{
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  {
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   "source": [
    "# Specifying Objectives"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6f47e3b",
   "metadata": {},
   "source": [
    "Below, we show how to specify an objective when dealing with a COP (Constraint Optimization Problem) instance. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ab301e7",
   "metadata": {},
   "source": [
    "**Remark.** In PyCSP$^3$, which is currently targeted to XCSP$^3$-core, we currently only consider mono-objective optimization. Multi-objective optimization will be addressed in the future."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40a6abde",
   "metadata": {},
   "source": [
    "To see how it works, we need first to import the library PyCSP$^3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c1b9eff6",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pycsp3 import *"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c361b8ea",
   "metadata": {},
   "source": [
    "For specifying an objective to optimize, you must call one of the two functions:\n",
    "- *minimize()*\n",
    "- *maximize()*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f6a8b808",
   "metadata": {},
   "source": [
    "We can pass a parameter to either functions, which can be:\n",
    "- a variable, as in: \n",
    "```\n",
    "minimize(v)\n",
    "```\n",
    "- an expression, as in:\n",
    "```\n",
    "minimize(v + w * w)\n",
    "```\n",
    "- a sum, as in:\n",
    "```\n",
    "minimize(Sum(x))\n",
    "```\n",
    "- a dot product, as in:\n",
    "```\n",
    "minimize([u,v,w] * [3, 2, 5])\n",
    "```\n",
    "- a generator, as in:\n",
    "```\n",
    "minimize(Sum((x[i] > 1) * c[i] for i in range(n))) \n",
    "```\n",
    "- a minimum, as in:\n",
    "```\n",
    "minimize(Minimum(x))\n",
    "```\n",
    "- a maximum, as in:\n",
    "```\n",
    "minimize(Maximum(x))\n",
    "```\n",
    "- a number of distinct values, as in:\n",
    "```\n",
    "minimize(NValues(x))\n",
    "```\n",
    "- ..."
   ]
  },
  {
   "cell_type": "raw",
   "id": "0c0c1d6c",
   "metadata": {},
   "source": [
    "For our illustration, let us declare an array x of 6 variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "bcf8a597",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = VarArray(size=6, dom=range(10))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1614c37e",
   "metadata": {},
   "source": [
    "We can decide to minimize the sum of values assigned to the variables of x. We just write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "91ed5c76",
   "metadata": {},
   "outputs": [],
   "source": [
    "minimize(\n",
    "    Sum(x)\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c8adb151",
   "metadata": {},
   "source": [
    "To control things, we can display the internal representation of the objective as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "6f0d4cff",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "minimize(sum(list:[x[0], x[1], x[2], x[3], x[4], x[5]]))\n"
     ]
    }
   ],
   "source": [
    "print(objective())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a68bf09",
   "metadata": {},
   "source": [
    "If ever we decide to maximize the number of distinct values assigned to the variables of x, because any new call to *minimize()* or *maximize()* overwrites the previous objective (if one was posted), we can write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c742abab",
   "metadata": {},
   "outputs": [],
   "source": [
    "maximize(\n",
    "    NValues(x)\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19c17f6a",
   "metadata": {},
   "source": [
    "We can control that we have a new objective."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "5dc43b57",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "maximize(nValues(list:[x[0], x[1], x[2], x[3], x[4], x[5]]))\n"
     ]
    }
   ],
   "source": [
    "print(objective())"
   ]
  }
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