L10.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Lecture 10 - Optimization\n",
    "\n",
    "C&C Problem 13.10-13.11. Find minimum of: $$ f(x) = 3 + 6x + 5x^2 + 3x^3 + 4x^4 $$ using:\n",
    "- Bisection with initial bracket $x_l=-2$, $x_u=1$\n",
    "- Newton's method with $x_0=1$\n",
    "- Newton's method with finite difference estimates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1800x1200 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "plt.rcParams['figure.dpi'] = 300\n",
    "\n",
    "def f(x):\n",
    "    return 3 + 6*x + 5*x**2 + 3*x**3 + 4*x**4\n",
    "\n",
    "def fp(x):\n",
    "    return 6 + 10*x + 9*x**2 + 16*x**3\n",
    "\n",
    "def fpp(x):\n",
    "    return 10 + 18*x + 48*x**2\n",
    "\n",
    "x = np.linspace(-2,1,100)\n",
    "plt.plot(x, f(x))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the simplest case, we can just use our existing root finding methods to find the minimum, $f^\\prime(x)=0$. For example bisection from L8, where `f` is renamed `fp` to reflect that we are optimizing instead of root finding."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimate of minimum: x=-0.5867, f(x)=1.0690\n",
      "Iterations: 31\n"
     ]
    }
   ],
   "source": [
    "def bisection(fp, xl, xu, tol=1e-8):\n",
    "    \n",
    "    # check for valid initial bracket\n",
    "    assert fp(xl) * fp(xu) < 0, 'Choose xl,xu such that f(xl) * f(xu) < 0'\n",
    "    err = 9999\n",
    "    errors = [] # store errors to plot later\n",
    "    \n",
    "    while err > tol:\n",
    "        xr = (xl + xu) / 2\n",
    "        test = fp(xl) * fp(xr)\n",
    "        if test < 0:\n",
    "            xu = xr\n",
    "        elif test > 0:\n",
    "            xl = xr\n",
    "        err = np.abs(fp(xr))\n",
    "        errors.append(err)\n",
    "    \n",
    "    return xr, errors\n",
    "\n",
    "xb, errors_b = bisection(fp, -2, 1)\n",
    "\n",
    "print('Estimate of minimum: x=%0.4f, f(x)=%0.4f' % (xb, f(xb)))\n",
    "print('Iterations: %d' % len(errors_b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Newton's method, modified for optimization by renaming `f` to `fp`, and `fp` to `fpp`. Note this function is only for the 1-dimensional case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def newton(fp, fpp, x, tol=1e-8):\n",
    "\n",
    "    err = 9999\n",
    "    errors = [] # store errors to plot later\n",
    "    \n",
    "    while err > tol:\n",
    "        x = x - fp(x)/fpp(x)\n",
    "        err = np.abs(fp(x))\n",
    "        errors.append(err)\n",
    "    \n",
    "    return x, errors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimate of minimum: x=-0.5867, f(x)=1.0690\n",
      "Iterations: 6\n"
     ]
    }
   ],
   "source": [
    "xn, errors_n = newton(fp, fpp, 1)\n",
    "\n",
    "print('Estimate of minimum: x=%0.4f, f(x)=%0.4f' % (xn, f(xn)))\n",
    "print('Iterations: %d' % len(errors_n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Newton's method converges faster than bisection.\n",
    "\n",
    "In the last part of the problem, we substitute finite (centered) differences for the first and second derivatives in Newton's method. This is a similar idea to the Secant method for root finding. The difference formulas are defined with a step size proportional to the value of $x$, for example $\\delta=0.01$ is a 1% change. Note these functions are not specific to this problem and will work for any $f(x)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fp_cd(f, x, d=0.01): # f prime centered difference\n",
    "    return (f(x + d*x) - f(x - d*x)) / (2*d*x)\n",
    "\n",
    "def fpp_cd(f, x, d=0.01): # f double-prime centered difference\n",
    "    return (f(x + d*x) - 2*f(x) + f(x - d*x)) / (d*x)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test these against the analytical derivatives at $x=1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First derivative analytical:  41\n",
      "First derivative numerical:  41.00190000000019\n"
     ]
    }
   ],
   "source": [
    "print('First derivative analytical: ', fp(1))\n",
    "print('First derivative numerical: ', fp_cd(f,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Second derivative analytical:  76\n",
      "Second derivative numerical:  76.00079999996012\n"
     ]
    }
