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    "# Constrained optimization using scipy\n",
    "\n",
    "**Randall Romero Aguilar, PhD**\n",
    "\n",
    "This demo is based on the original Matlab demo accompanying the  <a href=\"https://mitpress.mit.edu/books/applied-computational-economics-and-finance\">Computational Economics and Finance</a> 2001 textbook by Mario Miranda and Paul Fackler.\n",
    "\n",
    "Original (Matlab) CompEcon file: **demopt08.m**\n",
    "\n",
    "Running this file requires the Python version of CompEcon. This can be installed with pip by running\n",
    "\n",
    "    !pip install compecon --upgrade\n",
    "\n",
    "<i>Last updated: 2021-Oct-01</i>\n",
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## About\n",
    "\n",
    "The problem is\n",
    "\n",
    "\\begin{equation*}\n",
    "\\max\\{-x_0^2 - (x_1-1)^2 - 3x_0 + 2\\}\n",
    "\\end{equation*}\n",
    "\n",
    "subject to\n",
    "\n",
    "\\begin{align*}\n",
    "4x_0 + x_1 &\\leq 0.5\\\\\n",
    "x_0^2 + x_0x_1 &\\leq 2.0\\\\\n",
    "x_0 &\\geq 0 \\\\\n",
    "x_1 &\\geq 0\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using scipy\n",
    "\n",
    "The **scipy.optimize.minimize** function minimizes functions subject to equality constraints, inequality constraints, and bounds on the choice variables. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.optimize import minimize\n",
    "\n",
    "np.set_printoptions(precision=4,suppress=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* First, we define the objective function, changing its sign so we can minimize it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x[0]**2 + (x[1]-1)**2 + 3*x[0] - 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Second, we specify the inequality constraints using a tuple of two dictionaries (one per constraint), writing each of them in the form $g_i(x) \\geq 0$, that is\n",
    "\\begin{align*}\n",
    "0.5 - 4x_0 - x_1 &\\geq 0\\\\\n",
    "2.0 - x_0^2 - x_0x_1 &\\geq 0\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "cons = ({'type': 'ineq', 'fun': lambda x: 0.5 - 4*x[0] - x[1]},\n",
    "       {'type': 'ineq', 'fun': lambda x: 2.0 - x[0]**2 - x[0]*x[1]})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Third, we specify the bounds on $x$:\n",
    "\\begin{align*}\n",
    "0 &\\leq x_0 \\leq \\infty\\\\\n",
    "0 &\\leq x_1 \\leq \\infty\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "bnds = ((0, None), (0, None))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Finally, we minimize the problem, using the SLSQP method, starting from $x=[0,1]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "     fun: -1.7499999999999876\n",
      "     jac: array([ 3., -1.])\n",
      " message: 'Optimization terminated successfully'\n",
      "    nfev: 10\n",
      "     nit: 3\n",
      "    njev: 3\n",
      "  status: 0\n",
      " success: True\n",
      "       x: array([0. , 0.5])\n"
     ]
    }
   ],
   "source": [
    "x0 = [0.0, 1.0]\n",
    "res = minimize(f, x0, method='SLSQP', bounds=bnds, constraints=cons)\n",
    "print(res)"
   ]
  }
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