{
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Name:  **Your name here**  \n",
    "UID:  **Your student ID num here**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homework 7:  Dual methods \n",
    "Upload a pdf version of your solution to gradescope.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 1:  The conjugate and the dual\n",
    "Consider the \"monotropic\" program\n",
    "\\begin{align}\n",
    "\\text{minimize} & \\quad  \\|x\\|_\\infty \\\\\n",
    "\\text{subject to} & \\quad Ax=b . \\nonumber\n",
    "\\end{align}\n",
    "\n",
    "Write this as an unconstrained (or implicitly constrained) problem using the characteristic function of the zero vector $\\chi_0(z) .$  This function is zero if its argument is zero, and infinite otherwise.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Your answer here...`\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### a) What is the conjugate of $f(z)= \\|z\\|_\\infty$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Your answer here...`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### b) What is the conjugate of $g(z)=\\chi_0(z)$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Your answer here...`\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### c) Using the conjugate functions, write down the dual of the monotropic problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Your answer here...`\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 2:  Linear programming\n",
    "Consider the linear program\n",
    "\\begin{align*}\n",
    "\\text{minimize} \\quad &  c^T x \\\\\n",
    "\\text{subject to} \\quad & Ax=b\\\\\n",
    " & x\\ge 0.\n",
    "\\end{align*}\n",
    "\n",
    "#### a) Write the optimality conditions for this problem (i.e., the KKT system).  Your solution should contain 5 equations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Your answer here...`\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### b) Write the Lagrangian for this problem.\n",
    "Use $\\lambda$ and $\\eta$ are your lagrange multipliers for the equality and inequality constraints, respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Your answer here...`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### c) Minimize out the primal variables in the Lagrangian, and write the dual formulation of this linear program."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Your answer here...`\n",
    "\n",
    " \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 3: Sensitivity bound\n",
    "Consider the problem\n",
    "\\begin{align}\n",
    "\\text{minimize} \\quad &  f(x) \\\\\n",
    "\\text{subject to}\\quad & g(x) \\le 0.\n",
    "\\end{align}\n",
    "Let $x_0$ be a solution to this problem, and $\\lambda_0$ be the corresponding optimal Lagrange multiplier. Now, define a perturbed problem\n",
    "  \\begin{align}\n",
    "\\text{minimize}\\quad &  f(x) \\\\\n",
    "\\text{subject to}\\quad & g(x) \\le \\epsilon\n",
    "\\end{align}\n",
    "where $\\epsilon$ is a vector. Let $x_\\epsilon$ be a solution to the perturbed problem.  Note, if we put large negative values in $\\epsilon$, then  the constraint set gets smaller, and we expect the corresponding value of $f(x_\\epsilon)$ to increase.  \n",
    "\n",
    "Prove the \"sensitivity bound\"\n",
    "$$ f(x_0) -   \\lambda_0^T \\epsilon \\le f(x_\\epsilon). $$\n",
    "This bound shows that the Lagrange multipliers determine how much the objective increases as the vector $\\epsilon$ becomes more negative."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Your answer here...`"
   ]
  }
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