---------------------------------------------------
lines 6-59 of file: example/user/fix_constraint.cpp
---------------------------------------------------

{xrst_begin fix_constraint.cpp}

Using Constraints: Example and Test
###################################

Model
*****

.. math::

    \B{p}( y_i | \theta , u ) \sim \B{N} ( u_i + \theta_i , 1 )

.. math::

    \B{p}( u_i | \theta ) \sim \B{N} ( 0 , 1 )

.. math::

    \B{p}( \theta ) \sim \B{U} ( - \infty , + \infty )

where :math:`\B{U} ( - \infty ,  + \infty )` is the improper uniform prior
on :math:`[- \infty , + \infty ]`.
It follows that the Laplace approximation is exact and

.. math::

    \B{p}( y_i | \theta ) \sim \B{N} ( \theta_i , 2 )

The corresponding objective for the fixed effects is equivalent to:

.. math::

    \frac{1}{2} \sum_{i=0}^{N-1} ( y_i - \theta_i )^2

For this problem we add the explicit constraint

.. math::

    \frac{1}{2} \sum_i \theta_i^2 = 1;

The corresponding Lagrangian is

.. math::

    L( \theta , \lambda ) =
        \frac{1}{2} \sum_{i=0}^{N-1} ( y_i - \theta_i )^2
            + \lambda \left( \frac{1}{2} \sum_i \theta_i^2 - 1 \right)

{xrst_literal
    // BEGIN C++
    // END C++
}

{xrst_end fix_constraint.cpp}