abs_min_quad.cpp¶

abs_min_quad: Example and Test¶

Purpose¶

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

\[f( x_0, x_1 ) = ( x_0^2 + x_1^2 ) / 2 + | x_0 - 5 | + | x_1 + 5 |\]

For this case, the abs_min_quad object should be equal to the function itself. In addition, the function is convex and abs_min_quad should find its global minimizer. The minimizer of this function is \(x_0 = 1\), \(x_1 = -1\).

Source¶

# include <cppad/cppad.hpp>
# include "abs_min_quad.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_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     = 2;     // size of x
   size_t m     = 1;     // size of y
   size_t s     = 2 ;    // number of data points and absolute values
   //
   // 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;
   sum += ad_x[0] * ad_x[0] / 2.0 + abs( ad_x[0] - 5 );
   sum += ad_x[1] * ad_x[1] / 2.0 + abs( ad_x[1] + 5 );
   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 the point x_hat = 0
   d_vector x_hat(n);
   for(size_t j = 0; j < n; j++)
      x_hat[j] = 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
   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_quad iterations
   maxitr[1] = 35; // maximum number of qp_interior iterations

   // set Hessian equal to identity matrix I
   d_vector hessian(n * n);
   for(size_t i = 0; i < n; i++)
   {  for(size_t j = 0; j < n; j++)
         hessian[i * n + j] = 0.0;
      hessian[i * n + i] = 1.0;
   }

   // minimize the approxiamtion for f (which is equal to f for this case)
   d_vector delta_x(n);
   ok &= CppAD::abs_min_quad(
      level, n, m, s,
      g_hat, g_jac, hessian, bound, epsilon, maxitr, delta_x
   );

   // check that the solution
   ok &= CppAD::NearEqual( delta_x[0], +1.0, eps99, eps99 );
   ok &= CppAD::NearEqual( delta_x[1], -1.0, eps99, eps99 );

   return ok;
}