5_unconstrained_algorithms/line_search_ex1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Armijo line search examples\n",
    "\n",
    "Consider the problem \n",
    "\n",
    "\\begin{equation*}\n",
    "\t\\begin{aligned}\n",
    "\t\t& \\underset{x_1,x_2}{\\text{minimize}}\n",
    "\t\t& & f(x_1,x_2) = 4x_1^2 + 3x_1x_2 + x_2^2\\\\\n",
    "\t\\end{aligned}\n",
    "\\end{equation*}\n",
    "\n",
    "having a gradient $$ \\nabla f(\\mathbf{x}) = \\begin{bmatrix} 8x_1 + 3x_2 \\\\ 3x_1 + 2x_2 \\end{bmatrix}$$\n",
    "\n",
    "and an optimal solution $\\mathbf{x}^* = (0,0)$\n",
    "\n",
    "We define the function `f` and its gradient `grad_f` using the below inline functions "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "f = lambda x: 4*x[0,:]**2 + 3*x[0,:]**1*x[1,:]**1 + x[1,:]**2\n",
    "grad_f = lambda x: np.array([[8*x[0,:] + 3*x[1,:]], [3*x[0,:] + 2*x[1,:]]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also generate a grid of points where $-3 \\leq x_1 \\leq 3$ and $-3 \\leq x_2 \\leq 2$ to visualize this function using a contour map"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import cm\n",
    "\n",
    "X1,X2 = np.meshgrid(np.linspace(-3,3,40),np.linspace(-3,2,40))\n",
    "X = np.vstack((X1.reshape([1,-1]),X2.reshape([1,-1]))) # flatten grids side-by-side\n",
    "\n",
    "Z = f(X)\n",
    "Z_grad = grad_f(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us choose a starting point \n",
    "$$\\mathbf{x}_0 = \\left(1.0,-2.0\\right)$$\n",
    "This gives a steepest descent direction \n",
    "$$\\mathbf{d}_0=-\\nabla f(\\mathbf{x}_0) = \\begin{bmatrix}-2 & 1\\end{bmatrix}^\\mathrm{T}$$\n",
    "\n",
    "We plot the function (given by isocontours), the initial point $\\mathbf{x}_0$, the true minimizer $\\mathbf{x}^*$, the gradient vector $\\nabla f(\\mathbf{x}_k)$ and the steepest decent direction $-\\nabla f(\\mathbf{x}_k)$ below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 700x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x0 = np.array([1,-2])\n",
    "grad_f0 = grad_f(x0[:,None]).squeeze()\n",
    "d0 = -grad_f0\n",
    "\n",
    "origin = np.array([x0]*2).T # origin point\n",
    "vectors = np.array([grad_f0, -grad_f0])\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(7,6))\n",
    "# plot\n",
    "ax.contourf(X1, X2, Z.reshape(X1.shape),cmap=cm.coolwarm,levels=25,zorder=-1)\n",
    "Q = ax.quiver(*origin, vectors[0:1,0], vectors[0:1,1], color=['r',], scale=12, zorder=1)\n",
    "ax.quiverkey(Q, 0.1, 0.2, 2, r\"$\\nabla f(\\mathbf{x}_0)$\", labelpos='S', coordinates='axes')\n",
    "Q = ax.quiver(*origin, vectors[1:2,0], vectors[1:2,1], color=['b'], scale=12, zorder=1)\n",
    "ax.quiverkey(Q, 0.1, 0.1, 2, r\"$\\mathbf{d}_k = -\\nabla f(\\mathbf{x}_0)$\", labelpos='S', coordinates='axes')\n",
