Computing Jacobian and Hessian of Bender's Reduced Objective

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@This is cppad-20221105 documentation. Here is a link to its current documentation . Computing Jacobian and Hessian of Bender's Reduced Objective
Syntax



# include <cppad/cppad.hpp>

BenderQuad(x, y, fun, g, gx, gxx)

See Also
opt_val_hes

Problem
The type ADvector cannot be determined form the arguments above (currently the type ADvector must be CPPAD_TESTVECTOR(Base) .) This will be corrected in the future by requiring Fun to define Fun::vector_type which will specify the type ADvector .

Purpose
We are given the optimization problem @[@ \begin{array}{rcl} {\rm minimize} & F(x, y) & {\rm w.r.t.} \; (x, y) \in \B{R}^n \times \B{R}^m \end{array} @]@ that is convex with respect to @(@ y @)@. In addition, we are given a set of equations @(@ H(x, y) @)@ such that @[@ H[ x , Y(x) ] = 0 \;\; \Rightarrow \;\; F_y [ x , Y(x) ] = 0 @]@ (In fact, it is often the case that @(@ H(x, y) = F_y (x, y) @)@.) Furthermore, it is easy to calculate a Newton step for these equations; i.e., @[@ 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 @[@ 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 @[@ G^{(1)} (x) = \partial_x F [x, Y(x) ] @]@

x
The BenderQuad argument x has prototype



    const BAvector &x

(see BAvector below) and its size must be equal to n . It specifies the point at which we evaluating the reduced objective function and its derivatives.

y
The BenderQuad argument y has prototype



    const BAvector &y

and its size must be equal to m . It must be equal to @(@ Y(x) @)@; i.e., it must solve the problem in @(@ y @)@ for this given value of @(@ x @)@ @[@ \begin{array}{rcl} {\rm minimize} & F(x, y) & {\rm w.r.t.} \; y \in \B{R}^m \end{array} @]@

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

fun.f
The BenderQuad argument fun supports the syntax



    f = fun.f(x, y)

The fun.f argument x has prototype



    const ADvector &x

(see ADvector below) and its size must be equal to n . The fun.f argument y has prototype



    const ADvector &y

and its size must be equal to m . The fun.f result f has prototype



    ADvector f

and its size must be equal to one. The value of f is @[@ f = F(x, y) @]@.

fun.h
The BenderQuad argument fun supports the syntax



    h = fun.h(x, y)

The fun.h argument x has prototype



    const ADvector &x

and its size must be equal to n . The fun.h argument y has prototype



    const BAvector &y

and its size must be equal to m . The fun.h result h has prototype



    ADvector h

and its size must be equal to m . The value of h is @[@ h = H(x, y) @]@.

fun.dy
The BenderQuad argument fun supports the syntax



    dy = fun.dy(x, y, h)



x

The fun.dy argument x has prototype



    const BAvector &x

and its size must be equal to n . Its value will be exactly equal to the BenderQuad argument x and values depending on it can be stored as private objects in f and need not be recalculated.

The fun.dy argument y has prototype



    const BAvector &y

and its size must be equal to m . Its value will be exactly equal to the BenderQuad argument y and values depending on it can be stored as private objects in f and need not be recalculated.

The fun.dy argument h has prototype



    const ADvector &h

and its size must be equal to m .

dy

The fun.dy result dy has prototype



    ADvector dy

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

g
The argument g has prototype



    BAvector &g

and has size one. The input value of its element does not matter. On output, it contains the value of @(@ G (x) @)@; i.e., @[@ g[0] = G (x) @]@

gx
The argument gx has prototype



    BAvector &gx

and has size @(@ n @)@. The input values of its elements do not matter. On output, it contains the Jacobian of @(@ G (x) @)@; i.e., for @(@ j = 0 , \ldots , n-1 @)@, @[@ gx[ j ] = G^{(1)} (x)_j @]@

gxx
The argument gx has prototype



    BAvector &gxx

and has size @(@ n \times n @)@. The input values of its elements do not matter. On output, it contains the Hessian of @(@ G (x) @)@; i.e., for @(@ i = 0 , \ldots , n-1 @)@, and @(@ j = 0 , \ldots , n-1 @)@, @[@ gxx[ i * n + j ] = G^{(2)} (x)_{i,j} @]@

BAvector
The type BAvector must be a SimpleVector class. We use Base to refer to the type of the elements of BAvector ; i.e.,



    BAvector::value_type

ADvector
The type ADvector must be a SimpleVector class with elements of type AD<Base> ; i.e.,



    ADvector::value_type

must be the same type as



    AD< BAvector::value_type >

.

Example
The file bender_quad.cpp contains an example and test of this operation.

Input File: include/cppad/core/bender_quad.hpp