{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "338e56f7",
   "metadata": {},
   "source": [
    "<!--NOTEBOOK_HEADER-->\n",
    "*This notebook contains material from [CBE60499](https://ndcbe.github.io/CBE60499);\n",
    "content is available [on Github](git@github.com:ndcbe/CBE60499.git).*\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c780a3c",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [3.2 Mathematics Primer](https://ndcbe.github.io/CBE60499/03.02-Math-Primer-2.html) | [Contents](toc.html) | [Tag Index](tag_index.html) | [3.4 Newton-type Methods for Unconstrained Optimization](https://ndcbe.github.io/CBE60499/03.04-Netwon-Methods.html) ><p><a href=\"https://colab.research.google.com/github/ndcbe/CBE60499/blob/master/docs/03.03-Optimality.ipynb\"> <img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a><p><a href=\"https://ndcbe.github.io/CBE60499/03.03-Optimality.ipynb\"> <img align=\"left\" src=\"https://img.shields.io/badge/Github-Download-blue.svg\" alt=\"Download\" title=\"Download Notebook\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 1,
     "link": "[3.3 Unconstrained Optimality Conditions](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3-Unconstrained-Optimality-Conditions)",
     "section": "3.3 Unconstrained Optimality Conditions"
    }
   },
   "source": [
    "# 3.3 Unconstrained Optimality Conditions\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "nbpages": {
     "level": 1,
     "link": "[3.3 Unconstrained Optimality Conditions](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3-Unconstrained-Optimality-Conditions)",
     "section": "3.3 Unconstrained Optimality Conditions"
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from scipy import linalg\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib import cm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.3.1 Local and Global Solutions](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.1-Local-and-Global-Solutions)",
     "section": "3.3.1 Local and Global Solutions"
    }
   },
   "source": [
    "## 3.3.1 Local and Global Solutions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.3.2 Necessary Conditions for Optimality](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.2-Necessary-Conditions-for-Optimality)",
     "section": "3.3.2 Necessary Conditions for Optimality"
    }
   },
   "source": [
    "## 3.3.2 Necessary Conditions for Optimality"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.3.3 Sufficient Conditions for Optimality](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.3-Sufficient-Conditions-for-Optimality)",
     "section": "3.3.3 Sufficient Conditions for Optimality"
    }
   },
   "source": [
    "## 3.3.3 Sufficient Conditions for Optimality"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.3.4 Example 2.19 in Biegler (2010)](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4-Example-2.19-in-Biegler-(2010))",
     "section": "3.3.4 Example 2.19 in Biegler (2010)"
    }
   },
   "source": [
    "## 3.3.4 Example 2.19 in Biegler (2010)\n",
    "\n",
    "**Main Idea**: Use the necessary and sufficient conditions to classify a candidate solution.\n",
    "\n",
    "![Book](figures/ex2-19.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.1 The Test Function](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.1-The-Test-Function)",
     "section": "3.3.4.1 The Test Function"
    }
   },
   "source": [
    "### 3.3.4.1 The Test Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.1 The Test Function](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.1-The-Test-Function)",
     "section": "3.3.4.1 The Test Function"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(x*) =  -5.089256907976166 \n",
      "\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "## Define Python function\n",
    "def my_f(x,verbose=False):\n",
    "    ''' Evaluate function given above at point x\n",
    "\n",
    "    Inputs:\n",
    "        x - vector with 2 elements\n",
    "        \n",
    "    Outputs:\n",