    "ax.plot([x0[0],x0[0] + 10*d0[0]], [x0[1],x0[1] + 10*d0[1]], color=\"k\", label=\"line search\", linewidth=0.5, linestyle=\"--\",zorder=2)\n",
    "ax.scatter([0,], [0,], marker=\"o\", color=\"g\", edgecolors=\"k\", linestyle=\"None\",label=\"$\\mathbf{x}^*$\",zorder=3)\n",
    "ax.scatter([x0[0],], [x0[1],], marker=\"o\", color=\"m\", edgecolors=\"k\", linestyle=\"None\",label=\"$\\mathbf{x}_0$\",zorder=3)\n",
    "ax.set_xlabel(\"$x_1$\")\n",
    "ax.set_ylabel(\"$x_2$\")\n",
    "ax.axis('equal')\n",
    "ax.set_xlim((-2,2))\n",
    "ax.set_ylim((-3,2))\n",
    "ax.legend()\n",
    "fig.canvas.draw()\n",
    "# fig.savefig(\"linesearch_ex1_contours.pdf\",bbox_inches=\"tight\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets us examine how the function changes its value along the search direction $\\mathbf{d}_k$\n",
    "\n",
    "$$\\phi(\\alpha) = f(\\mathbf{x}_{k+1}) = f(\\mathbf{x}_k + \\alpha \\mathbf{d}_k)$$\n",
    "\n",
    "We also note that\n",
    "$$\n",
    "\\phi'(\\alpha) = f'(\\mathbf{x}_k + \\alpha \\mathbf{d}_k) = \\nabla f(\\mathbf{x}_k + \\alpha \\mathbf{d}_k)^\\mathrm{T} \\mathbf{d}_k,\n",
    "$$ \n",
    "from the chain rule\n",
    "\n",
    "Let $\\mathbf{x}_{1}$ be the next iterate, let us examine how $f(\\mathbf{x}_{1})$ changes along the search direction $\\mathbf{d}_0=-\\nabla^{\\mathrm{T}}f(\\mathbf{x_0})$ at the current iterate $\\mathbf{x}_0$. We derive an expression for $\\phi(\\alpha)$ at $\\mathbf{x_0}$\n",
    "\n",
    "$$\\phi(\\alpha) = f(\\mathbf{x}_{1}) = f(\\mathbf{x}_0 + \\alpha \\mathbf{d}_0)$$\n",
    "\n",
    "and its derivative\n",
    "\n",
    "$$\\phi'(\\alpha) = \\nabla f(\\mathbf{x}_1 + \\alpha \\mathbf{d}_0)^\\mathrm{T} \\mathbf{d}_0$$\n",
    "\n",
    "We use the below symbolic expression to find the analytical equations representing $\\phi(\\alpha)$ and $\\phi'(\\alpha)$ by substituting $\\mathbf{x}_0 + \\alpha \\mathbf{d}_0$ into $f$ and $\\nabla f$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4 x_{1}^{2} + 3 x_{1} x_{2} + x_{2}^{2}$"
      ],
      "text/plain": [
       "4*x_1**2 + 3*x_1*x_2 + x_2**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}8 x_{1} + 3 x_{2}\\\\3 x_{1} + 2 x_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[8*x_1 + 3*x_2],\n",
       "[3*x_1 + 2*x_2]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 11 \\alpha^{2} - 5 \\alpha + 2$"
      ],
      "text/plain": [
       "11*\\alpha**2 - 5*\\alpha + 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 - 13 \\alpha\\\\- 4 \\alpha - 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2 - 13*\\alpha],\n",
       "[-4*\\alpha - 1]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 22 \\alpha - 5$"
      ],
      "text/plain": [
       "22*\\alpha - 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{5}{22}$"
      ],
      "text/plain": [
       "5/22"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "import sympy.physics.vector as spv\n",
