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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
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   "source": [
    "from __future__ import division\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
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   "source": [
    "# NOTE: make sure that for the following, the\n",
    "# input x is either a 1-dimensional numpy.array\n",
    "# or a list \n",
    "\n",
    "# sphere function\n",
    "def sphere(x, alpha=0.5):\n",
    "    return alpha * np.sum(np.array(x)**2)\n",
    "\n",
    "# derivative of sphere function\n",
    "def sphere_der(x, alpha=0.5):\n",
    "    return 2*alpha*np.array(x)\n",
    "\n",
    "# ellipsoid with condition number 10^alpha\n",
    "# make sure to have python 3 or 'division' imported\n",
    "def elli(x, alpha=1):\n",
    "    s = np.array(x)\n",
    "    n = len(s)   # problem dimension\n",
    "    sum = 0\n",
    "    for i in range(n):\n",
    "        sum = sum + s[i]*s[i]*(10**alpha)**(i/(n-1))\n",
    "    sum = sum / 2\n",
    "    return sum\n",
    "\n",
    "# derivative of ellipsoid function\n",
    "def elli_der(x, alpha=1):\n",
    "    s = np.array(x)\n",
    "    n = len(s)   # problem dimension\n",
    "    der = np.zeros(n)    \n",
    "    for i in range(n):\n",
    "        der[i] = s[i] * (10**alpha)**(i/(n-1))\n",
    "    return der\n",
    "\n",
    "# Hessian matrix of ellipsoid function at solution x\n",
    "def elli_hess(x, alpha=1):\n",
    "    n = len(x)\n",
    "    H = np.eye(n)\n",
    "    for i in range(n):\n",
    "        H[i,i] = (10**alpha)**(i/(n-1))\n",
    "    return H\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# plotting some level sets of sphere function:\n",
    "delta = 0.025\n",
    "x = np.arange(-2.0, 2.0, delta)\n",
    "y = np.arange(-2.0, 2.0, delta)\n",
    "X, Y = np.meshgrid(x, y)\n",
    "f = np.zeros((X.shape[0], X.shape[1]))\n",
    "for i in range(X.shape[0]):\n",
    "    for j in range(X.shape[1]):\n",
    "        f[i][j] = sphere([X[i][j], Y[i][j]], alpha=2)\n",
    "        \n",
    "plt.figure(figsize=(6,6))\n",
    "plt.contour(X, Y, f, levels=0.2*np.array(range(1, 20)))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# plotting some level sets of ellipsoid function for varying alpha:\n",
    "delta = 0.025\n",
    "x = np.arange(-2.0, 2.0, delta)\n",
    "y = np.arange(-2.0, 2.0, delta)\n",
    "X, Y = np.meshgrid(x, y)\n",
    "felli_m1 = np.zeros((X.shape[0], X.shape[1])) # for plot with alpha=-1\n",
    "felli_0 = np.zeros((X.shape[0], X.shape[1])) # for plot with alpha=0\n",
    "felli_1 = np.zeros((X.shape[0], X.shape[1])) # for plot with alpha=1\n",
    "felli_3 = np.zeros((X.shape[0], X.shape[1])) # for plot with alpha=3\n",
    "for i in range(X.shape[0]):\n",
    "    for j in range(X.shape[1]):\n",
    "        felli_m1[i][j] = elli([X[i][j], Y[i][j]], alpha=-1)\n",
    "        felli_0[i][j] = elli([X[i][j], Y[i][j]], alpha=0)        \n",
    "        felli_1[i][j] = elli([X[i][j], Y[i][j]], alpha=1)\n",
    "        felli_3[i][j] = elli([X[i][j], Y[i][j]], alpha=3)\n",
    "        \n",
    "        \n",
    "plt.figure(figsize=(16,4))\n",
    "plt.subplot(141) # first plot of a 1x4 grid\n",
    "plt.contour(X, Y, felli_m1, levels=0.2*np.array(range(1, 20)))\n",
    "plt.title(\"Ellipsoid with alpha=-1\")\n",
    "plt.xlabel(\"$x_1$\")\n",
    "plt.ylabel(\"$x_2$\")\n",
    "plt.axis('equal')\n",
    "plt.subplot(142) # second plot of a 1x4 grid\n",
    "plt.contour(X, Y, felli_0, levels=0.2*np.array(range(1, 20)))\n",
    "plt.title(\"Ellipsoid with alpha=0\")\n",
    "plt.xlabel(\"$x_1$\")\n",
    "plt.ylabel(\"$x_2$\")\n",
    "plt.axis('equal')\n",
    "plt.subplot(143) # third plot of a 1x4 grid\n",
    "plt.contour(X, Y, felli_1, levels=0.2*np.array(range(1, 20)))\n",
    "plt.title(\"Ellipsoid with alpha=1\")\n",
    "plt.xlabel(\"$x_1$\")\n",
    "plt.ylabel(\"$x_2$\")\n",
    "plt.axis('equal')\n",
    "plt.subplot(144) # forth plot of a 1x4 grid\n",
    "plt.contour(X, Y, felli_3, levels=0.2*np.array(range(1, 20)))\n",
    "plt.title(\"Ellipsoid with alpha=3\")\n",
    "plt.xlabel(\"$x_1$\")\n",
    "plt.ylabel(\"$x_2$\")\n",
    "plt.axis('equal')\n",
    "plt.show()"
   ]
  },
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