    "        f - function value (scalar)\n",
    "    '''\n",
    "    # Constants\n",
    "    a = np.array([0.3, 0.6, 0.2])\n",
    "    b = np.array([5, 26, 3])\n",
    "    c = np.array([40, 1, 10])\n",
    "    \n",
    "    # Intermediates. Recall Python indicies start at 0\n",
    "    u = x[0] - 0.8\n",
    "    s = np.sqrt(1-u)\n",
    "    s2 = np.sqrt(1+u)\n",
    "    v = x[1] -(a[0] + a[1]*u**2*s-a[2]*u)\n",
    "    alpha = -b[0] + b[1]*u**2*s2+ b[2]*u # September 5, 2018: changed 's' to 's2'\n",
    "    beta = c[0]*v**2*(1-c[1]*v)/(1+c[2]*u**2)\n",
    "    f = alpha*np.exp(-beta)\n",
    "    \n",
    "    if verbose:\n",
    "        print(\"##### my_f at x = \",x, \"#####\")\n",
    "        print(\"u = \",u)\n",
    "        print(\"sqrt(1-u) = \",s)\n",
    "        print(\"sqrt(1+u) = \",s2)\n",
    "        print(\"v = \",v)\n",
    "        print(\"alpha = \",alpha)\n",
    "        print(\"beta = \",beta)\n",
    "        print(\"f(x) = \",f)\n",
    "        print(\"##### Done. #####\\n\")\n",
    "    \n",
    "    return f\n",
    "\n",
    "## Define candidate point and check against value reported in book\n",
    "xtest = np.array([0.7395, 0.3144])\n",
    "ftest = my_f(xtest)\n",
    "print(\"f(x*) = \",my_f(xtest),\"\\n\")\n",
    "\n",
    "## Make 3D plot to visualize\n",
    "x1 = np.arange(0.0,1.1,0.05)\n",
    "x2 = np.arange(0.0,1.0,0.05)\n",
    "\n",
    "# Create a matrix of all points to sample\n",
    "X1, X2 = np.meshgrid(x1, x2)\n",
    "n1 = len(x1)\n",
    "n2 = len(x2)\n",
    "\n",
    "# Notice the order. This was wrong in quadratic.ipynb and has been corrected in quadratic_update.ipynb\n",
    "F = np.zeros([n2, n1])\n",
    "\n",
    "xtemp = np.zeros(2)\n",
    "\n",
    "# Evaluate f(x) over grid\n",
    "for i in range(0,n1):\n",
    "    xtemp[0] = x1[i]\n",
    "    for j in range(0,n2):\n",
    "        xtemp[1] = x2[j]\n",
    "        F[j,i] = my_f(xtemp)\n",
    "\n",
    "# Create 3D figure\n",
    "fig = plt.figure()\n",
    "ax = fig.gca(projection='3d')\n",
    "\n",
    "# Plot f(x)\n",
    "surf = ax.plot_surface(X1, X2, F, linewidth=0,cmap=cm.coolwarm,antialiased=True)\n",
    "\n",
    "# Add candidate point\n",
    "ax.scatter(xtest[0],xtest[1],ftest,s=50,color=\"green\",depthshade=True)\n",
    "\n",
    "# Draw vertical line through stationary point to help visualization\n",
    "# Maximum value in array\n",
    "fmax = np.amax(F)\n",
    "fmin = np.amin(F)\n",
    "ax.plot([xtest[0], xtest[0]], [xtest[1], xtest[1]], [fmin,fmax],color=\"green\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.1 The Test Function](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.1-The-Test-Function)",
     "section": "3.3.4.1 The Test Function"
    }
   },
   "source": [
    "**Discussion**: Based on the 3D plot, is this function convex?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.2 Finite Difference Gradient and Hessian](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.2-Finite-Difference-Gradient-and-Hessian)",
     "section": "3.3.4.2 Finite Difference Gradient and Hessian"
    }
   },
   "source": [
    "### 3.3.4.2 Finite Difference Gradient and Hessian\n",
    "\n",
    "We need the calculate $\\nabla f(x)$ and $\\nabla^2 f(x)$ to analyze a given point. Below are functions that implement a central finite difference. Another option it to analytical calculate the derivatives or use automatic differentiation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.2 Finite Difference Gradient and Hessian](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.2-Finite-Difference-Gradient-and-Hessian)",
     "section": "3.3.4.2 Finite Difference Gradient and Hessian"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "***** my_grad at x =  [0.7395 0.3144] *****\n",
      "e[ 0 ] =  [1.e-06 0.e+00]\n",
      "f(x + e[ 0 ]) =  -5.089256904035737\n",
      "f(x - e[ 0 ]) =  -5.089256911839596\n",
      "e[ 1 ] =  [0.e+00 1.e-06]\n",
      "f(x + e[ 1 ]) =  -5.089256892701673\n",
      "f(x - e[ 1 ]) =  -5.089256922857939\n",
      "***** Done. ***** \n",
