authors/marcos-dv/geometry/largest_small_polygon.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cvozYYJ3vA5s"
      },
      "source": [
        "# Largest small polygon\n",
        "[![largest_small_polygon.ipynb](https://img.shields.io/badge/github-%23121011.svg?logo=github)](https://github.com/ampl/colab.ampl.com/blob/master/authors/marcos-dv/geometry/largest_small_polygon.ipynb) [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/ampl/colab.ampl.com/blob/master/authors/marcos-dv/geometry/largest_small_polygon.ipynb) [![Kaggle](https://kaggle.com/static/images/open-in-kaggle.svg)](https://kaggle.com/kernels/welcome?src=https://github.com/ampl/colab.ampl.com/blob/master/authors/marcos-dv/geometry/largest_small_polygon.ipynb) [![Gradient](https://assets.paperspace.io/img/gradient-badge.svg)](https://console.paperspace.com/github/ampl/colab.ampl.com/blob/master/authors/marcos-dv/geometry/largest_small_polygon.ipynb) [![Open In SageMaker Studio Lab](https://studiolab.sagemaker.aws/studiolab.svg)](https://studiolab.sagemaker.aws/import/github/ampl/colab.ampl.com/blob/master/authors/marcos-dv/geometry/largest_small_polygon.ipynb) [![Hits](https://h.ampl.com/https://github.com/ampl/colab.ampl.com/blob/master/authors/marcos-dv/geometry/largest_small_polygon.ipynb)](https://colab.ampl.com)\n",
        "\n",
        "Description: lecture about models for the Largest Small Polygon Problem\n",
        "\n",
        "Tags: geometry, non-linear, amplpy, ipopt, educational\n",
        "\n",
        "Notebook author: Marcos Dominguez Velad <<marcos@ampl.com>>\n",
        "\n",
        "Model author: Marcos Dominguez Velad <<marcos@ampl.com>> et al.\n",
        "\n",
        "References:\n",
        "1. Bingane, C. (2022). *Largest small polygons: A sequential convex optimization approach*. Optimization Letters, 1-13.\n",
        "2. Graham, R. L. (1975). *The largest small hexagon*. Journal of Combinatorial Theory, Series A, 18(2), 165-170.\n",
        "3. Reinhardt, K. (1922). *Extremale Polygone gegebenen Durchmessers*. Jahresbericht der Deutschen Mathematiker-Vereinigung, 31, 251-270.\n",
        "4. [Netlib repository](https://netlib.org/ampl/models/)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "fd_HVI1VvA5x"
      },
      "outputs": [],
      "source": [
        "# Install dependencies\n",
        "%pip install -q amplpy"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "b213Q6XzvA5y"
      },
      "outputs": [],
      "source": [
        "# Google Colab & Kaggle integration\n",
        "from amplpy import AMPL, ampl_notebook\n",
        "\n",
        "ampl = ampl_notebook(\n",
        "    modules=[\"open\", \"knitro\"],  # modules to install\n",
        "    license_uuid=\"default\",  # license to use\n",
        ")  # instantiate AMPL object and register magics"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RxZ4DFi1vA50"
      },
      "source": [
        "A small polygon is a unit diameter polygon. Finding the polygon with maximum area is known as the \"Largest Small Polygon\" (LSP) problem. We are proposing some non-linear formulations to find the small polygon that maximizes the area for a given number of sides to solve the problem computationally. This is an open geometry problem, so we are providing a computational and approximate way to tackle it. \n",
        "\n",
        "## Problem statement\n",
        "\n",
        "Given a number *n*, find the n-sided polygon that has diameter one and that has the largest area among all diameter-one n-gons. The diameter of a polygon is the maximum distance between any two vertices in the polygon, and distances between points are assumed to be Euclidean distances.\n",
        "\n",
