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lines 6-72 of file: example/abs_normal/simplex_method.cpp
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{xrst_begin simplex_method.cpp}
{xrst_spell
   rlr
}

abs_normal simplex_method: Example and Test
###########################################

Problem
*******
Our original problem is

.. math::

   \R{minimize} \; | u - 1| \; \R{w.r.t} \; u \in \B{R}

We reformulate this as the following problem

.. math::

   \begin{array}{rlr}
      \R{minimize}      & v             & \R{w.r.t} \; (u,v) \in \B{R}^2 \\
      \R{subject \; to} &  u - 1 \leq v \\
                         &  1 - u \leq v
   \end{array}

We know that the value of :math:`v` at the solution is greater than
or equal zero. Hence we can reformulate this problem as

.. math::

   \begin{array}{rlr}
   \R{minimize}      & v             & \R{w.r.t} \; ( u_- , u_+ , v) \in \B{R}_+^3 \\
   \R{subject \; to} & u_+ - u_- - 1  \leq v \\
                     &  1 - u_+ + u_- \leq v
   \end{array}

This is equivalent to

.. math::

   \begin{array}{rlr}
      \R{minimize}
      & (0, 0, 1) \cdot ( u_+, u_- , v)^T  & \R{w.r.t} \; (u,v) \in \B{R}_+^3 \\
   \R{subject \; to}
      &
      \left( \begin{array}{ccc}
         +1 & -1 & -1 \\
         -1 & +1 & +1
      \end{array} \right)
      \left( \begin{array}{c} u_+ \\ u_- \\ v \end{array} \right)
      +
      \left( \begin{array}{c} -1 \\ 1 \end{array} \right)
      \leq
      0
   \end{array}

which is in the form expected by :ref:`simplex_method-name` .

Source
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{xrst_literal
   // BEGIN C++
   // END C++
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{xrst_end simplex_method.cpp}