\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
cppad_ipopt_nlp
Nonlinear Programming Using the CppAD Interface to Ipopt
Deprecated 2012-11-28
This interface to Ipopt is deprecated, use ipopt_solve instead.
Syntax
include
"cppad_ipopt_nlp.hpp"
cppad_ipopt_solution
solution ;cppad_ipopt_nlp
cppad_nlp (export LD_LIBRARY_PATH
= $LD_LIBRARY_PATH:
ipopt_library_pathsPurpose
The class cppad_ipopt_nlp
is used to solve nonlinear programming
problems of the form
This is done using Ipopt optimizer and CppAD Algorithmic Differentiation package.
cppad_ipopt namespace
All of the declarations for these routines
are in the cppad_ipopt
namespace
(not the CppAD
namespace).
For example; SizeVector below
actually denotes the type cppad_ipopt::SizeVector
.
ipopt_library_paths
If you are linking to a shared version of the Ipopt library,
you may have to add a path to the LD_LIBRARY_PATH
.
You can determine the directory you need to add using the command
pkg-config ipopt --libs
The output will have the following form
-L
dir-lipopt
You may need to add the directory %dir% to LD_LIBRARY_PATH%
; e.g.,
export LD_LIBRARY_PATH
=” dir :$LD_LIBRARY_PATH
“
fg(x)
The function \(fg : \B{R}^n \rightarrow \B{R}^{m+1}\) is defined by
Index Vector
We define an index vector as a vector of non-negative integers for which none of the values are equal; i.e., it is both a vector and a set. If \(I\) is an index vector \(|I|\) is used to denote the number of elements in \(I\) and \(\| I \|\) is used to denote the value of the maximum element in \(I\).
Projection
Given an index vector \(J\) and a positive integer \(n\) where \(n > \| J \|\), we use \(J \otimes n\) for the mapping \(( J \otimes n ) : \B{R}^n \rightarrow \B{R}^{|J|}\) defined by
for \(j = 0 , \ldots |J| - 1\).
Injection
Given an index vector \(I\) and a positive integer \(m\) where \(m > \| I \|\), we use \(m \otimes I\) for the mapping \(( m \otimes I ): \B{R}^{|I|} \rightarrow \B{R}^m\) defined by
Representation
In many applications, each of the component functions of \(fg(x)\) only depend on a few of the components of \(x\). In this case, expressing \(fg(x)\) in terms of simpler functions with fewer arguments can greatly reduce the amount of work required to compute its derivatives.
We use the functions \(r_k : \B{R}^{q(k)} \rightarrow \B{R}^{p(k)}\) for \(k = 0 , \ldots , K\) to express our representation of \(fg(x)\) in terms of simpler functions as follows
where \(\circ\) represents function composition, for \(k = 0 , \ldots , K - 1\), and \(\ell = 0 , \ldots , L(k)\), \(I_{k,\ell}\) and \(J_{k,\ell}\) are index vectors with \(| J_{k,\ell} | = q(k)\), \(\| J_{k,\ell} \| < n\), \(| I_{k,\ell} | = p(k)\), and \(\| I_{k,\ell} \| \leq m\).
Simple Representation
In the simple representation, \(r_0 (x) = fg(x)\), \(K = 1\), \(q(0) = n\), \(p(0) = m+1\), \(L(0) = 1\), \(I_{0,0} = (0 , \ldots , m)\), and \(J_{0,0} = (0 , \ldots , n-1)\).
SizeVector
The type SizeVector
is defined by the
cppad_ipopt_nlp.hpp
include file to be a
SimpleVector class with elements of type
size_t
.
NumberVector
The type NumberVector
is defined by the
cppad_ipopt_nlp.hpp
include file to be a
SimpleVector class with elements of type
Ipopt::Number
.
ADNumber
The type ADNumber
is defined by the
cppad_ipopt_nlp.hpp
include file to be a
an AD type that can be used to compute derivatives.