        "### Some theoretical results\n",
        "\n",
        "The LSP problem has been solved for odd number of sides. In 1922, Reinhardt showed that the optimal solution for $n = 2k+1$ is the regular n-gon. The area of the regular unit n-gon is:\n",
        "\n",
        "$$\\frac{n}{2}(sin\\frac{\\pi}{n}-tan\\frac{\\pi}{2n}) \\text{ ,   when n is odd}$$\n",
        "\n",
        "$$\\frac{n}{8}sin\\frac{2\\pi}{n} \\text{ ,   when n is even}$$\n",
        "\n",
        "These are lower bounds for our problem, but the even one might be improved from the regular polygon of $n-1$ sides:\n",
        "\n",
        "* Take the odd regular $n-1$-gon. Add a vertex (at distance 1) along the bisectriz of any angle.\n",
        "\n",
        "* The obtained polygon will be larger than the even regular $n$-gon. This fact proves that the regular polygon is not optimal for $n \\geq 6$.\n",
        "\n",
        "Although LSP has not been solved for general even number of sides, there are some conjectures related to the solutions and variants of the problem that assume some kind of symmetry in terms of the *diameter graph* of the polygon (the diameter graph of a small polygon is the graph with the vertices of the polygon, where vertices are connected if and only if the distance between them is equal to 1).\n",
        "\n",
        "## Model with polar coordinates\n",
        "\n",
        "The problem can be stated and solved by describing the problem in an straightforward way, formulating a non linear problem in polar coordinates. We are proposing a similar model to the one in the [netlib repository](https://netlib.org/ampl/models/), an AMPL model (by David M. Gay) that started as a GAMS model by Francisco J. Prieto. This model solves the general *n* instance of the problem.\n",
        "\n",
        "### Variables\n",
        "\n",
        "If $(x_i, y_i)$ are the cartesian coordinates (real numbers) of the points, they will be the vertices of a small polygon if the distances between them are less or equal to 1.\n",
        "\n",
        "* For all $i = 1..n, j = 1..i-1$, \n",
        "\n",
        "$$\\sqrt{(x_i-x_j)^2+(y_i-y_j)^2} \\leq 1$$\n",
        "\n",
        "That is equivalent to:\n",
        "\n",
        "$$(x_i-x_j)^2+(y_i-y_j)^2 \\leq 1$$\n",
        "\n",
        "\n",
        "### Compute area\n",
        "\n",
        "Given the *n* vertices of a polygon, we can compute the area of the polygon by using the [Shoelace formula](https://en.wikipedia.org/wiki/Shoelace_formula), like a 2D determinant:\n",
        "\n",
        "<figure>\n",
        "  <img src=\"https://drive.google.com/uc?id=1tqWpAcXnsFxzY0Tw7y38DrUQn-CW0ESK\" width=\"600\" height=\"360\">\n",
        "</figure>\n",
        "\n",
        "Example:\n",
        "\n",
        "The area of the polygon defined with the vertices\n",
        "{(-1,0), (0.2,1), (2,0.7), (2,-1)} is\n",
        "\n",
        "<img src=\"https://drive.google.com/uc?id=1WAtQLPap-GUOLQFtld-EP3Kfk8RQvpsy\" width=\"200\" height=\"140\">\n",
        "\n",
        "$$\\frac{1}{2}|-1\\times1+0.2\\times0.7+2\\times(-1)+2\\times0-0\\times0.2-1\\times2-0.7\\times2-(-1)\\times(-1)| = 3.63$$\n",
        "\n",
        "The formula works when the points are sorted counter clock-wise or clock-wise, then, in order to use the formula is is necessary to ensure the order of the points along the perimeter. How could we impose an order among the vertices, so first vertex (x1, y1) is connected to the second (x2, y2), which at the same time is connected to (x3, y3)... and so on until closing the polygon?\n",
        "\n",
        "$$\\frac{1}{2}\\sum_{i=1}^{n-1}(x_i \\cdot y_{i+1} - y_i \\cdot x_{i+1})+\\frac{1}{2}(x_n \\cdot y_1 - y_n \\cdot x_1)$$\n",
