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 (
      n , m , x_i , x_l , x_u , g_l , g_u , & fg_info , & solution
)
export LD_LIBRARY_PATH = $LD_LIBRARY_PATH: ipopt_library_paths

Purpose¶

The class cppad_ipopt_nlp is used to solve nonlinear programming problems of the form

\[\begin{split}\begin{array}{rll} {\rm minimize} & f(x) \\ {\rm subject \; to} & g^l \leq g(x) \leq g^u \\ & x^l \leq x \leq x^u \end{array}\end{split}\]

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

\begin{eqnarray} fg_0 (x) & = & f(x) \\ fg_1 (x) & = & g_0 (x) \\ & \vdots & \\ fg_m (x) & = & g_{m-1} (x) \end{eqnarray}

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

\[[ J \otimes n ] (x)_j = x_{J(j)}\]

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

\[\begin{split}[ m \otimes I ] (y)_i = \left\{ \begin{array}{ll} y_k & {\rm if} \; i = I(k) \; {\rm for \; some} \; k \in \{ 0 , \cdots, |I|-1 \} \\ 0 & {\rm otherwise} \end{array} \right.\end{split}\]

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

\[fg(x) = \sum_{k=0}^{K-1} \; \sum_{\ell=0}^{L(k) - 1} [ (m+1) \otimes I_{k,\ell} ] \; \circ \; r_k \; \circ \; [ J_{k,\ell} \otimes n ] \; (x)\]

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& J
      )

The 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 `cppad_ipopt_nlp` object.
unknown	The status returned by the optimizer is not defined in the Ipopt documentation for `finalize_solution` .
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 `nan` or `inf` ) from the users function fg_info . `eval` or from the CppAD evaluations of its derivatives (see the Ipopt option `check_derivatives_for_naninf` ).
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 $f(x)$.