abs_min_linear.cpp¶

abs_min_linear: Example and Test¶

Purpose¶

The function \(f : \B{R}^3 \rightarrow \B{R}\) defined by

\begin{eqnarray} f( x_0, x_1 ) & = & | d_0 - x_0 | + | d_1 - x_0 | + | d_2 - x_0 | \\ & + & | d_3 - x_1 | + | d_4 - x_1 | + | d_5 - x_1 | \\ \end{eqnarray}

is affine, except for its absolute value terms. For this case, the abs_normal approximation should be equal to the function itself. In addition, the function is convex and abs_min_linear should find its global minimizer. The minimizer of this function is \(x_0 = \R{median}( d_0, d_1, d_2 )\) and \(x_1 = \R{median}( d_3, d_4, d_5 )\)

Source¶

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

namespace {
   CPPAD_TESTVECTOR(double) join(
      const CPPAD_TESTVECTOR(double)& x ,
      const CPPAD_TESTVECTOR(double)& u )
   {  size_t n = x.size();
      size_t s = u.size();
      CPPAD_TESTVECTOR(double) xu(n + s);
      for(size_t j = 0; j < n; j++)
         xu[j] = x[j];
      for(size_t j = 0; j < s; j++)
         xu[n + j] = u[j];
      return xu;
   }
}
bool abs_min_linear(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 dpx   = 3;          // number of data points per x variable
   size_t level = 0;          // level of tracing
   size_t n     = 2;          // size of x
   size_t m     = 1;          // size of y
   size_t s     = dpx * n;    // number of data points and absolute values
   // data points
   d_vector  data(s);
   for(size_t i = 0; i < s; i++)
      data[i] = double(s - i) + 5.0 - double(i % 2) / 2.0;
   //
   // record the function f(x)
   ad_vector ad_x(n), ad_y(m);
   for(size_t j = 0; j < n; j++)
      ad_x[j] = double(j + 1);
   Independent( ad_x );
   AD<double> sum = 0.0;
   for(size_t j = 0; j < n; j++)
      for(size_t k = 0; k < dpx; k++)
         sum += abs( data[j * dpx + k] - ad_x[j] );
   ad_y[0] = sum;
   ADFun<double> f(ad_x, ad_y);

   // 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;

   // --------------------------------------------------------------------
   // Choose a point x_hat
   d_vector x_hat(n);
   for(size_t j = 0; j < n; j++)
      x_hat[j] = double(0.0);

   // value of a_hat = a(x_hat)
   d_vector a_hat = a.Forward(0, x_hat);

   // (x_hat, a_hat)
   d_vector xu_hat = join(x_hat, a_hat);

   // value of g[ x_hat, a_hat ]
   d_vector g_hat = g.Forward(0, xu_hat);

   // Jacobian of g[ x_hat, a_hat ]
   d_vector g_jac = g.Jacobian(xu_hat);

   // trust region bound (make large enough to include solutuion)
   d_vector bound(n);
   for(size_t j = 0; j < n; j++)
      bound[j] = 10.0;

   // convergence criteria
   d_vector epsilon(2);
   double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
   epsilon[0]   = eps99;
   epsilon[1]   = eps99;

   // maximum number of iterations
   s_vector maxitr(2);
   maxitr[0] = 10; // maximum number of abs_min_linear iterations
   maxitr[1] = 35; // maximum number of qp_interior iterations

   // minimize the approxiamtion for f, which is equal to f because
   // f is affine, except for absolute value terms
   d_vector delta_x(n);
   ok &= CppAD::abs_min_linear(
      level, n, m, s, g_hat, g_jac, bound, epsilon, maxitr, delta_x
   );

   // number of data points per variable is odd
   ok &= dpx % 2 == 1;

   // check that the solution is the median of the corresponding data`
   for(size_t j = 0; j < n; j++)
   {  // data[j * dpx + 0] , ... , data[j * dpx + dpx - 1] corresponds to x[j]
      // the median of this data has index j * dpx + dpx / 2
      size_t j_median = j * dpx + (dpx / 2);
      //
      ok &= CppAD::NearEqual( delta_x[j], data[j_median], eps99, eps99 );
   }

   return ok;
}