5_unconstrained_algorithms/line_search_ex2.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) = \\left(1 - x_{1}\\right)^{2} + \\left(1 - x_{2}\\right)^{2} + \\frac{\\left(- x_{1}^{2} + 2 x_{2}\\right)^{2}}{2}\\\\\n",
    "\t\\end{aligned}\n",
    "\\end{equation*}\n",
    "\n",
    "having a gradient $$ \\nabla f(\\mathbf{x}) = \\left[\\begin{matrix}- 2 x_{1} \\left(- x_{1}^{2} + 2 x_{2}\\right) + 2 x_{1} - 2\\\\- 2 x_{1}^{2} + 6 x_{2} - 2\\end{matrix}\\right]$$\n",
    "\n",
    "and an optimal solution $\\mathbf{x}^* = (1.21,0.824)$\n",
    "\n",
    "We define the function `f` and its gradient `grad_f` using the below inline functions \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "f = lambda x: ((1 - x[0,:] )**2) + ((1 - x[1,:] )**2) + 0.5*((2*x[1,:] - x[0,:]**2)**2)\n",
    "grad_f = lambda x: np.array([[2*x[0,:]*(x[0,:]**2 - 2*x[1,:]) + 2*x[0,:] - 2], [-2*x[0,:]**2 + 6*x[1,:] - 2]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also generate a grid of points where $-3 \\leq x_1 \\leq 3$ and $-2 \\leq x_2 \\leq 5$ to visualize this function using a contour map"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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(-2,5,40))\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",
    "$$\\mathbf{d}_0=-\\nabla f(\\mathbf{x}_0) = \\begin{bmatrix}-2 & -8\\end{bmatrix}^\\mathrm{T}$$\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": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "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",
    "x_opt = [1.2134116627622296, 0.8241226211091329]\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=50, 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=50, 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([x_opt[0],], [x_opt[1],], 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((-3,3))\n",
    "ax.set_ylim((-1,3))\n",
    "ax.legend()\n",
    "fig.canvas.draw()\n",
    "# fig.savefig(\"linesearch_ex2_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 its derivative is (from the chain rule)\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",
    "\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",
    "$$\\phi'(\\alpha) = \\nabla f(\\mathbf{x}_0 + \\alpha \\mathbf{d}_0)^\\mathrm{T} \\mathbf{d}_0$$\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": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(1 - x_{1}\\right)^{2} + \\left(1 - x_{2}\\right)^{2} + \\frac{\\left(- x_{1}^{2} + 2 x_{2}\\right)^{2}}{2}$"
      ],
      "text/plain": [
       "(1 - x_1)**2 + (1 - x_2)**2 + (-x_1**2 + 2*x_2)**2/2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- 2 x_{1} \\left(- x_{1}^{2} + 2 x_{2}\\right) + 2 x_{1} - 2\\\\- 2 x_{1}^{2} + 6 x_{2} - 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-2*x_1*(-x_1**2 + 2*x_2) + 2*x_1 - 2],\n",
       "[               -2*x_1**2 + 6*x_2 - 2]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 8 \\alpha^{4} + 80 \\alpha^{3} + 256 \\alpha^{2} - 68 \\alpha + \\frac{19}{2}$"
      ],
      "text/plain": [
       "8*\\alpha**4 + 80*\\alpha**3 + 256*\\alpha**2 - 68*\\alpha + 19/2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- 16 \\alpha^{3} - 88 \\alpha^{2} - 32 \\alpha + 2\\\\- 8 \\alpha^{2} - 56 \\alpha + 8\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-16*\\alpha**3 - 88*\\alpha**2 - 32*\\alpha + 2],\n",
       "[                -8*\\alpha**2 - 56*\\alpha + 8]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 32 \\alpha^{3} + 240 \\alpha^{2} + 512 \\alpha - 68$"
      ],
      "text/plain": [
       "32*\\alpha**3 + 240*\\alpha**2 + 512*\\alpha - 68"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "0.12532689288652915"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def get_real(roots, last=False):\n",
    "    reals = [root for root in roots if root.is_real]\n",
    "    if last:\n",
    "        return reals[-1]\n",
    "    else:\n",
    "        return reals[0]\n",
    "\n",
    "import sympy as sym\n",
    "import sympy.physics.vector as spv\n",
    "from sympy import pprint, latex, simplify\n",
    "from sympy.plotting import plot\n",
    "\n",
    "a = sym.Symbol(r\"\\alpha\", real=True)\n",
    "x1 = sym.Symbol(r\"x_1\")\n",
    "x2 = sym.Symbol(r\"x_2\")\n",
    "\n",
