From 7e2e500b0e03081e0a9cac335c256290c5e94110 Mon Sep 17 00:00:00 2001
From: Russ Tedrake <russt@mit.edu>
Date: Sun, 13 Jun 2021 20:02:44 -0400
Subject: [PATCH] add notes on sdp relaxation of non-convex quadratic
 constraints

---
 README.md          |  3 +-
 data               |  2 +-
 optimization.html  | 50 +++++++++++++++++++++++---
 optimization.ipynb | 90 +++++++++++++++++++++++++++++++++++++++++++++-
 4 files changed, 137 insertions(+), 8 deletions(-)

diff --git a/README.md b/README.md
index e78ac3d8..6104c083 100644
--- a/README.md
+++ b/README.md
@@ -66,7 +66,8 @@ the project view and selecting `Mark directory as > Excluded`.
 
 I use selection and then Ctrl+alt+P for manual wrapping. I updated the HTML
 style to use 2 for tabs/indent/continuation, and added `p`
-to `Other > Do not indent children of`.
+to `Other > Do not indent children of`. I removed `h1` from the "indent before",
+and unchecked "keep line breaks in text".
 
 Additional information for Docker Hub users
 -------------------------------------------
diff --git a/data b/data
index 0cd7e488..3dab7b6b 160000
--- a/data
+++ b/data
@@ -1 +1 @@
-Subproject commit 0cd7e488a3b2ba8b969cf52d4d09e331a169a753
+Subproject commit 3dab7b6be0fb243fc18fbd9d9d633f826b7f48bd
diff --git a/optimization.html b/optimization.html
index 48940b6e..6d889d8c 100644
--- a/optimization.html
+++ b/optimization.html
@@ -12,7 +12,7 @@
     <script src="https://hypothes.is/embed.js" async></script>
     <script type="text/javascript" src="htmlbook/book.js"></script>
 
-    <script src="htmlbook/mathjax-config.js" defer></script>
+    <script src="htmlbook/mathjax-config.js" defer></script> 
     <script type="text/javascript" id="MathJax-script" defer
       src="htmlbook/MathJax/es5/tex-chtml.js">
     </script>
@@ -30,9 +30,9 @@
 <div data-type="titlepage">
   <header>
     <h1><a href="index.html" style="text-decoration:none;">Underactuated Robotics</a></h1>
-    <p data-type="subtitle">Algorithms for Walking, Running, Swimming, Flying, and Manipulation</p>
+    <p data-type="subtitle">Algorithms for Walking, Running, Swimming, Flying, and Manipulation</p> 
     <p style="font-size: 18px;"><a href="http://people.csail.mit.edu/russt/">Russ Tedrake</a></p>
-    <p style="font-size: 14px; text-align: right;">
+    <p style="font-size: 14px; text-align: right;"> 
       &copy; Russ Tedrake, 2021<br/>
       Last modified <span id="last_modified"></span>.</br>
       <script>
@@ -45,8 +45,8 @@ <h1><a href="index.html" style="text-decoration:none;">Underactuated Robotics</a
 
 <p><b>Note:</b> These are working notes used for <a
 href="http://underactuated.csail.mit.edu/Spring2021/">a course being taught
-at MIT</a>. They will be updated throughout the Spring 2021 semester.  <a
-href="https://www.youtube.com/channel/UChfUOAhz7ynELF-s_1LPpWg">Lecture videos are available on YouTube</a>.</p>
+at MIT</a>. They will be updated throughout the Spring 2021 semester.  <a 
+href="https://www.youtube.com/channel/UChfUOAhz7ynELF-s_1LPpWg">Lecture  videos are available on YouTube</a>.</p> 
 
 <table style="width:100%;"><tr style="width:100%">
   <td style="width:33%;text-align:left;"><a class="previous_chapter" href=multibody.html>Previous Chapter</a></td>
@@ -198,6 +198,46 @@ <h1><a href="index.html" style="text-decoration:none;">Underactuated Robotics</a
       Example: Balancing force control on Atlas
     </subsection>
     <subsection><h1>Semidefinite Programming and Linear Matrix Inequalities</h1>
+
+      <example>
+        <h1>Semidefinite programming relaxation of non-convex quadratic
+          constraints</h1>
+
+        <p>Consider the problem: \begin{align*} \min_x \quad & \| x - a \|^2 \\
+        \text{subject to} \quad & \| x - b \| \ge 1 \end{align*} We can write
+        this as \begin{align*} \min_{x,y} \quad & y - 2ax + a^2 \\ \text{subject
+        to} \quad & y - 2bx + b^2 \ge 1 \\ & y \ge x^2 \end{align*} where we
+        write $y \ge x^2$ as the semidefinite constraint $\begin{bmatrix} y & x
+        \\ x & 1 \end{bmatrix}.$</p>
+
+        <figure><img width="80%" src="data/sdp_relaxation_quadratic.svg"/>
+        <figcaption>Optimization landscape for $a=.8, b=.5.$</figcaption>
+        </figure>
+
+        <p>I've plotted the feasible region and the objective with an arrow. As
+        you can see, the feasible region is the epigraph of $y=x^2$ intersected
+        with the linear constraint. Now here is the key point: for linear
+        objectives, the optimal solution will lie on the boundary only if the
+        cost is directly orthogonal to the objective; otherwise it will lie at a
+        vertex. So in this case, the solution will only lie on the interior if
+        $a = b;$ for every other value, this relaxation will give the optimal
+        solution. Note that we could have equally well written a quadratic
+        equality constraint.</p>
+
+        <p>I find this incredibly clever, and only frustrating that it is
+        somewhat particular to quadratics. (The fact that the entire exterior of
+        the quadratic region could be an optimal solution does not hold if we
+        try to use the same approach for e.g. being outside of a polytope).</p>
+
+        <todo>Update SHA</todo>
+        <p><a target="optimization"
+          href="https://colab.research.google.com/github/RussTedrake/underactuated/blob/master/optimization.ipynb#scrollTo=fhc5mrE9_V81"><img
+          src="https://colab.research.google.com/assets/colab-badge.svg"
+          alt="Open In Colab"/></a>
+            </p>
+
+      </example>
+
     </subsection>
 
