + \infty` or
:math:`G_{j,j} > 0.0`.

epsilon
*******
This argument is the convergence criteria;
see :ref:`qp_box@KKT Conditions` below.
It must be greater than zero.

maxitr
******
This is the maximum number of
:ref:`qp_interior-name` iterations to try before giving up
on convergence.

xin
***
This argument has size *n* and is the initial point for the algorithm.
It must strictly satisfy the constraints; i.e.,

   *a* < *xin* , *xin* < *b* , *C* * *xin* ``-`` *c*  < 0

xout
****
This argument has size is *n* and
the input value of its elements does no matter.
Upon return it is the primal variables
:math:`x` corresponding to the problem solution.

ok
**
If the return value *ok* is true, convergence is obtained; i.e.,

.. math::

   | F ( x , y_a, s_a, y_b, s_b, y_c, s_c ) |_\infty < \varepsilon

where :math:`|v|_\infty` is the infinity norm of the vector :math:`v`,
:math:`\varepsilon` is *epsilon* ,
:math:`x` is equal to *xout* ,
:math:`y_a, s_a \in \B{R}_+^n`,
:math:`y_b, s_b \in \B{R}_+^n` and
:math:`y_c, s_c \in \B{R}_+^m`.

KKT Conditions
**************
Give a vector :math:`v \in \B{R}^m` we define
:math:`D(v) \in \B{R}^{m \times m}` as the corresponding diagonal matrix.
We also define :math:`1_m \in \B{R}^m` as the vector of ones.
We define

.. math::

   F ( x , y_a, s_a, y_b, s_b, y_c, s_c )
   =
   \left(
   \begin{array}{c}
   g + G x - y_a + y_b + y_c^T C         \\
   a + s_a - x                           \\
   x + s_b - b                           \\
   C x + c + s_c                         \\
   D(s_a) D(y_a) 1_m                     \\
   D(s_b) D(y_b) 1_m                     \\
   D(s_c) D(y_c) 1_m
   \end{array}
   \right)

where
:math:`x \in \B{R}^n`,
:math:`y_a, s_a \in \B{R}_+^n`,
:math:`y_b, s_b \in \B{R}_+^n` and
:math:`y_c, s_c \in \B{R}_+^m`.
The KKT conditions for a solution of this problem is

.. math::

   F ( x , y_a, s_a, y_b, s_b, y_c, s_c ) = 0

{xrst_toc_hidden
   example/abs_normal/qp_box.cpp
   example/abs_normal/qp_box.xrst
}
Example
*******
The file :ref:`qp_box.cpp-name` contains an example and test of
``qp_box`` .

{xrst_end qp_box}