    "f_sym = ((1 - x1 )**2) + ((1 - x2 )**2) + ((2*x2 - x1**2)**2)/2\n",
    "display(f_sym)\n",
    "# print(latex(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",
    "# print(latex(gradf_sym))\n",
    "# find minimizer\n",
    "# roots = sym.solve(gradf_sym, (x1, x2))\n",
    "# print([float(x) for x in simplify(roots[-1])])\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",
    "# print(latex(f_sym.simplify()))\n",
    "\n",
    "gradf_sym = gradf_sym.subs({x1:x0[0]+d0[0]*a, x2: x0[1]+d0[1]*a})\n",
    "display(gradf_sym.applyfunc(simplify))\n",
    "# print(latex(gradf_sym.applyfunc(simplify)))\n",
    "\n",
    "df_sym=gradf_sym.dot(sym.Matrix([d0[0],d0[1]]))\n",
    "display(df_sym.simplify())\n",
    "# print(latex(df_sym.simplify()))\n",
    "\n",
    "# differentiate and set equal to 0 to find alpha_min\n",
    "roots = sym.solve(df_sym,  a)\n",
    "\n",
    "a_min = float(get_real(roots))\n",
    "display(a_min)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "substitution gives the following expressions for $\\phi(\\alpha)$ and $\\phi'(\\alpha)$\n",
    "\n",
    "$$\n",
    "\\phi(\\alpha) = f(\\mathbf{x}_{1}) = f(\\mathbf{x}_0 + \\alpha \\mathbf{d}_0) = f\\left(\\left[\\begin{matrix}- 4 \\alpha - 2 \\left(- 2 \\alpha - 1\\right) \\left(- 16 \\alpha - \\left(- 2 \\alpha - 1\\right)^{2} + 4\\right) - 4\\\\- 48 \\alpha - 2 \\left(- 2 \\alpha - 1\\right)^{2} + 10\\end{matrix}\\right]\\right)= 8 \\alpha^{4} + 80 \\alpha^{3} + 256 \\alpha^{2} - 68 \\alpha + \\frac{19}{2}\n",
    "$$\n",
    "\n",
    "We also note that the gradient of the above functions is given by\n",
    "\n",
    "$$\n",
    "\\phi'(\\alpha) = \\nabla f(\\mathbf{x}_1 + \\alpha \\mathbf{d}_0)^\\mathrm{T} \\mathbf{d}_0 = \\nabla f^\\mathrm{T} \\left(\\left[\\begin{matrix}- 4 \\alpha - 2 \\left(- 2 \\alpha - 1\\right) \\left(- 16 \\alpha - \\left(- 2 \\alpha - 1\\right)^{2} + 4\\right) - 4\\\\- 48 \\alpha - 2 \\left(- 2 \\alpha - 1\\right)^{2} + 10\\end{matrix}\\right]\\right) \\begin{bmatrix}-2 \\\\ -8\\end{bmatrix}= 32 \\alpha^{3} + 240 \\alpha^{2} + 512 \\alpha - 68\n",
    "$$\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": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "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(get_real(sym.solve(f_sym - f_sym.subs(a,0)), last=True))\n",
    "b_plot = float(get_real(sym.solve(f_sym - f_sym.subs(a,0) - e*df_sym.subs(a,0)*a,a), last=True))\n",
    "c_plot = float(get_real(sym.solve(f_sym.subs(a,2*a) - f_sym.subs(a,0) - 2*e*df_sym.subs(a,0)*a,a), last=True))\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",
    "ypad = (phi(a_plot) - phi(a_min)) * 0.1\n",
    "\n",
    "ax.vlines([a_plot,b_plot,c_plot],phi(a_min)-ypad,phi(a_plot)+ypad,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((0,0.9))\n",
    "ax.set_ylim((phi(a_min)-ypad,phi(a_plot)+ypad))\n",
    "ax.legend(ncol=1,loc='upper left',bbox_to_anchor=(1.01, 1.00))\n",
    "fig.canvas.draw()\n",
    "# fig.savefig(\"linesearch_ex2_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": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "c67df87f4f894f11883ff338c883e628",
       "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 = ((1 - x1 )**2) + ((1 - x2 )**2) + ((2*x2 - x1**2)**2)/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(get_real(sym.solve(df_sym,  a), last=False))\n",
    "    # find bounds for alpha\n",
    "    a_plot = float(get_real(sym.solve(f_sym - f_sym.subs(a,0)), last=True))\n",
    "    b_plot = float(get_real(sym.solve(f_sym - f_sym.subs(a,0) - e*df_sym.subs(a,0)*a,a), last=True))\n",
    "    c_plot = float(get_real(sym.solve(f_sym.subs(a,2*a) - f_sym.subs(a,0) - 2*e*df_sym.subs(a,0)*a,a), last=True))\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((-3,3))\n",
    "    axs[0].set_ylim((-1,3))\n",
    "    axs[0].legend()\n",
    "    axs[0].set_title(r\"$\\mathbf{x}_{k+1}$ = (%.2f,%.2f)\" %(x_next[0],x_next[1]))\n",
    "\n",
    "    ypad = (phi(a_plot) - phi(a_min)) * 0.1\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)-ypad,phi(a_plot)+ypad,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)-ypad,phi(a_plot)+ypad))\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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