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  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "bbdb5d29",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy.optimize import minimize"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "102c0edd",
   "metadata": {},
   "source": [
    "Example:\n",
    "\n",
    "Objective Function:\n",
    "$$\n",
    "\\min x^2_1 + x_1 \\times x_2\n",
    "$$\n",
    "s.t. Constraints:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "x^3_1 + x_1 \\times x_2 & = 100 \\\\\n",
    "x^2_1 + x_2 \\leq 50 \\\\\n",
    "-100 \\leq x_1, x_2 & \\le 100\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f3a4de91",
   "metadata": {},
   "outputs": [],
   "source": [
    "# define the objective function and constraints\n",
    "def objective_fcn(x):\n",
    "    # parameters:\n",
    "    x1 = x[0]\n",
    "    x2 = x[1]\n",
    "    return x1**2 + x1 * x2\n",
    "\n",
    "def equality_constraint(x):\n",
    "    x1 = x[0]\n",
    "    x2 = x[1]\n",
    "    return x1**3 + x1 * x2 - 100   # Remember that we need to put everything from the RHS to the LHS so its = 0\n",
    "\n",
    "def inequality_constraint(x):\n",
    "    x1 = x[0]\n",
    "    x2 = x[1]\n",
    "    return x1**2 + x2 - 50\n",
    "\n",
    "# Note: could also make all of these lambda functions\n",
    "\n",
    "bounds_x1 = (-100, 100)\n",
    "bounds_x2 = (-100, 100)\n",
    "# we need to make it into a list\n",
    "bounds = [bounds_x1, bounds_x2]\n",
    "\n",
    "# we need to define the equality types \n",
    "constraint1 = {'type': 'eq', 'fun': equality_constraint}\n",
    "constraint2 = {'type': 'ineq', 'fun': inequality_constraint}\n",
    "\n",
    "constraints = [constraint1, constraint2]\n",
    "\n",
    "# finally, give an initial value for x0. As long as it's reasonable, any pairs of numbers in between the\n",
    "# bounds should be ok\n",
    "x0 = [1,1]\n",
    "\n",
    "# now we apply the minimization function. \n",
    "# scipy.optimize.minimize(fun, x0, method, bounds, constraints)\n",
    "\n",
    "result = minimize(objective_fcn, x0, method='SLSQP', bounds=bounds, constraints=constraints)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "3c945da9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "     fun: 96.00000027660178\n",
       "     jac: array([50.00000095,  2.        ])\n",
       " message: 'Optimization terminated successfully'\n",
       "    nfev: 32\n",
       "     nit: 10\n",
       "    njev: 10\n",
       "  status: 0\n",
       " success: True\n",
       "       x: array([ 1.99999997, 46.00000084])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b637a70",
   "metadata": {},
   "source": [
    "The result returns:\n",
    "* fun = values of objective function\n",
    "* jac = jacobian\n",
    "* message = description of the cause of the termination\n",
    "* nfev, njev = No. of evaluations of the objective functions and of its Jacobian\n",
    "* nit = NO. of iterations\n",
    "* status, success = Termination and exit status\n",
    "* x = Solution of the optimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5b710a15",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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