\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

min_nso_quad.cpp

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abs_normal min_nso_quad: Example and Test

Purpose

We minimize the function \(f : \B{R}^3 \rightarrow \B{R}\) defined by

\begin{eqnarray} f( x_0, x_1, x_2 ) & = & x_0^2 + 2 (x_0 + x_1)^2 + | x_2 | \end{eqnarray}

Discussion

This routine uses abs_min_quad which uses qp_box , a quadratic programming algorithm. It is mean to be compared with min_nso_linear.cpp which uses a linear programing algorithm for the same problem. To see this comparison, set level = 1 is both examples.

Source

# include <cppad/cppad.hpp>
# include "min_nso_quad.hpp"

bool min_nso_quad(void)
{  bool ok = true;
   //
   using CppAD::AD;
   using CppAD::ADFun;
   //
   typedef CPPAD_TESTVECTOR(size_t)       s_vector;
   typedef CPPAD_TESTVECTOR(double)       d_vector;
   typedef CPPAD_TESTVECTOR( AD<double> ) ad_vector;
   //
   size_t level = 0;    // level of tracing
   size_t n     = 3;    // size of x
   size_t m     = 1;    // size of y
   size_t s     = 1;    // number of data points and absolute values
   //
   // start recording the function f(x)
   ad_vector ax(n), ay(m);
   for(size_t j = 0; j < n; j++)
      ax[j] = double(j + 1);
   Independent( ax );
   //
   ay[0]  =  ax[0] * ax[0];
   ay[0] += 2.0 * (ax[0] + ax[1]) * (ax[0] + ax[1]);
   ay[0] += fabs( ax[2] );
   ADFun<double> f(ax, ay);
   //
   // create its abs_normal representation in g, a
   ADFun<double> g, a;
   f.abs_normal_fun(g, a);

   // check dimension of domain and range space for g
   ok &= g.Domain() == n + s;
   ok &= g.Range()  == m + s;

   // check dimension of domain and range space for a
   ok &= a.Domain() == n;
   ok &= a.Range()  == s;

   // epsilon
   d_vector epsilon(2);
   double eps = 1e-3;
   epsilon[0] = eps;
   epsilon[1] = eps;

   // maxitr
   s_vector maxitr(3);
   maxitr[0] = 100;
   maxitr[1] = 20;
   maxitr[2] = 20;

   // b_in
   double b_in = 1.0;

   // call min_nso_quad
   d_vector x_in(n), x_out(n);
   for(size_t j = 0; j < n; j++)
      x_in[j]  = double(j + 1);

   //
   ok &= CppAD::min_nso_quad(
      level, f, g, a, epsilon, maxitr, b_in, x_in, x_out
   );
   //
   for(size_t j = 0; j < n; j++)
      ok &= std::fabs( x_out[j] ) < eps;

   return ok;
}