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# ifndef CPPAD_BENDER_QUAD_INCLUDED
# define CPPAD_BENDER_QUAD_INCLUDED

/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-07 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the 
                    Common Public License Version 1.0.

A copy of this license is included in the COPYING file of this distribution.
Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
-------------------------------------------------------------------------- */
/*
$begin BenderQuad$$
$spell
        cppad.hpp
        BAvector
        gx
        gxx
        CppAD
        Fy
        dy
        Jacobian
        ADvector
        const
        dg
        ddg
$$


$index Hessian, Bender$$
$index Jacobian, Bender$$
$index BenderQuad$$
$section Computing Jacobian and Hessian of Bender's Reduced Objective$$

$head Syntax$$
$syntax%%
%# include <cppad/cppad.hpp>
BenderQuad(%x%, %y%, %fun%, %g%, %gx%, %gxx%)%$$  

$head Problem$$
The type $cref/ADvector/BenderQuad/ADvector/$$ cannot be determined
form the arguments above 
(currently the type $italic ADvector$$ must be 
$syntax%CPPAD_TEST_VECTOR<%Base%>%$$.)
This will be corrected in the future by requiring $italic Fun$$
to define $syntax%%Fun%::vector_type%$$ which will specify the
type $italic ADvector$$.

$head Purpose$$
We are given the optimization problem
$latex \[
\begin{array}{rcl}
        {\rm minimize} & F(x, y) & {\rm w.r.t.} \; (x, y) \in \R^n \times \R^m
\end{array}
\] $$
that is convex with respect to $latex y$$.
In addition, we are given a set of equations $latex H(x, y)$$
such that 
$latex \[
        H[ x , Y(x) ] = 0 \;\; \Rightarrow \;\; F_y [ x , Y(x) ] = 0
\] $$
(In fact, it is often the case that $latex H(x, y) = F_y (x, y)$$.)
Furthermore, it is easy to calculate a Newton step for these equations; i.e.,
$latex \[
        dy = - [ \partial_y H(x, y)]^{-1} H(x, y)
\] $$
The purpose of this routine is to compute the 
value, Jacobian, and Hessian of the reduced objective function
$latex \[
        G(x) = F[ x , Y(x) ]
\] $$
Note that if only the value and Jacobian are needed, they can be
computed more quickly using the relations
$latex \[
        G^{(1)} (x) = \partial_x F [x, Y(x) ]
\] $$ 

$head x$$
The $code BenderQuad$$ argument $italic x$$ has prototype
$syntax%
        const %BAvector% &%x%
%$$
(see $xref/BenderQuad/BAvector/BAvector/$$ below)
and its size must be equal to $italic n$$.
It specifies the point at which we evaluating 
the reduced objective function and its derivatives.


$head y$$
The $code BenderQuad$$ argument $italic y$$ has prototype
$syntax%
        const %BAvector% &%y%
%$$
and its size must be equal to $italic m$$.
It must be equal to $latex Y(x)$$; i.e., 
it must solve the problem in $latex y$$ for this given value of $latex x$$
$latex \[
\begin{array}{rcl}
        {\rm minimize} & F(x, y) & {\rm w.r.t.} \; y \in \R^m
\end{array}
\] $$

$head fun$$
The $code BenderQuad$$ object $italic fun$$ 
must support the member functions listed below.
The $syntax%AD<%Base%>%$$ arguments will be variables for
a tape created by a call to $cref%Independent%$$ from $code BenderQuad$$
(hence they can not be combined with variables corresponding to a 
different tape). 

$subhead fun.f$$
The $code BenderQuad$$ argument $italic fun$$ supports the syntax
$syntax%
        %f% = %fun%.f(%x%, %y%)
%$$
The $syntax%%fun%.f%$$ argument $italic x$$ has prototype
$syntax%
        const %ADvector% &%x%
%$$
(see $xref/BenderQuad/ADvector/ADvector/$$ below)
and its size must be equal to $italic n$$.
The $syntax%%fun%.f%$$ argument $italic y$$ has prototype
$syntax%
        const %ADvector% &%y%
%$$
and its size must be equal to $italic m$$.
The $syntax%%fun%.f%$$ result $italic f$$ has prototype
$syntax%
        %ADvector% %f%
%$$
and its size must be equal to one.
The value of $italic f$$ is
$latex \[
        f = F(x, y)
\] $$.