   ],
   "source": [
    "print('Second derivative analytical: ', fpp(1))\n",
    "print('Second derivative numerical: ', fpp_cd(f,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can use them in Newton's method. Only `f` is required as input."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimate of minimum: x=-0.5867, f(x)=1.0690\n",
      "Iterations: 6\n"
     ]
    }
   ],
   "source": [
    "def newton_cd(f, x, tol=1e-8):\n",
    "\n",
    "    err = 9999\n",
    "    errors = [] # store errors to plot later\n",
    "    \n",
    "    while err > tol:\n",
    "        x = x - fp_cd(f,x) / fpp_cd(f,x)\n",
    "        err = np.abs(fp_cd(f,x))\n",
    "        errors.append(err)\n",
    "    \n",
    "    return x, errors\n",
    "\n",
    "xcd, errors_cd = newton_cd(f, 1)\n",
    "\n",
    "print('Estimate of minimum: x=%0.4f, f(x)=%0.4f' % (xcd, f(xcd)))\n",
    "print('Iterations: %d' % len(errors_cd))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The iteration count is the same as the analytical Newton's method. However, note that each iteration of the finite difference version is performing more function evaluations to estimate the derivatives. It would be possible to make this more efficient by computing the `f(x +/- d*x)` terms only once and using them to calculate both `fp` and `fpp`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 16.20\n",
    "\n",
    "Find location (2-dimensional) of peak pollutant concentration in a channel: \n",
    "$$ c(x,y) = 7.7 + 0.15x + 0.22y - 0.05x^2 - 0.016y^2 - 0.007xy $$\n",
    "\n",
    "Over the domain: $-10 \\leq x \\leq 10$, $0 \\leq y \\leq 20$. Solve with:\n",
    "- Gradient ascent with $\\alpha=0.01$\n",
    "- Newton's method\n",
    "- `scipy.optimize.minimize`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def c(x,y):\n",
    "    return 7.7 + 0.15*x + 0.22*y - 0.05*x**2 - 0.016*y**2 - 0.007*x*y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First look at the function using a filled contour plot. The `meshgrid` function creates the set of all 2D `(x,y)` points from the ranges of `x` and `y`. These are assigned to temporary variables because they are not used later in the optimization. Last, the concentration values come from calling `c(x,y)`. The `contourf` function has many options, see docs [here](https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.contourf.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1800x1200 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_temp, y_temp = np.meshgrid(np.linspace(-10,10,100), np.linspace(0,20,100))\n",
    "plt.contourf(x_temp, y_temp, c(x_temp, y_temp))\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we write gradient ascent, we will need the gradient function. This could also be defined using the numerical derivatives."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def c_grad(x,y):\n",
    "    dfdx = 0.15 - 0.1*x - 0.007*y\n",
    "    dfdy = 0.22 - 0.032*y - 0.007*x\n",
    "    return np.array([dfdx, dfdy])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# use vector norm to measure error in gradient\n",
    "from scipy.linalg import norm\n",
    "\n",
    "def gradient_ascent(f_grad, x, y, alpha=0.01, tol=1e-8):\n",
    "    err = 9999\n",
    "    errors = []\n",
    "    \n",
    "    while err > tol:\n",
    "        grad = f_grad(x,y)\n",
    "        x = x + alpha * grad[0]\n",
    "        y = y + alpha * grad[1]\n",
    "        err = np.abs(norm(grad,2))\n",
    "        errors.append(err)\n",
    "    \n",
    "    return x, y, errors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimate of minimum: x=1.0346, y=6.6487, c(x,y)=8.5089\n",
      "Iterations: 54405\n"
     ]
    }
   ],
   "source": [
    "x, y, errors = gradient_ascent(c_grad, 5, 15)\n",
    "print('Estimate of minimum: x=%0.4f, y=%0.4f, c(x,y)=%0.4f' % (x, y, c(x,y)))\n",
    "print('Iterations: %d' % len(errors))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The initial point $(x_0,y_0)$ should not matter for this problem, because it looks concave. The local maximum will be the global maximum. Our function converges to what looks like a reasonable point, based on the contour plot. However, it takes quite a few iterations to get there. This is sensitive to the choice of $\\alpha$. For example, $\\alpha=10$ will converge in ~50 iterations instead.\n",