ADVector
The type ADVector
is defined by the
cppad_ipopt_nlp.hpp
include file to be a
SimpleVector class with elements of type
ADNumber
.
n
The argument n has prototype
size_t
n
It specifies the dimension of the argument space; i.e., \(x \in \B{R}^n\).
m
The argument m has prototype
size_t
m
It specifies the dimension of the range space for \(g\); i.e., \(g : \B{R}^n \rightarrow \B{R}^m\).
x_i
The argument x_i has prototype
const NumberVector&
x_i
and its size is equal to \(n\). It specifies the initial point where Ipopt starts the optimization process.
x_l
The argument x_l has prototype
const NumberVector&
x_l
and its size is equal to \(n\). It specifies the lower limits for the argument in the optimization problem; i.e., \(x^l\).
x_u
The argument x_u has prototype
const NumberVector&
x_u
and its size is equal to \(n\). It specifies the upper limits for the argument in the optimization problem; i.e., \(x^u\).
g_l
The argument g_l has prototype
const NumberVector&
g_l
and its size is equal to \(m\). It specifies the lower limits for the constraints in the optimization problem; i.e., \(g^l\).
g_u
The argument g_u has prototype
const NumberVector&
g_u
and its size is equal to \(n\). It specifies the upper limits for the constraints in the optimization problem; i.e., \(g^u\).
fg_info
The argument fg_info has prototype
FG_info fg_info
where the class FG_info is derived from the
base class cppad_ipopt_fg_info
.
Certain virtual member functions of fg_info are used to
compute the value of \(fg(x)\).
The specifications for these member functions are given below:
fg_info.number_functions
This member function has prototype
virtual size_t cppad_ipopt_fg_info::number_functions
(void
)
If K has type size_t
, the syntax
K = fg_info .
number_functions
()
sets K to the number of functions used in the representation of \(fg(x)\); i.e., \(K\) in the Representation above.
The cppad_ipopt_fg_info
implementation of this function
corresponds to the simple representation mentioned above; i.e.
K = 1 .
fg_info.eval_r
This member function has the prototype
virtual ADVector cppad_ipopt_fg_info::eval_r
(size_t
k ,const ADVector&
u ) = 0;
Thus it is a pure virtual function and must be defined in the derived class FG_info .
This function computes the value of \(r_k (u)\)
used in the Representation
for \(fg(x)\).
If k in \(\{0 , \ldots , K-1 \}\) has type size_t
,
u is an ADVector
of size q ( k )
and r is an ADVector
of size p ( k )
the syntax
r = fg_info .
eval_r
( k , u )
set r to the vector \(r_k (u)\).
fg_info.retape
This member function has the prototype
virtual bool cppad_ipopt_fg_info::retape
(size_t
k )
If k in \(\{0 , \ldots , K-1 \}\) has type size_t
,
and retape has type bool
,
the syntax
retape = fg_info .
retape
( k )
sets retape to true or false.
If retape is true,
cppad_ipopt_nlp
will retape the operation sequence
corresponding to \(r_k (u)\) for
every value of u .
An cppad_ipopt_nlp
object
should use much less memory and run faster if retape is false.
You can test both the true and false cases to make sure
the operation sequence does not depend on u .
The cppad_ipopt_fg_info
implementation of this function
sets retape to true
(while slower it is also safer to always retape).
fg_info.domain_size
This member function has prototype
virtual size_t cppad_ipopt_fg_info::domain_size
(size_t
k )
If k in \(\{0 , \ldots , K-1 \}\) has type size_t
,
and q has type size_t
, the syntax
q = fg_info .
domain_size
( k )
sets q to the dimension of the domain space for \(r_k (u)\); i.e., \(q(k)\) in the Representation above.
The cppad_ipopt_h_base
implementation of this function
corresponds to the simple representation mentioned above; i.e.,
\(q = n\).
fg_info.range_size
This member function has prototype
virtual size_t cppad_ipopt_fg_info::range_size
(size_t
k )
If k in \(\{0 , \ldots , K-1 \}\) has type size_t
,
and p has type size_t
, the syntax
p = fg_info .
range_size
( k )
sets p to the dimension of the range space for \(r_k (u)\); i.e., \(p(k)\) in the Representation above.