      "\n",
      "[0.00390193 0.01507813]\n"
     ]
    }
   ],
   "source": [
    "## Gradient\n",
    "def my_grad(x,verbose=True):\n",
    "    '''\n",
    "    Calculate gradient of function my_f using central difference formula\n",
    "    \n",
    "    Inputs:\n",
    "        x - point for which to evaluate gradient\n",
    "        \n",
    "    Outputs:\n",
    "        grad - gradient (vector)\n",
    "        \n",
    "    Assumptions:\n",
    "        1. my_f is defined\n",
    "        2. input x has the correct number of elements for my_f\n",
    "        3. my_f is continous and differentiable    \n",
    "    '''\n",
    "    \n",
    "    eps = 1E-6\n",
    "    n = len(x)\n",
    "    grad = np.zeros(n)\n",
    "\n",
    "    if(verbose):\n",
    "        print(\"***** my_grad at x = \",x,\"*****\")\n",
    "    \n",
    "    for i in range(0,n):\n",
    "        \n",
    "        # Create vector of zeros except eps in position i\n",
    "        e = np.zeros(n)\n",
    "        e[i] = eps\n",
    "        \n",
    "        # Finite difference formula\n",
    "        my_f_plus = my_f(x + e)\n",
    "        my_f_minus = my_f(x - e)\n",
    "        \n",
    "        # Diagnostics\n",
    "        if(verbose):\n",
    "            print(\"e[\",i,\"] = \",e)\n",
    "            print(\"f(x + e[\",i,\"]) = \",my_f_plus)\n",
    "            print(\"f(x - e[\",i,\"]) = \",my_f_minus)\n",
    "        \n",
    "        \n",
    "        grad[i] = (my_f_plus - my_f_minus)/(2*eps)\n",
    "    \n",
    "    if(verbose):\n",
    "        print(\"***** Done. ***** \\n\")\n",
    "    \n",
    "    return grad\n",
    "\n",
    "grad = my_grad(xtest)\n",
    "print(grad)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.2 Finite Difference Gradient and Hessian](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.2-Finite-Difference-Gradient-and-Hessian)",
     "section": "3.3.4.2 Finite Difference Gradient and Hessian"
    }
   },
   "source": [
    "**Discussion**: According to the book, $\\nabla f(x_{test}) = 0$. Is the above answer reasonable?\n",
    "\n",
    "**Note**: Before fixing the mistake in `my_f`, the gradient was `[-0.28182046  0.01506107]`. The version we discussed in class had the mistake in `my_f`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.2 Finite Difference Gradient and Hessian](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.2-Finite-Difference-Gradient-and-Hessian)",
     "section": "3.3.4.2 Finite Difference Gradient and Hessian"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 76.99973992 108.3413359 ]\n",
      " [108.3413359  392.7191905 ]]\n"
     ]
    }
   ],
   "source": [
    "## Hessian\n",
    "def my_hes(x):\n",
    "    '''\n",
    "    Calculate gradient of function my_f using central difference formula and my_grad\n",
    "    \n",
    "    Inputs:\n",
    "        x - point for which to evaluate gradient\n",
    "        \n",
    "    Outputs:\n",
    "        H - Hessian (matrix)\n",
    "        \n",
    "    Assumptions:\n",
    "        1. my_f and my_grad is defined\n",
    "        2. input x has the correct number of elements for my_f\n",
    "        3. my_f is continous and twice differentiable\n",
    "        4. No mistakes in my_grad\n",
    "    '''\n",
    "    \n",
    "    eps = 1E-6\n",
    "    n = len(x)\n",
    "    H = np.zeros([n,n])\n",
    "    \n",
    "    for i in range(0,n):\n",
    "        # Create vector of zeros except eps in position i\n",
    "        e = np.zeros(n)\n",
    "        e[i] = eps\n",
    "        \n",
    "        # Evaluate gradient twice\n",
    "        grad_plus = my_grad(x + e,verbose=False)\n",
    "        grad_minus = my_grad(x - e,verbose=False)\n",
    "        \n",
    "        # Notice we are building the Hessian by column (or row)\n",
    "        H[:,i] = (grad_plus - grad_minus)/(2*eps)\n",
    "    \n",
    "    '''\n",
    "    Note: This is not very efficient. You can probably this of several \n",
    "    performance improvements. We will learn more practical ways to approximate \n",