    "from sympy import pprint\n",
    "from sympy.plotting import plot\n",
    "\n",
    "a = sym.Symbol(r\"\\alpha\")\n",
    "x1 = sym.Symbol(r\"x_1\")\n",
    "x2 = sym.Symbol(r\"x_2\")\n",
    "\n",
    "f_sym = 4*x1**2 + 3*x1**1*x2**1 + x2**2\n",
    "display(f_sym)\n",
    "\n",
    "# try the gradient approach\n",
    "gradf_sym = sym.Matrix([sym.diff(f_sym,x1),sym.diff(f_sym,x2)])\n",
    "display(gradf_sym)\n",
    "\n",
    "# substitute direction\n",
    "f_sym = f_sym.subs({x1:x0[0]+d0[0]*a, x2: x0[1]+d0[1]*a})\n",
    "display(f_sym.simplify())\n",
    "gradf_sym = gradf_sym.subs({x1:x0[0]+d0[0]*a, x2: x0[1]+d0[1]*a})\n",
    "display(gradf_sym)\n",
    "\n",
    "df_sym=gradf_sym.dot(sym.Matrix([d0[0],d0[1]]))\n",
    "display(df_sym.simplify())\n",
    "\n",
    "# differentiate and set equal to 0 to find alpha_min\n",
    "a_min = sym.solve(df_sym,  a)[0]\n",
    "display(a_min)\n",
    "a_min = float(a_min)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "We will plot the following function\n",
    "\n",
    "$$\n",
    "\\phi(\\alpha) = f(\\mathbf{x}_{1}) = f(\\mathbf{x}_0 + \\alpha \\mathbf{d}_0) = f\\left(\\begin{bmatrix}1-2\\alpha \\\\ -2 + \\alpha\\end{bmatrix}\\right)= 13\\alpha^2 - 10\\alpha + 4\n",
    "$$\n",
    "\n",
    "We also note that the gradient of the above functions is given by\n",
    "\n",
    "$$\n",
    "\\phi'(\\alpha) = \\nabla f(\\mathbf{x}_1 + \\alpha \\mathbf{d}_0)^\\mathrm{T} \\mathbf{d}_0 = \\nabla f^\\mathrm{T} \\left(\\begin{bmatrix}1-2\\alpha \\\\ -2 + \\alpha\\end{bmatrix}\\right) \\begin{bmatrix}-2 \\\\ 1\\end{bmatrix}= 22\\alpha - 5\n",
    "$$\n",
    "\n",
    "with $\\alpha^*={5}/{22}$ from FONCs and SOSCs.\n",
    "\n",
    "Armijo's rule for line search specifies the following two conditions on $\\alpha$ to ensure sufficient descent (given in terms of the first-order Taylor series expansion at $\\alpha=0$)\n",
    "\n",
    "\\begin{align*}\n",
    "    \\phi(\\alpha) & \\le \\phi(0) + \\epsilon\\phi'(0)\\alpha = \\psi(\\alpha;\\epsilon)\\\\\n",
    "    \\phi(2\\alpha) & \\ge \\phi(0) + 2\\epsilon\\phi'(0)\\alpha = \\psi(\\alpha;2\\epsilon)\n",
    "\\end{align*}\n",
    "\n",
    "We visualize these conditions along the search direction given by the plot below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alpha = np.linspace(0,1.0,100)\n",
    "phi = lambda alpha: f(x0[:,None] + alpha*d0[:, None])\n",
    "phi_prime = lambda alpha: grad_f(x0[:,None] + alpha*d0[:, None]).T @ d0[:, None]\n",
    "\n",
    "e = 0.1\n",
    "\n",
    "# find bounds for alpha\n",
    "a_plot = float(sym.solve(f_sym - f_sym.subs(a,0))[1])\n",
    "b_plot = float(sym.solve(f_sym - f_sym.subs(a,0) - e*df_sym.subs(a,0)*a,a)[1])\n",
    "c_plot = float(sym.solve(f_sym.subs(a,2*a) - f_sym.subs(a,0) - 2*e*df_sym.subs(a,0)*a,a)[1])\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "# plot\n",