    "\n",
    "Can we do better with Newton's method? First we have to define the Hessian for the two-dimensional problem. This is the analytical version, which we would probably not do for more complex problems. Instead finite differences or another approximation could be used. For this problem the function $c(x,y)$ is quadratic, the Hessian contains constant values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def c_H(x,y):\n",
    "    d2fdx2 = -0.1\n",
    "    d2fdy2 = -0.032\n",
    "    d2fdxdy = -0.007 # equal to d2fdydx\n",
    "    return np.array([[d2fdx2, d2fdxdy], [d2fdxdy, d2fdy2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.linalg import inv # matrix inverse\n",
    "\n",
    "def newton_multivariate(f_grad, f_H, x, y, tol=1e-8):\n",
    "    err = 9999\n",
    "    errors = []\n",
    "    \n",
    "    # group x,y into the same vector to do matrix math\n",
    "    x = np.array([x,y])\n",
    "    \n",
    "    while err > tol:\n",
    "        # our gradient function is still set up for separate x,y inputs\n",
    "        grad = f_grad(x[0],x[1]) \n",
    "        H = f_H(x,y)\n",
    "        # the -inv(H) holds for all min/max problems.\n",
    "        # the second derivatives take care of the sign, unlike in gradient ascent\n",
    "        x = x - inv(H) @ grad\n",
    "        err = np.abs(norm(grad,2))\n",
    "        errors.append(err)\n",
    "    \n",
    "    return x[0], x[1], errors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimate of minimum: x=1.0346, y=6.6487, c(x,y)=8.5089\n",
      "Iterations: 2\n"
     ]
    }
   ],
   "source": [
    "xn, yn, errors_n = newton_multivariate(c_grad, c_H, 5, 15)\n",
    "print('Estimate of minimum: x=%0.4f, y=%0.4f, c(x,y)=%0.4f' % (xn, yn, c(xn,yn)))\n",
    "print('Iterations: %d' % len(errors_n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function is quadratic, so Newton's method converges quickly. In general, there is still a risk of diverging, just as in the one-dimensional root finding problems. \n",
    "\n",
    "Other notes:\n",
    "- This problem only has one global maximum. For a multi-modal function, these methods will return the first optimum they find. It's up to us to compare their function values and possibly second derivatives to determine which one is the global optimum.\n",
    "- We did not use the range constraints because the methods are following the gradient, and will not run into the boundary. For a general problem we may need to incorporate these.\n",
    "\n",
    "Last, built-in Scipy `minimize`. \n",
    "- We have to modify our function `c(x,y)` to take a vector input instead.\n",
    "- We want to maximize, not minimize, so return the negative of `c(x,y)`\n",
    "- Full docs [here](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html). If no gradient function is given, it is estimated numerically. The default method is BFGS, which estimates the inverse Hessian so it is not needed as an input."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.optimize import minimize\n",
    "\n",
    "def c(x):\n",
    "    c = 7.7 + 0.15*x[0] + 0.22*x[1] - 0.05*x[0]**2 - 0.016*x[1]**2 - 0.007*x[0]*x[1]\n",
    "    return -c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      fun: -8.508949539785746\n",
      " hess_inv: array([[ 9.96400441, -1.81612532],\n",
      "       [-1.81612532, 30.88265912]])\n",
      "      jac: array([ 2.02655792e-06, -1.31130219e-06])\n",
      "  message: 'Optimization terminated successfully.'\n",
      "     nfev: 33\n",
      "      nit: 8\n",
      "     njev: 11\n",
      "   status: 0\n",
      "  success: True\n",
      "        x: array([1.0346144, 6.648644 ])\n"
     ]
    }
   ],
   "source": [
    "res = minimize(c, x0=[5,15])\n",
    "\n",
    "print(res)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result object contains all of the information we need: whether the method converged, how many iterations it took, the objective function, and the decision variables. We can access these using `res.fun` and `res.x`. In this case notice we are maximizing the negative, so the optimal objective function value is `8.509`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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