    "    the Hessian for optimization algorithms in the next few lectures.\n",
    "    '''\n",
    "    \n",
    "    return H\n",
    "\n",
    "H = my_hes(xtest)\n",
    "print(H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.2 Finite Difference Gradient and Hessian](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.2-Finite-Difference-Gradient-and-Hessian)",
     "section": "3.3.4.2 Finite Difference Gradient and Hessian"
    }
   },
   "source": [
    "**Primary Discussion**: How does this compare to the answer given in the book? Which elements are very close to the given answer? Which have greater error? Why does this make sense?\n",
    "\n",
    "**Secondary Discussion**: What happens if you set `eps` to different values in `my_grad` and `my_hes`? Is `H` still symmetric?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.3 Analysis of Optimality Conditions](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.3-Analysis-of-Optimality-Conditions)",
     "section": "3.3.4.3 Analysis of Optimality Conditions"
    }
   },
   "source": [
    "### 3.3.4.3 Analysis of Optimality Conditions\n",
    "\n",
    "**Activity**\n",
    "1. Calculate the eigenvalues of H.\n",
    "2. Is $x_{test}$ a\n",
    "    * stationary point\n",
    "    * local maximizer\n",
    "    * strict global maximizer\n",
    "    * global maximizer\n",
    "    * local minimizer\n",
    "    * strict local minimizer\n",
    "    * global minimizer\n",
    "    \n",
    "Answer Yes, No, or Possibly for each bullet point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[3.3.4.3 Analysis of Optimality Conditions](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.4.3-Analysis-of-Optimality-Conditions)",
     "section": "3.3.4.3 Analysis of Optimality Conditions"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 43.39788177+0.j 426.32104865+0.j]\n"
     ]
    }
   ],
   "source": [
    "# YOUR SOLUTION HERE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.3.5 Continuous Optimization Algorithms](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.5-Continuous-Optimization-Algorithms)",
     "section": "3.3.5 Continuous Optimization Algorithms"
    }
   },
   "source": [
    "## 3.3.5 Continuous Optimization Algorithms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.3.5 Continuous Optimization Algorithms](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.5-Continuous-Optimization-Algorithms)",
     "section": "3.3.5 Continuous Optimization Algorithms"
    }
   },
   "source": [
    "* Gradient-based Algorithms\n",
    "   * Conjugate gradient methods\n",
    "   * Sequential quadratic programming methods\n",
    "   * Sequential linear programming methods\n",
    "   * Interior point methods\n",
    "* Derivative-free optimization\n",
    "   * Deterministic algorithms\n",
    "      * Nelder–Mead method\n",
    "   * Stochastic algorithms\n",
    "       * Simulated Annealing\n",
    "       * Genetic Algorithms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[3.3.5 Continuous Optimization Algorithms](https://ndcbe.github.io/CBE60499/03.03-Optimality.html#3.3.5-Continuous-Optimization-Algorithms)",
     "section": "3.3.5 Continuous Optimization Algorithms"
    }
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "72d2d201",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [3.2 Mathematics Primer](https://ndcbe.github.io/CBE60499/03.02-Math-Primer-2.html) | [Contents](toc.html) | [Tag Index](tag_index.html) | [3.4 Newton-type Methods for Unconstrained Optimization](https://ndcbe.github.io/CBE60499/03.04-Netwon-Methods.html) ><p><a href=\"https://colab.research.google.com/github/ndcbe/CBE60499/blob/master/docs/03.03-Optimality.ipynb\"> <img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a><p><a href=\"https://ndcbe.github.io/CBE60499/03.03-Optimality.ipynb\"> <img align=\"left\" src=\"https://img.shields.io/badge/Github-Download-blue.svg\" alt=\"Download\" title=\"Download Notebook\"></a>"
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