    "ax.plot(alpha, phi(alpha), color=\"k\", linewidth=1.5, linestyle=\"-\",zorder=-1,label=r\"$\\phi(\\alpha)$\")\n",
    "ax.plot(alpha, [phi(0),]*len(alpha), color=\"r\", linewidth=0.5, linestyle=\"-\",label=r\"$\\psi_A(\\alpha)$\",zorder=-1)\n",
    "ax.plot(alpha, phi(0) + e*alpha*phi_prime(0).squeeze(), color=\"b\", linewidth=0.5, linestyle=\"-\",label=r\"$\\psi_B(\\alpha) = \\phi(0)+\\epsilon\\phi'(0)\\alpha$\",zorder=-1)\n",
    "ax.plot(alpha, phi(0) + 2*e*alpha*phi_prime(0).squeeze(), color=\"g\", linewidth=0.5, linestyle=\"-\",label=r\"$\\psi_B(\\alpha) = \\phi(0)+2\\epsilon\\phi'(0)\\alpha$\",zorder=-1)\n",
    "ax.plot(alpha, phi(2*alpha), color=\"g\", linewidth=0.5, linestyle=\"-\",zorder=-1,label=r\"$\\phi(2\\alpha)$\")\n",
    "\n",
    "ax.scatter([a_min,], [phi(a_min),], marker=\"o\", color=\"m\", edgecolors=\"k\", linestyle=\"None\",label=r\"$\\alpha^*$\",zorder=1)\n",
    "ax.scatter([a_plot,], [phi(a_plot),], marker=\"o\", color=\"r\", edgecolors=\"k\", linestyle=\"None\",label=r\"A, $\\phi(\\alpha) = \\phi(0)$\",zorder=1)\n",
    "ax.scatter([b_plot,], [phi(b_plot),], marker=\"o\", color=\"b\", edgecolors=\"k\", linestyle=\"None\",label=r\"B, $\\phi(\\alpha) = \\phi(0) + \\epsilon\\phi'(0)\\alpha$\",zorder=1)\n",
    "ax.scatter([c_plot,], [phi(c_plot),], marker=\"o\", color=\"g\", edgecolors=\"k\", linestyle=\"None\",label=r\"C, $\\phi(2\\alpha) = \\phi(0) + 2\\epsilon\\phi'(0)\\alpha$\",zorder=1)\n",
    "\n",
    "ax.vlines([a_plot,b_plot,c_plot],1.1,2.4,color=[\"r\",\"b\",\"g\"], linewidth=0.5, linestyle=\"--\")\n",
    "\n",
    "ax.set_xlabel(r\"$\\alpha$\")\n",
    "ax.set_ylabel(r\"$\\phi(\\alpha)$\")\n",
    "ax.set_xlim((0,0.6))\n",
    "ax.set_ylim((1.1,2.4))\n",
    "ax.legend(ncol=1,loc='upper left',bbox_to_anchor=(1.01, 1.00))\n",
    "fig.canvas.draw()\n",
    "# fig.savefig(\"linesearch_ex1_projection.pdf\",bbox_inches=\"tight\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Case A: If $\\phi(\\alpha)$ exceeds $\\phi(0)$, i.e., $f(\\mathbf{x}_0 + \\alpha\\mathbf{d}_0) > f(\\mathbf{x}_0)$, then no descent occurs and this represents an upper limit on the possible values of $\\alpha < \\alpha_A$, where $\\alpha_A$ is the root of $\\phi(\\alpha) = \\phi(0)$.\n",
    "\n",
    "* Case B: $\\alpha$ must be such that $\\phi(\\alpha) \\le \\phi(0) + \\epsilon\\phi'(0)\\alpha$, to not be considered too large. This is a stricter upper bound on $\\alpha < \\alpha_B$, where $\\alpha_B$ is the root of $\\phi(\\alpha) = \\phi(0) + \\epsilon\\phi'(0)\\alpha$.\n",
    "\n",
    "* Case B: $\\alpha$ must be such that $\\phi(2\\alpha) \\ge \\phi(0) + 2\\epsilon\\phi'(0)\\alpha$, to not be considered too small. This is a lower bound on $\\alpha > \\alpha_C$, where $\\alpha_C$ is the root of $\\phi(2\\alpha) = \\phi(0) + 2\\epsilon\\phi'(0)\\alpha$.\n",
    "\n",
