\(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\R}[1]{ {\rm #1} }\)

mixed_optimize_fixed_2.py

View page source

The Ipopt Example Problem: Example and Test

Problem

This optimization problem is take from the ipopt interfaces documentation.

\begin{align*} \R{minimize} \; & x_0 x_3 ( x_0 + x_1 + x_2) + x_2 & \R{\; w.r.t \;} x \in \B{R}^4 \\ \R{subject \; to} \; & 25 \leq x_0 x_1 x_2 x_3 \\ & 40 = x_0^2 + x_1^2 + x_2^2 + x_3^2 \\ & 1 \leq x \leq 5 \end{align*}

The starting point for the optimization is \((1,5,5,1)\) and the optimal solution is

\[( \; 1.00000000, \; 4.74299963, \; 3.82114998, \; 1.37940829 \; )\]

def optimize_fixed_2() :
   import cppad_py
   import numpy
   ok         = True
   inf        = numpy.inf
   #
   # define f(x) = x_0 x_3 ( x_0 + x_1 + x_2) + x_2
   x_start = numpy.array([ 1, 5, 5, 1 ], dtype=float )
   ax      = cppad_py.independent(x_start)
   av_0    = ax[0] * ax[3] * ( ax[0] + ax[1] + ax[2] ) + ax[2]
   av      = numpy.array( [ av_0 ] )
   f       = cppad_py.d_fun(ax, av)
   #
   # define g_0 (x) = x_0 * x_1 * x_2 * x_3
   #        g_1 (x) = x_0^2 + x_1^2 + x_2^2 + x_3^2
   ax       = cppad_py.independent(x_start)
   av_0     = ax[0] * ax[1] * ax[2] * ax[3]
   av_1     = ax[0]*ax[0] + ax[1]*ax[1] + ax[2]*ax[2] + ax[3]*ax[3]
   av       = numpy.array( [av_0, av_1] )
   g        = cppad_py.d_fun(ax, av)
   #
   x_lower  = numpy.array( [1, 1, 1, 1], dtype=float )
   x_upper  = numpy.array( [5, 5, 5, 5], dtype=float )
   g_lower  = numpy.array( [  25, 40], dtype=float )
   g_upper  = numpy.array( [ inf, 40], dtype=float )
   #
   mixed_obj = cppad_py.mixed(
      fixed_init     = x_start,
      fix_likelihood = f,
      fix_constraint = g,
   )
   #
   options  = 'String  sb    yes\n'     # suppress optimizer banner
   options += 'Integer print_level 0\n' # suppress optimizer trace
   solution  = mixed_obj.optimize_fixed(
      fixed_ipopt_options   = options,
      fixed_lower           = x_lower,
      fixed_upper           = x_upper,
      fixed_in              = x_start,
      fix_constraint_lower  = g_lower,
      fix_constraint_upper  = g_upper,
   )
   x_opt   = solution.fixed_opt
   x_check =  numpy.array( [1.00000000, 4.74299963, 3.82114998, 1.37940829] )
   ok      = ok and numpy.all( numpy.abs( x_opt - x_check ) < 1e-7 )
   return ok