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lines 6-91 of file: example/user/optimize_fixed.cpp
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{xrst_begin optimize_fixed.cpp}

Optimize Fixed Effects: Example and Test
########################################

Model
*****

.. math::

    \B{p}( y_i | \theta , u ) \sim \B{N} ( u_i + \theta_0 , \theta_1^2 )

.. math::

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

.. math::

    \B{p}( \theta ) \sim \B{N} ( 4 , 1 )

It follows that the Laplace approximation is exact and

.. math::

    \B{p}( y_i | \theta ) \sim \B{N} \left( \theta_0 , 1 + \theta_1^2 \right)

The constraints on the fixed effect are

.. math::

    - \infty \leq \theta_0 \leq + \infty
    \R{\; and \;}
    0.1 \leq \theta_1 \leq 100

Objective
*********
The corresponding objective for the fixed effects is equivalent to:

.. math::

    F( \theta ) = \frac{1}{2} \left[
        ( \theta_0 - 4 )^2 + ( \theta_1 - 4 )^2 +
            N \log \left( 1 + \theta_1^2 \right) +
            ( 1 + \theta_1^2)^{-1} \sum_{i=0}^{N-1} ( y_i - \theta_0 )^2
    \right]

First Order Partials
********************
The first order partial derivatives of the objective are:

.. math::

    F_0 ( \theta )
    =
    ( \theta_0 - 4 )
    -
    ( 1 + \theta_1^2)^{-1} \sum_{i=0}^{N-1} ( y_i - \theta_0 )

.. math::

    F_1 ( \theta )
    =
    ( \theta_1 - 4 )
    +
    N \left( 1 + \theta_1^2 \right)^{-1} \theta_1
    -
    ( 1 + \theta_1^2)^{-2} \theta_1 \sum_{i=0}^{N-1} ( y_i - \theta_0 )^2

Optimizer Trace
***************
This example uses the optimizer trace information; see
:ref:`fixed_solution@trace_vec` .

Optimizer Warm Start
********************
This example uses the optimizer warm start information; see
:ref:`fixed_solution@warm_start` .

Source Code
***********
{xrst_literal
    // BEGIN C++
    // END C++
}

{xrst_end optimize_fixed.cpp}