     <subsection id=sums_of_squares><h1>Sums-of-squares optimization</h1>
diff --git a/optimization.ipynb b/optimization.ipynb
index 291d2de7..6dbad2e7 100644
--- a/optimization.ipynb
+++ b/optimization.ipynb
@@ -32,9 +32,97 @@
     "    urlretrieve(f\"http://underactuated.csail.mit.edu/scripts/setup/setup_underactuated_colab.py\",\n",
     "                \"setup_underactuated_colab.py\")\n",
     "    from setup_underactuated_colab import setup_underactuated\n",
-    "    setup_underactuated(underactuated_sha='560c2adace05eb20ebd78377582015d5b2d3859a', drake_version='0.27.0', drake_build='release')"
+    "    setup_underactuated(underactuated_sha='560c2adace05eb20ebd78377582015d5b2d3859a', drake_version='0.27.0', drake_build='release')\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "source": [
+    "# SDP relaxation of non-convex quadratic constraints\n",
+    "\n",
+    "Consider the problem:\n",
+    "\n",
+    "\\begin{align*}\n",
+    "\\min_x \\quad & \\| x - a \\|^2 \\\\\n",
+    "\\text{subject to} \\quad & \\| x - b \\| \\ge 1\n",
+    "\\end{align*}\n",
+    "\n",
+    "We can write this as\n",
+    "\n",
+    "\\begin{align*}\n",
+    "\\min_{x,y} \\quad & y - 2ax + a^2 \\\\\n",
+    "\\text{subject to} \\quad & y - 2bx + b^2 \\ge 1 \\\\\n",
+    "& y \\ge x^2\n",
+    "\\end{align*}\n",
+    "\n",
+    "where we write $y \\ge x^2$ as the semidefinite constraint\n",
+    "$\\begin{bmatrix} y & x \\\\ x & 1 \\end{bmatrix}.$  I've plotted the feasible\n",
+    "region and the objective with an arrow.  As you know, for linear\n",
+    "objectives, the optimal solution will lie on the boundary only if the cost is\n",
+    "directly orthogonal to the objective; otherwise it will lie at a vertex.  So in\n",
+    "this case, the solution will only lie on the interior if $a = b;$ for every\n",
+    "other value, this relaxation will give the optimal solution.  (Try changing $a$ to see)."
+   ],
+   "metadata": {
+    "collapsed": false
+   }
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "outputs": [
+    {
+     "data": {
+      "text/plain": "<Figure size 432x288 with 1 Axes>",
+      "image/png": "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\n"
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "fig, ax = plt.subplots()\n",
+    "a = .8\n",
+    "b = .5\n",
+    "x = np.linspace(-2, 2)\n",
+    "plt.fill(x, np.maximum(x*x, 1 + 2*b*x - b*b), color='lightgray')\n",
+    "ax.plot(x, x*x, 'k')\n",
+    "ax.plot(x, 1 + 2*b*x - b*b, 'k')\n",
+    "qx = 0\n",
+    "qy = 3\n",
+    "scale=4\n",
+    "ax.quiver(qx, qy, 2*a, -1, scale=10, zorder=2, color='b')\n",
+    "\n",
+    "ax.set_xlim(-2, 2)\n",
+    "ax.set_ylim(0, 4)\n",
+    "ax.set_xlabel('x')\n",
+    "ax.set_ylabel('y')\n",
+    "\n",
+    "from pydrake.all import MathematicalProgram, Solve\n",
+    "\n",
+    "prog = MathematicalProgram()\n",
+    "x = prog.NewContinuousVariables(1, 'x')[0]\n",
+    "y = prog.NewContinuousVariables(1, 'y')[0]\n",
+    "\n",
+    "prog.AddPositiveSemidefiniteConstraint(np.array([[y, x],[x, 1]]))\n",
+    "prog.AddLinearConstraint(y - 2*b*x + b*b >= 1)\n",
+    "prog.AddLinearCost(y - 2*a*x + a*a)\n",
+    "\n",
+    "result = Solve(prog)\n",
+    "ax.plot(result.GetSolution(x), result.GetSolution(y), 'b*', markersize=15);"
+   ],
+   "metadata": {
+    "collapsed": false,
+    "pycharm": {
+     "name": "#%%\n"
+    }
+   }
+  },
   {
    "cell_type": "markdown",
    "metadata": {},