        "\n",
        "### Polar coordinates to sort vertices\n",
        "\n",
        "We need to sort vertices somehow. Intuitively, we could sort them by angle respect to any point in the plane.\n",
        "\n",
        "We are using polar coordinates to sort vertices by angle in a straightforward way. We are sorting respect to (0,0). Each cartesian point $(x,y)$ distinct to (0,0) is matched to a polar point $(r, t)$, where $r$ is the radius of the point respect to the origin, and $t$ the angle respect to the $X$ axis, with $0 < r$, and $0 \\leq t < 2\\pi$.\n",
        "\n",
        "$$x = r \\cdot cos(t)$$\n",
        "$$y = r \\cdot sin(t)$$\n",
        "\n",
        "We can substitute $x$ and $y$ as a function of $r$ and $t$, adding sorting constraints as:\n",
        "\n",
        "* For all $i = 2..n$,\n",
        "$$t_i >= t_{i-1}$$\n",
        "\n",
        "Two vertices having the same angle would cause a suboptimal solution as there would be three points in the same line (origin vertex and two others).\n",
        "\n",
        "### Complete model\n",
        "\n",
        "Substitute $x$ and $y$ by their polar transformation coordinates. Notice that is is assumed vertex (x1,y1) = (0,0) = (r1, t1), as we can always fix one of the vertices.\n",
        "\n",
        "* Variables, for all $i = 1..n$:\n",
        "$$r_i >= 0$$\n",
        "$$0 <= t_i <= \\pi$$\n",
        "\n",
        "With the $\\pi$ bound over $t$ we force the polygon to happen over the first semiplane (y >= 0).\n",
        "\n",
        "* Objective: maximize area,\n",
        "\n",
        "$$\\frac{1}{2}\\sum_{i=1}^{n-1}(r_i \\cdot cos(t_i) \\cdot r_{i+1} \\cdot sin(t_{i+1}))-(r_i \\cdot sin(t_i) \\cdot r_{i+1} \\cdot cos(t_{i+1}))+\\frac{1}{2}(r_n \\cdot cos(t_n) \\cdot r_1 \\cdot sin(t_1))-(r_n \\cdot sin(t_n) \\cdot r_1 \\cdot cos(t_1))$$\n",
        "\n",
        "We can simplify the previous expression. Rearrange each term:\n",
        "\n",
        "$$r_i \\cdot cos(t_i) \\cdot r_{i+1} \\cdot sin(t_{i+1})-r_i \\cdot sin(t_i) \\cdot r_{i+1} \\cdot cos(t_{i+1}) = $$\n",
        "$$ = r_i r_{i+1} (cos(t_i) \\cdot sin(t_{i+1})-sin(t_i) \\cdot cos(t_{i+1})$$\n",
        "Sine of the sum of $t_{i+1}$, $-t_{i}$:\n",
        "$$r_i r_{i+1} sin(t_{i+1}-t_i)$$\n",
        "\n",
        "The objective can be rewritten as:\n",
        "$$\\frac{1}{2}\\sum_{i=1}^{n-1}(r_i r_{i+1} sin(t_{i+1}-t_i))+\\frac{1}{2}(r_1 r_n \\cdot sin(t_1-t_n))$$\n",
        "\n",
        "* Constraints:\n",
        "\n",
        "1. Distance between vertices. For all $i = 1..n, j = i+1..n$, \n",
        "\n",
        "$$(r_i \\cdot cos(t_i)-r_j \\cdot cos(t_j))^2+(r_i \\cdot sin(t_i)^2-r_j \\cdot sin(t_j))^2 \\leq 1$$\n",
        "\n",
        "$$(r_i \\cdot cos(t_i)-r_j \\cdot cos(t_j))^2 = r_i^2 cos^2(t_i) + r_j^2 cos^2(t_j) - 2r_ir_jcos(t_i)cos(t_j)$$\n",
        "$$(r_i \\cdot sin(t_i)-r_j \\cdot sin(t_j))^2 = r_i^2 sin^2(t_i) + r_j^2 sin^2(t_j) - 2r_ir_jsin(t_i)sin(t_j)$$\n",
        "\n",
        "Sum both:\n",
        "\n",
        "$$r_i^2 + r_j^2 - 2r_ir_j(cos(t_i)cos(t_j) + sin(t_i)sin(t_j)) = r_i^2 + r_j^2 - 2r_ir_j cos(t_i-t_j)$$\n",
        "\n",
        "So the constraints could be rewritten as:\n",
        "\n",
        "$$r_i^2 + r_j^2 - 2r_ir_j cos(t_i-t_j) \\leq 1$$\n",
        "\n",
        "This way, non-linear expressions in the model are simpler, there are less sines and cosines, so it will be easier to find better solutions more efficiently.\n",
        "\n",
        "2. Angle order. For all $i = 2..n$, \n",
        "\n",
        "$$t_i >= t_{i-1}$$\n",
        "\n",
        "Finally, we can fix one of the vertices:\n",
        "\n",
        "$$r_1 = 0, \\ t_1 = 0$$\n",