    "Any $\\alpha \\in (\\alpha_C,\\alpha_B)$ is considered an acceptable step size that provides sufficient descent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interactive example\n",
    "\n",
    "Let us now look at this example by varying $\\alpha$ and $\\epsilon$ interactively. We will observe:\n",
    "* The effect of $\\alpha$ on the next iterate $\\mathbf{x}_1$ at $k=0$.\n",
    "* The effect of $\\epsilon$ on the acceptable range of values for alpha $(\\alpha_C,\\alpha_B)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "579fd349e9a947b9aa6945d499101d82",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.1, description='α', max=0.5, step=0.01), FloatSlider(value=0.1, desc…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "import ipywidgets as widgets\n",
    "\n",
    "def plot_func(alpha_next,e):\n",
    "    # input parameters\n",
    "    # alpha_next = 0.25\n",
    "    # e = 0.1\n",
    "\n",
    "    phi = lambda alpha: f(x0[:,None] + alpha*d0[:, None])\n",
    "    phi_prime = lambda alpha: grad_f(x0[:,None] + alpha*d0[:, None]).T @ d0[:, None]\n",
    "\n",
    "    x0 = np.array([1.0,-2.0])\n",
    "    grad_f0 = grad_f(x0[:,None]).squeeze()\n",
    "    d0 = -grad_f0\n",
    "    x_next = x0 + alpha_next*d0\n",
    "\n",
    "    # symbolic calculations\n",
    "    f_sym = 4*x1**2 + 3*x1**1*x2**1 + x2**2\n",
    "\n",
    "    # try the gradient approach\n",
    "    gradf_sym = sym.Matrix([sym.diff(f_sym,x1),sym.diff(f_sym,x2)])\n",
    "\n",
    "    # substitute direction\n",
    "    f_sym = f_sym.subs({x1:x0[0]+d0[0]*a, x2: x0[1]+d0[1]*a})\n",
    "    gradf_sym = gradf_sym.subs({x1:x0[0]+d0[0]*a, x2: x0[1]+d0[1]*a})\n",
    "    df_sym=gradf_sym.dot(sym.Matrix([d0[0],d0[1]]))\n",
    "\n",
    "    # differentiate and set equal to 0 to find alpha_min\n",
    "    a_min = float(sym.solve(df_sym,  a)[0])\n",
    "    # find bounds for alpha\n",
    "    a_plot = float(sym.solve(f_sym - f_sym.subs(a,0))[1])\n",
    "    b_plot = float(sym.solve(f_sym - f_sym.subs(a,0) - e*df_sym.subs(a,0)*a,a)[1])\n",
    "    c_plot = float(sym.solve(f_sym.subs(a,2*a) - f_sym.subs(a,0) - 2*e*df_sym.subs(a,0)*a,a)[1])\n",
    "\n",
    "    fig, axs = plt.subplots(nrows=1,ncols=2,figsize=(15,6))\n",
    "    # Contour map of f\n",
    "    axs[0].contourf(X1, X2, Z.reshape(X1.shape),cmap=cm.coolwarm,levels=25,zorder=-1)\n",
    "    axs[0].plot([x0[0],x0[0] + 10*d0[0]], [x0[1],x0[1] + 10*d0[1]], color=\"k\", label=\"search direction\", linewidth=0.5, linestyle=\"--\",zorder=2)\n",
    "    axs[0].scatter([x0[0],], [x0[1],], marker=\"o\", color=\"m\", edgecolors=\"k\", linestyle=\"None\",label=\"$\\mathbf{x}_k$\",zorder=3)\n",
    "    axs[0].scatter([x_next[0],], [x_next[1],], marker=\"s\", color=\"orange\", edgecolors=\"k\", linestyle=\"None\",label=r\"$\\mathbf{x}_{k+1} = \\mathbf{x}_{k} + \\alpha_k \\mathbf{d}_k$\",zorder=3)\n",