The cppad_ipopt_h_base
implementation of this function
corresponds to the simple representation mentioned above; i.e.,
\(p = m+1\).
fg_info.number_terms
This member function has prototype
virtual size_t cppad_ipopt_fg_info::number_terms
(size_t
k )
If k in \(\{0 , \ldots , K-1 \}\) has type size_t
,
and L has type size_t
, the syntax
L = fg_info .
number_terms
( k )
sets L to the number of terms in representation for this value of k ; i.e., \(L(k)\) in the Representation above.
The cppad_ipopt_h_base
implementation of this function
corresponds to the simple representation mentioned above; i.e.,
\(L = 1\).
fg_info.index
This member function has prototype
virtual void cppad_ipopt_fg_info::index
(size_t
k , size_t
ell , SizeVector&
I , SizeVector&
JThe argument
k
has type size_t
and is a value between zero and \(K-1\) inclusive.
The argument
ell
has type size_t
and is a value between zero and \(L(k)-1\) inclusive.
The argument
I
is a SimpleVector with elements
of type size_t
and size greater than or equal to \(p(k)\).
The input value of the elements of I does not matter.
The output value of
the first \(p(k)\) elements of I
must be the corresponding elements of \(I_{k,ell}\)
in the Representation above.
The argument
J
is a SimpleVector with elements
of type size_t
and size greater than or equal to \(q(k)\).
The input value of the elements of J does not matter.
The output value of
the first \(q(k)\) elements of J
must be the corresponding elements of \(J_{k,ell}\)
in the Representation above.
The cppad_ipopt_h_base
implementation of this function
corresponds to the simple representation mentioned above; i.e.,
for \(i = 0 , \ldots , m\),
I [ i ] = i ,
and for \(j = 0 , \ldots , n-1\),
J [ j ] = j .
solution
After the optimization process is completed, solution contains the following information:
status
The status field of solution has prototype
cppad_ipopt_solution::solution_status
solution .status
It is the final Ipopt status for the optimizer. Here is a list of the possible values for the status:
status |
Meaning |
not_defined |
The optimizer did not return a final status to this |
unknown |
The status returned by the optimizer is not defined in the Ipopt
documentation for |
success |
Algorithm terminated successfully at a point satisfying the convergence tolerances (see Ipopt options). |
maxiter_exceeded |
The maximum number of iterations was exceeded (see Ipopt options). |
stop_at_tiny_step |
Algorithm terminated because progress was very slow. |
stop_at_acceptable_point |
Algorithm stopped at a point that was converged, not to the ‘desired’ tolerances, but to ‘acceptable’ tolerances (see Ipopt options). |
local_infeasibility |
Algorithm converged to a non-feasible point (problem may have no solution). |
user_requested_stop |
This return value should not happen. |
diverging_iterates |
It the iterates are diverging. |
restoration_failure |
Restoration phase failed, algorithm doesn’t know how to proceed. |
error_in_step_computation |
An unrecoverable error occurred while Ipopt tried to compute the search direction. |
invalid_number_detected |
Algorithm received an invalid number (such as |
internal_error |
An unknown Ipopt internal error occurred. Contact the Ipopt authors through the mailing list. |
x
The x
field of solution has prototype
NumberVector
solution .x
and its size is equal to \(n\). It is the final \(x\) value for the optimizer.
z_l
The z_l
field of solution has prototype
NumberVector
solution .z_l
and its size is equal to \(n\). It is the final Lagrange multipliers for the lower bounds on \(x\).
z_u
The z_u
field of solution has prototype
NumberVector
solution .z_u
and its size is equal to \(n\). It is the final Lagrange multipliers for the upper bounds on \(x\).
g
The g
field of solution has prototype
NumberVector
solution .g
and its size is equal to \(m\). It is the final value for the constraint function \(g(x)\).
lambda
The lambda
field of solution has prototype
NumberVector
solution .lambda
and its size is equal to \(m\). It is the final value for the Lagrange multipliers corresponding to the constraint function.
obj_value
The obj_value
field of solution has prototype
Number
solution .obj_value
It is the final value of the objective function \(f(x)\).