        "\n",
        "### AMPL model\n",
        "\n",
        "An `area_lb` parameter is introduced so we can add better lower bounds for the area. This way it is easier to deal with suboptimal and degenerated solutions. By default, the lower bound is 0."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "g2hQVY8VvA56",
        "outputId": "a03a64b9-30fc-48fe-870b-cb066c6e3bef"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Overwriting polar_polygon.mod\n"
          ]
        }
      ],
      "source": [
        "%%writefile polar_polygon.mod\n",
        "\n",
        "# Number of sides\n",
        "param N integer > 0;\n",
        "set I := 1..N;\n",
        "\n",
        "param pi := 4*atan(1.);\n",
        "param max_diam := 1;\n",
        "# area lower bound, might help to avoid degenerated solutions\n",
        "param area_lb default 0;\n",
        "\n",
        "# Variables\n",
        "var r{i in I} <= max_diam, >= 0\t# polar radius (distance to origin)\n",
        "\t\t:=  4*i*(N + 1 - i)/(N+1)**2; # initial guess\n",
        "\n",
        "var t{i in I} >= 0 <= pi # polar angle\n",
        "\t\t:= pi*i/N; # uniform distribution over the positive semicircle\n",
        "\n",
        "var area >= area_lb; # objective\n",
        "\n",
        "# Constraints\n",
        "# Distance between vertices\n",
        "s.t. vertices_distance{i in 1..N, j in i+1 .. N}:\n",
        "\tr[i]**2 + r[j]**2 - 2*r[i]*r[j]*cos(t[j]-t[i]) <= max_diam*max_diam;\n",
        "\n",
        "# Sort by polar angle\n",
        "s.t. sort_vertices{i in 2..N}:\n",
        "\tt[i] >= t[i-1];\n",
        "\n",
        "# Fixed vertex\n",
        "s.t. fix_angle: t[1] = 0;\n",
        "s.t. fix_radius:   r[1] = 0;\n",
        "s.t. area_formula: area = .5*sum{i in 1..N-1} r[i+1]*r[i]*sin(t[i+1]-t[i]) + .5*(r[1]*r[N]*sin(t[N]-t[1]));\n",
        "\n",
        "maximize total_area: area;"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LaawhvbEzRUE"
      },
      "source": [
        "We use `ipopt` (version 3.12.13) open source solver for cloud usage. The output is very verbose, so we set \"outlev=0\"."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "i29_flLkvA57",
        "outputId": "034be4b7-4b0e-41d2-a3d8-43d3e1e60486"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Ipopt 3.12.13: outlev=0\n",
            "\n",
            "\n",
            "******************************************************************************\n",
            "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
            " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
            "         For more information visit http://projects.coin-or.org/Ipopt\n",
            "******************************************************************************\n",
            "\n",
            " \n",
            "Ipopt 3.12.13: Optimal Solution Found\n",
            "\n",
            "suffix ipopt_zU_out OUT;\n",
            "suffix ipopt_zL_out OUT;\n"
          ]
        }
      ],
      "source": [
        "%%ampl_eval\n",
        "reset;\n",
        "model polar_polygon.mod;\n",
        "# option solver knitro; # try it locally\n",
        "option solver ipopt;\n",
        "option ipopt_options 'outlev=0';\n",
        "let N := 10;\n",
        "solve;"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "56uihJVdvA57"
      },
      "source": [
        "Now that we have the solution, we are going to plot it with `matplotlib`.\n",
        "\n",
        "We define the `solve_polar()` method to call AMPL and solve the problem for $N$ sides. `polar2cart()` transforms from polar to cartesian coordinates. `plot_polygon()` draws the result."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "BsHZmZOHvA58"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "\n",
