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   "source": [
    "Name: **Your name here**  \n",
    "UID: **Your student ID num here**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homework 4:  Convex Functions  \n",
    "\n",
    "Use either the slides, or Boyd $\\S$3.2 as a reference.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 1\n",
    "For this problem, you may use any of the results/rules from the lecture slides.\n",
    "\n",
    "### (a) Consider the function\n",
    " $$f(x) = \\begin{cases}\n",
    "0, \\text{ if } x<0\\\\\n",
    "\\infty , \\text{ if } x\\ge 0\\\\\n",
    "\\end{cases}.$$\n",
    "\n",
    "Is this function closed and proper?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Your work here*\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### (b) Consider the set of $n\\times n$ low-rank matrices $\\{A\\in\\mathbb{R}^{n\\times n} | \\, \\text{rank}(A)\\le k \\},$ where $0<k<n$.   Is this set convex?  Why?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Your work here*\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### (c) Suppose $\\{A_i\\}$ are matrices of identical size.  Define the set $$\\{X \\,\\,|\\,\\,  \\text{trace}(A_i^T X A_i) \\le 0, \\,\\, \\forall i \\}.$$\n",
    "Is this set of matrices convex?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Your work here*\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### (d) Let $C \\subset \\mathbb{R}^n$ be a compact (closed and bounded), non-empty convex set.  For a scalar $x,$ let C(x) be the set of all points in $C$ whose first coordinate is closer to x than any other point. More precisely: $C(x)=\\{a\\in C \\, |\\,  \\forall b \\in C , |a_1-x|\\le |b_1-x|  \\}.$  Is $C(x)$ a convex set for any choice of $x$? Why?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Your work here*\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### (e)  Let $\\|\\cdot\\|$ be the vector 2-norm.  Is the set $C = \\{(x,y)\\, |\\,\\, \\|x\\| \\le  y \\}$ convex?  Is it a cone?  Why?  What about the set $C = \\{(x,y)\\, |\\,\\, \\|x\\|^2 \\le  y \\}?$  Is this convex? Is this a  cone?  Here, $x$ is a vector and $y$ is a scalar."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Your work here*\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2\n",
    "### (a) Suppose the function $f(x,y)$ is convex.  Prove that \n",
    "  $$g(x) = \\min_y f(x,y)$$\n",
    "  is also convex, where $x,y\\in \\mathbb{R}^n.$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Your work here*\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### (b)  Suppose $f$ is strongly convex.  Show that the gradient of $f$ is strongly monotone, i.e., there is a constant $m>0$ with \n",
    "  $$ \\langle x-y,\\nabla f(x) - \\nabla f(y) \\rangle \\ge m\\|x-y\\|^2.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Your work here*\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 3\n",
    " Suppose that $g(x)$ is convex and $h(x)$ is concave (i.e. $-h(x)$ is convex).  Suppose we restrict both functions into a closed, convex set $C$ such that both $g(x)$ and $h(x)$ are always positive when $x \\in C.$  Prove that the function\n",
    "  $f(x) = g(x)/h(x)$ \n",
    "  is quasi-convex.  (note: It follows that every local minimum is also global)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Your work here*\n"
   ]
  }
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