$subhead fun.h$$
The $code BenderQuad$$ argument $italic fun$$ supports the syntax
$syntax%
        %h% = %fun%.h(%x%, %y%)
%$$
The $syntax%%fun%.h%$$ argument $italic x$$ has prototype
$syntax%
        const %ADvector% &%x%
%$$
and its size must be equal to $italic n$$.
The $syntax%%fun%.h%$$ argument $italic y$$ has prototype
$syntax%
        const %BAvector% &%y%
%$$
and its size must be equal to $italic m$$.
The $syntax%%fun%.h%$$ result $italic h$$ has prototype
$syntax%
        %ADvector% %h%
%$$
and its size must be equal to $italic m$$.
The value of $italic h$$ is
$latex \[
        h = H(x, y)
\] $$.

$subhead fun.dy$$
The $code BenderQuad$$ argument $italic fun$$ supports the syntax
$syntax%
        %dy% = %fun%.dy(%x%, %y%, %h%)

%x%
%$$
The $syntax%%fun%.dy%$$ argument $italic x$$ has prototype
$syntax%
        const %BAvector% &%x%
%$$
and its size must be equal to $italic n$$.
Its value will be exactly equal to the $code BenderQuad$$ argument 
$italic x$$ and values depending on it can be stored as private objects
in $italic f$$ and need not be recalculated.
$syntax%

%y%
%$$
The $syntax%%fun%.dy%$$ argument $italic y$$ has prototype
$syntax%
        const %BAvector% &%y%
%$$
and its size must be equal to $italic m$$.
Its value will be exactly equal to the $code BenderQuad$$ argument 
$italic y$$ and values depending on it can be stored as private objects
in $italic f$$ and need not be recalculated.
$syntax%

%h%
%$$
The $syntax%%fun%.dy%$$ argument $italic h$$ has prototype
$syntax%
        const %ADvector% &%h%
%$$
and its size must be equal to $italic m$$.
$syntax%

%dy%
%$$
The $syntax%%fun%.dy%$$ result $italic dy$$ has prototype
$syntax%
        %ADvector% %dy%
%$$
and its size must be equal to $italic m$$.
The return value $italic dy$$ is given by
$latex \[
        dy = - [ \partial_y H (x , y) ]^{-1} h
\] $$
Note that if $italic h$$ is equal to $latex H(x, y)$$,
$latex dy$$ is the Newton step for finding a zero
of $latex H(x, y)$$ with respect to $latex y$$;
i.e., 
$latex y + dy$$ is an approximate solution for the equation
$latex H (x, y + dy) = 0$$. 

$head g$$
The argument $italic g$$ has prototype
$syntax%
        %BAvector% &%g%
%$$
and has size one.
The input value of its element does not matter.
On output,
it contains the value of $latex G (x)$$; i.e.,
$latex \[
        g[0] = G (x)
\] $$

$head gx$$
The argument $italic gx$$ has prototype
$syntax%
        %BAvector% &%gx%
%$$
and has size $latex n $$.
The input values of its elements do not matter.
On output,
it contains the Jacobian of $latex G (x)$$; i.e.,
for $latex j = 0 , \ldots , n-1$$, 
$latex \[
        gx[ j ] = G^{(1)} (x)_j
\] $$

$head gxx$$
The argument $italic gx$$ has prototype
$syntax%
        %BAvector% &%gxx%
%$$
and has size $latex n \times n$$.
The input values of its elements do not matter.
On output,
it contains the Hessian of $latex G (x)$$; i.e.,
for $latex i = 0 , \ldots , n-1$$, and
$latex j = 0 , \ldots , n-1$$, 
$latex \[
        gxx[ i * n + j ] = G^{(2)} (x)_{i,j}
\] $$