        "# solve the problem for n sides, return solution (polar coords) and area\n",
        "def solve_polar(n, prev_sol=0):\n",
        "    ampl.reset()\n",
        "    ampl.read(\"polar_polygon.mod\")\n",
        "    ampl.param[\"N\"] = n\n",
        "    ampl.param[\"area_lb\"] = prev_sol\n",
        "    ampl.solve()\n",
        "\n",
        "    # Get result from AMPL\n",
        "    r = [ampl.var[\"r\"][i].value() for i in range(1, n + 1)]\n",
        "    t = [ampl.var[\"t\"][i].value() for i in range(1, n + 1)]\n",
        "\n",
        "    # To plot easier, repeat the first point in the last position\n",
        "    # (to close the polygon)\n",
        "    r.append(r[0])\n",
        "    t.append(t[0])\n",
        "\n",
        "    return r, t, ampl.obj[\"total_area\"].value()\n",
        "\n",
        "\n",
        "# Transformation from polar to cartesian\n",
        "def polar2cart(r, angle):\n",
        "    xs = np.array(r) * np.sin(np.array(angle))\n",
        "    ys = np.array(r) * np.cos(np.array(angle))\n",
        "    return xs, ys\n",
        "\n",
        "\n",
        "def plot_polygon(sides, xs, ys, area):\n",
        "    print(\"\\nSides: \", sides)\n",
        "    print(\"Area: \", area)\n",
        "    plt.figure()\n",
        "    plt.plot(xs, ys)\n",
        "    plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "HsX4aYwtR9Qq"
      },
      "source": [
        "In the below loop, solve the problem for all the sides between `min_sides` and `max_sides`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "9d6dO6k4vA59",
        "outputId": "04af511b-22e1-46dc-ef50-beab55d69082"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\n",
            "Problem size:  10\n",
            "Ipopt 3.12.13: outlev=0\n",
            "\n",
            "\n",
            "******************************************************************************\n",
            "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
            " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
            "         For more information visit http://projects.coin-or.org/Ipopt\n",
            "******************************************************************************\n",
            "\n",
            " \n",
            "Ipopt 3.12.13: Optimal Solution Found\n",
            "\n",
            "suffix ipopt_zU_out OUT;\n",
            "suffix ipopt_zL_out OUT;\n",
            "\n",
            "Problem size:  11\n",
            "Ipopt 3.12.13: outlev=0\n",
            "\n",
            "\n",
            "******************************************************************************\n",
            "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
            " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
            "         For more information visit http://projects.coin-or.org/Ipopt\n",
            "******************************************************************************\n",
            "\n",
            " \n",
            "Ipopt 3.12.13: Optimal Solution Found\n",
            "\n",
            "suffix ipopt_zU_out OUT;\n",
            "suffix ipopt_zL_out OUT;\n",
            "\n",
            "Problem size:  12\n",
            "Ipopt 3.12.13: outlev=0\n",
            "\n",
            "\n",
            "******************************************************************************\n",
            "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
            " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
            "         For more information visit http://projects.coin-or.org/Ipopt\n",