    "    axs[0].set_xlabel(\"$x_1$\")\n",
    "    axs[0].set_ylabel(\"$x_2$\")\n",
    "    axs[0].axis('equal')\n",
    "    axs[0].set_xlim((-2,2))\n",
    "    axs[0].set_ylim((-3,2))\n",
    "    axs[0].legend()\n",
    "    axs[0].set_title(r\"$\\mathbf{x}_{k+1}$ = (%.2f,%.2f)\" %(x_next[0],x_next[1]))\n",
    "\n",
    "    # Projection of f onto search direction\n",
    "    axs[1].plot(alpha, phi(alpha), color=\"k\", linewidth=1.5, linestyle=\"-\",zorder=-1,label=r\"$\\phi(\\alpha)$\")\n",
    "    axs[1].plot(alpha, [phi(0),]*len(alpha), color=\"r\", linewidth=0.5, linestyle=\"-\",label=r\"$\\psi_A(\\alpha)$\",zorder=-1)\n",
    "    axs[1].plot(alpha, phi(0) + e*alpha*phi_prime(0).squeeze(), color=\"b\", linewidth=0.5, linestyle=\"-\",label=r\"$\\psi_B(\\alpha) = \\phi(0)+\\epsilon\\phi'(0)\\alpha$\",zorder=-1)\n",
    "    axs[1].plot(alpha, phi(0) + 2*e*alpha*phi_prime(0).squeeze(), color=\"g\", linewidth=0.5, linestyle=\"-\",label=r\"$\\psi_B(\\alpha) = \\phi(0)+2\\epsilon\\phi'(0)\\alpha$\",zorder=-1)\n",
    "    axs[1].plot(alpha, phi(2*alpha), color=\"g\", linewidth=0.5, linestyle=\"-\",zorder=-1,label=r\"$\\phi(2\\alpha)$\")\n",
    "\n",
    "    axs[1].scatter([a_min,], [phi(a_min),], marker=\"o\", color=\"m\", edgecolors=\"k\", linestyle=\"None\",label=r\"$\\alpha^*$\",zorder=1)\n",
    "    axs[1].scatter([a_plot,], [phi(a_plot),], marker=\"o\", color=\"r\", edgecolors=\"k\", linestyle=\"None\",label=r\"A, $\\phi(\\alpha) = \\phi(0)$\",zorder=1)\n",
    "    axs[1].scatter([b_plot,], [phi(b_plot),], marker=\"o\", color=\"b\", edgecolors=\"k\", linestyle=\"None\",label=r\"B, $\\phi(\\alpha) = \\phi(0) + \\epsilon\\phi'(0)\\alpha$\",zorder=1)\n",
    "    axs[1].scatter([c_plot,], [phi(c_plot),], marker=\"o\", color=\"g\", edgecolors=\"k\", linestyle=\"None\",label=r\"C, $\\phi(2\\alpha) = \\phi(0) + 2\\epsilon\\phi'(0)\\alpha$\",zorder=1)\n",
    "    axs[1].scatter([alpha_next,], [phi(alpha_next),], marker=\"s\", color=\"orange\", edgecolors=\"k\", linestyle=\"None\",label=r\"$\\alpha_k$\",zorder=1)\n",
    "    axs[1].vlines([a_plot,b_plot,c_plot],phi(a_min)-0.1,phi(a_plot)+0.1,color=[\"r\",\"b\",\"g\"], linewidth=0.5, linestyle=\"--\")\n",
    "    axs[1].set_title(\"%.2f $\\leq alpha \\leq$ %.2f\" %(c_plot,b_plot))\n",
    "\n",
    "    axs[1].set_xlabel(r\"$\\alpha$\")\n",
    "    axs[1].set_ylabel(r\"$\\phi(\\alpha)$\")\n",
    "    axs[1].set_xlim((0,0.9))\n",
    "    axs[1].set_ylim((phi(a_min)-0.1,phi(a_plot)+0.1))\n",
    "    axs[1].legend(ncol=1,loc='upper left',bbox_to_anchor=(1.01, 1.00));\n",
    "\n",
    "interact(plot_func, alpha_next = widgets.FloatSlider(value=0.1,min=0.0,max=.5,step=0.01,description=u\"\\u03B1\"),\n",
    "    e = widgets.FloatSlider(value=0.1,min=0.01,max=0.5,step=0.05,description=u\"\\u03B5\"));\n"
   ]
  }
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