{
  "nbformat_minor": 0, 
  "nbformat": 4, 
  "cells": [
    {
      "execution_count": null, 
      "cell_type": "code", 
      "source": [
        "%matplotlib inline"
      ], 
      "outputs": [], 
      "metadata": {
        "collapsed": false
      }
    }, 
    {
      "source": [
        "\nConstraint optimization: visualizing the geometry\n==================================================\n\nA small figure explaining optimization with constraints\n\n"
      ], 
      "cell_type": "markdown", 
      "metadata": {}
    }, 
    {
      "execution_count": null, 
      "cell_type": "code", 
      "source": [
        "import numpy as np\nimport pylab as pl\nfrom scipy import optimize\n\nx, y = np.mgrid[-2.9:5.8:.05, -2.5:5:.05]\nx = x.T\ny = y.T\n\nfor i in (1, 2):\n    # Create 2 figure: only the second one will have the optimization\n    # path\n    pl.figure(i, figsize=(3, 2.5))\n    pl.clf()\n    pl.axes([0, 0, 1, 1])\n\n    contours = pl.contour(np.sqrt((x - 3)**2 + (y - 2)**2),\n                        extent=[-3, 6, -2.5, 5],\n                        cmap=pl.cm.gnuplot)\n    pl.clabel(contours,\n            inline=1,\n            fmt='%1.1f',\n            fontsize=14)\n    pl.plot([-1.5, -1.5,  1.5,  1.5, -1.5],\n            [-1.5,  1.5,  1.5, -1.5, -1.5], 'k', linewidth=2)\n    pl.fill_between([ -1.5,  1.5],\n                    [ -1.5, -1.5],\n                    [  1.5,  1.5],\n                    color='.8')\n    pl.axvline(0, color='k')\n    pl.axhline(0, color='k')\n\n    pl.text(-.9, 4.4, '$x_2$', size=20)\n    pl.text(5.6, -.6, '$x_1$', size=20)\n    pl.axis('equal')\n    pl.axis('off')\n\n# And now plot the optimization path\naccumulator = list()\n\ndef f(x):\n    # Store the list of function calls\n    accumulator.append(x)\n    return np.sqrt((x[0] - 3)**2 + (x[1] - 2)**2)\n\n\n# We don't use the gradient, as with the gradient, L-BFGS is too fast,\n# and finds the optimum without showing us a pretty path\ndef f_prime(x):\n    r = np.sqrt((x[0] - 3)**2 + (x[0] - 2)**2)\n    return np.array(((x[0] - 3)/r, (x[0] - 2)/r))\n\noptimize.fmin_l_bfgs_b(f, np.array([0, 0]), approx_grad=1,\n                       bounds=((-1.5, 1.5), (-1.5, 1.5)))\n\naccumulated = np.array(accumulator)\npl.plot(accumulated[:, 0], accumulated[:, 1])\n\npl.show()"
      ], 
      "outputs": [], 
      "metadata": {
        "collapsed": false
      }
    }
  ], 
  "metadata": {
    "kernelspec": {
      "display_name": "Python 2", 
      "name": "python2", 
      "language": "python"
    }, 
    "language_info": {
      "mimetype": "text/x-python", 
      "nbconvert_exporter": "python", 
      "name": "python", 
      "file_extension": ".py", 
      "version": "2.7.12", 
      "pygments_lexer": "ipython2", 
      "codemirror_mode": {
        "version": 2, 
        "name": "ipython"
      }
    }
  }
}