            "******************************************************************************\n",
            "\n",
            " \n",
            "Ipopt 3.12.13: Optimal Solution Found\n",
            "\n",
            "suffix ipopt_zU_out OUT;\n",
            "suffix ipopt_zL_out OUT;\n",
            "\n",
            "Sides:  10\n",
            "Area:  0.7491373270801958\n"
          ]
        },
        {
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        },
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\n",
            "Sides:  11\n",
            "Area:  0.7587484213086267\n"
          ]
        },
        {
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        },
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\n",
            "Sides:  12\n",
            "Area:  0.760729846642839\n"
          ]
        },
        {
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "min_sides = 10\n",
        "max_sides = 12\n",
        "\n",
        "# save previous solutions to plot\n",
        "solutions = {}\n",
        "# save last optimal value to be used as lower bound\n",
        "prev_sol = 0\n",
        "\n",
        "for sides in range(min_sides, max_sides + 1):\n",
        "    print(\"\\nProblem size: \", sides)\n",
        "    solutions[sides] = solve_polar(sides, prev_sol)\n",
        "    prev_sol = solutions[sides][2]\n",
        "\n",
        "for sides in range(min_sides, max_sides + 1):\n",
        "    radius, angle, area = solutions[sides]\n",
        "    xs, ys = polar2cart(radius, angle)\n",
        "    plot_polygon(sides, xs, ys, area)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DPYJ_aDJvA5-"
      },
      "source": [
        "## Second model with cartesian coordinates\n",
        "\n",
        "There is a way to deal with orientation without using polar coordinates. It is easy to check if three points within the plane are clockwise or counter-clockwise oriented. We are going to undo the polar transformation and use new angle constraints, so that the objective and the distance constraints get simplified.\n",
        "\n",
        "### Three points orientation\n",
        "\n",
        "Given three points, $A=(A_x, A_y)$, $B=(B_x,B_y)$, $C=(C_x,C_y)$, we can compute the slopes of the vectors $AB$ and $BC$.\n",
        "\n",
        "<img src=\"https://drive.google.com/uc?export=view&id=1Hoz9MGwWOkqp1gKuxtAgPN_eniDgXl1v\" width=\"500\" height=\"390\">\n",
        "\n",
        "$$AB_s = \\frac{B_y-A_y}{B_x-A_x}$$\n",
        "$$BC_s = \\frac{C_y-B_y}{C_x-B_x}$$\n",
        "\n",
        "If both slopes are positive, BC's slope being less (greater) than AB's implies that the three points will be (counter-)clockwise oriented. The comparison could be flipped if some of the slopes is negative, but in the end, the following expression let's us know the orientation:\n",
        "\n",
        "$$AB_s-BC_s = \\frac{B_y-A_y}{B_x-A_x}-\\frac{C_y-B_y}{C_x-B_x} = \\frac{(B_y-A_y)(C_x-B_x)-(C_y-B_y)(B_x-A_x)}{(B_x-A_x)(C_x-B_x)}$$\n",
        "\n",
        "* If the numerator is positive, points are clockwise.\n",
        "* If the numerator is negative, points are counter-clockwise.\n",
        "* If the numerator is 0, points belong to the same line.\n",
        "\n",
        "### Orientation in the model\n",
        "\n",
        "As we are fixing one of the vertices (0,0), we can force all the other points to have the same orientation. For $i = 2..N-1$, we can force the same orientation for $A=(0,0)$, $B=(x_i,y_i)$, and $C=(x_{i+1},y_{i+1})$. This way, vertices are going to end up ordered. By using the previous formula, this is possible if we have the same sign in the following expression for each $i$:\n",
        "\n",