$head BAvector$$
The type $italic BAvector$$ must be a 
$xref/SimpleVector/$$ class. 
We use $italic Base$$ to refer to the type of the elements of 
$italic BAvector$$; i.e.,
$syntax%
        %BAvector%::value_type
%$$

$head ADvector$$
The type $italic ADvector$$ must be a 
$xref/SimpleVector/$$ class with elements of type 
$syntax%AD<%Base%>%$$; i.e.,
$syntax%
        %ADvector%::value_type
%$$
must be the same type as
$syntax%
        AD< %BAvector%::value_type >
%$$.


$head Example$$
$children%
        example/bender_quad.cpp
%$$
The file
$xref/BenderQuad.cpp/$$
contains an example and test of this operation.   
It returns true if it succeeds and false otherwise.


$end
-----------------------------------------------------------------------------
*/

namespace CppAD { // BEGIN CppAD namespace

template <class BAvector, class Fun>
void BenderQuad(
        const BAvector   &x     , 
        const BAvector   &y     , 
        Fun               fun   , 
        BAvector         &g     ,
        BAvector         &gx    ,
        BAvector         &gxx   )
{       // determine the base type
        typedef typename BAvector::value_type Base;

        // check that BAvector is a SimpleVector class
        CheckSimpleVector<Base, BAvector>();

        // declare the ADvector type
        typedef CPPAD_TEST_VECTOR< AD<Base> > ADvector;

        // size of the x and y spaces
        size_t n = x.size();
        size_t m = y.size();

        // check the size of gx and gxx
        CPPAD_ASSERT_KNOWN(
                g.size() == 1,
                "BenderQuad: size of the vector g is not equal to 1"
        );
        CPPAD_ASSERT_KNOWN(
                gx.size() == n,
                "BenderQuad: size of the vector gx is not equal to n"
        );
        CPPAD_ASSERT_KNOWN(
                gxx.size() == n * n,
                "BenderQuad: size of the vector gxx is not equal to n * n"
        );

        // some temporary indices
        size_t i, j;

        // variable versions x
        ADvector vx(n);
        for(j = 0; j < n; j++)
                vx[j] = x[j];
        
        // declare the independent variables
        Independent(vx);

        // evaluate h = H(x, y) 
        ADvector h(m);
        h = fun.h(vx, y);

        // evaluate dy (x) = Newton step as a function of x through h only
        ADvector dy(m);
        dy = fun.dy(x, y, h);

        // variable version of y
        ADvector vy(m);
        for(j = 0; j < m; j++)
                vy[j] = y[j] + dy[j];

        // evaluate G~ (x) = F [ x , y + dy(x) ] 
        ADvector gtilde(1);
        gtilde = fun.f(vx, vy);

        // AD function object that corresponds to G~ (x)
        ADFun<Base> Gtilde(vx, gtilde); 

        // value of G(x)
        g[0] = Value( gtilde[0] );

        // initial forward direction vector as zero
        BAvector dx(n);
        for(j = 0; j < n; j++)
                dx[j] = Base(0);

        // weight, first and second order derivative values
        BAvector dg(1), w(1), ddw(2 * n);
        w[0] = 1.;

        // Jacobian and Hessian of G(x) is equal Jacobian and Hessian of Gtilde
        for(j = 0; j < n; j++)
        {       // compute partials in x[j] direction
                dx[j] = Base(1);
                dg    = Gtilde.Forward(1, dx);
                gx[j] = dg[0];

                // restore the dx vector to zero
                dx[j] = Base(0);

                // compute second partials w.r.t x[j] and x[l]  for l = 1, n
                ddw = Gtilde.Reverse(2, w);
                for(i = 0; i < n; i++)
                        gxx[ i * n + j ] = ddw[ i * 2 + 1 ];
        }

        return;
}
        
} // END CppAD namespace

# endif

---