6.1. Assignment. Optimization - Minimization.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import optimize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return np.sin(x / 5.) * np.exp(x / 10.) + 5. * np.exp(-x / 2.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2gEVEJD2ZDss81cwWAx2Al81sdOpHvwVaAAPNbLqZTTOzBhnGKiIiGQi+eJqI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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f242d50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_plt = np.arange(1, 30, 0.1)\n",
    "plt.plot(x_plt, f(x_plt))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Default min, different starting points result in different local minima"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[array([ 4.13628821]), array([ 4.13628818]), array([ 25.88018554])]\n"
     ]
    }
   ],
   "source": [
    "print([\n",
    "    optimize.minimize(f, 0., bounds = [(1, 30)]).x,\n",
    "    optimize.minimize(f, (10,), bounds = ((1, 30),)).x,\n",
    "    optimize.minimize(f, (25,), bounds = ((1, 30),)).x\n",
    "])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "BFGS ends up in local minima"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1.75, -11.9)\n"
     ]
    }
   ],
   "source": [
    "print(\n",
    "    round(optimize.minimize(f, 2., method = 'BFGS').fun, 2),\n",
    "    round(optimize.minimize(f, 30., method = 'BFGS').fun, 2)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Differential evolution does not"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     fun: array([-11.89889467])\n",
      "     jac: array([ 0.])\n",
      " message: 'Optimization terminated successfully.'\n",
      "    nfev: 98\n",
      "     nit: 5\n",
      " success: True\n",
      "       x: array([ 25.88019281])\n"
     ]
    }
   ],
   "source": [
    "print(\n",
    "    optimize.differential_evolution(f, [(1, 30)])\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But takes more work, event with few function arguments"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5, 18], [5, 98]]\n"
     ]
    }
   ],
   "source": [
    "bfgs = optimize.minimize(f, 30., method = 'BFGS')\n",
    "diffev = optimize.differential_evolution(f, [(1, 30)])\n",
    "print([\n",
    "    [bfgs.nit, bfgs.nfev],\n",
    "    [diffev.nit, diffev.nfev]\n",
    "])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With natural numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def h(x):\n",
    "    return np.int_(np.sin(x / 5.) * np.exp(x / 10.) + 5. * np.exp(-x / 2.))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10eeeecd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_plt = np.arange(1, 30, 0.1)\n",
    "plt.plot(x_plt, h(x_plt))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      fun: -5\n",
       " hess_inv: array([[1]])\n",
       "      jac: array([ 0.])\n",
       "  message: 'Optimization terminated successfully.'\n",
       "     nfev: 3\n",
       "      nit: 0\n",
       "     njev: 1\n",
       "   status: 0\n",
       "  success: True\n",
       "        x: array([ 30.])"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.minimize(h, 30., method = 'BFGS')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "     fun: -11.0\n",
       " message: 'Optimization terminated successfully.'\n",
       "    nfev: 77\n",
       "     nit: 4\n",
       " success: True\n",
       "       x: array([ 25.55158893])"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.differential_evolution(h, [(1, 30)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}