        "$$(B_y-A_y)(C_x-B_x)-(C_y-B_y)(B_x-A_x) = y_i(x_{i+1}-x_i)-(y_{i+1}-y_i)x_i = $$\n",
        "$$ = y_i x_{i+1}- y_i x_i - x_i y_{i+1} + x_i y_i = y_i x_{i+1} - x_i y_{i+1}$$\n",
        "\n",
        "Then, we can simply add the following constraint to order our vertices counter-clockwise. For each $i = 2..N-1$:\n",
        "\n",
        "$$y_i x_{i+1} - x_i y_{i+1} < 0$$\n",
        "$$x_i y_{i+1} - y_i x_{i+1} > 0$$\n",
        "\n",
        "By simplicity, in the model we are using the non-strict inequality. Three vertices should not be places in the same segment as that is a suboptimal solution. It would be suboptimal as the point in the middle is pointless, but the polygon could be bigger by using it (just place it in the mediatrix of two other vertices).\n",
        "\n",
        "For further information about this formula and orientation of polygons, take a look to [Graham's scan Algorithm](https://en.wikipedia.org/wiki/Graham_scan), where the formula is used to compute convex hulls.\n",
        "\n",
        "### Area computation with triangles\n",
        "\n",
        "Now, we can use cartesian coordinates as variables, and simplify constraints and objective (by removing sines and cosines).\n",
        "\n",
        "If we project a ray from the fixed vertex of the polygon to each other vertex, we are generating triangles whose area is (half of the area of the parallelogram computed via scalar product):\n",
        "\n",
        "$$u_i = \\frac{1}{2}(x_i y_{i+1} - y_i x_{i+1})$$\n",
        "\n",
        "<img src=\"https://drive.google.com/uc?export=view&id=1FzeP6wz6h7ZdbS68aSVAdMrjE20ZAT4W\" width=\"400\" height=\"360\">\n",
        "\n",
        "The area of the polygon is equal to the sum of areas of these triangles, so we could add continuous variables $u_i$ for $i = 2..N-1$, where $u_i$ is the area of the triangle that contains vertices $(0,0)$, $(x_i,y_i)$, and $(x_{i+1},y_{i+1})$. These variables are non-negative, and at optimality, they should be equal to the area of their triangle.\n",
        "\n",
        "Then, the objective function is rewritten as:\n",
        "\n",
        "$$maximize \\quad \\sum_{i=2..N-1} u_i$$\n",
        "\n",
        "$u_i$ should not exceed the area:\n",
        "\n",
        "$$u_i \\leq \\frac{1}{2}(x_i y_{i+1} - y_i x_{i+1})$$\n",
        "$$2u_i \\leq x_i y_{i+1} - y_i x_{i+1}$$\n",
        "\n",
        "So we are replacing orientation constraints with:\n",
        "\n",
        "$$0 \\leq 2u_i \\leq x_i y_{i+1} - y_i x_{i+1}$$\n",
        "\n",
        "This idea is used in state-of-the-art models such as the one published by Christian Bingane (2022), *Largest small polygons: A sequential convex optimization approach*.\n",
        "\n",
        "### Final model\n",
        "\n",
        "We are placing the polygon in the half-plane $y \\geq 0$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "qW74os5DvA5-",
        "outputId": "b2584464-d5dd-4687-b09a-10472c29ca75"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Overwriting cartesian_polygon.mod\n"
          ]
        }
      ],
      "source": [
        "%%writefile cartesian_polygon.mod\n",
        "\n",
        "# Number of sides\n",
        "param N integer > 0;\n",
        "set I := 1..N;\n",
        "\n",
        "param max_diam := 1;\n",
        "# area lower bound, might help to avoid degenerated solutions\n",
        "param area_lb default 0;\n",
        "\n",
        "# Variables\n",
        "var x{i in I} <= max_diam, >= -max_diam\n",
        "\t\t:=  1.5*(N-i)/(N+1)-0.5; # initial guess that could be improved\n",
        "\t\t# :=  4*i*(N + 1 - i)/(N+1)**2; # initial guess that could be improved\n",
        "\n",
        "var y{i in I} >= 0, <= max_diam\n",
        "\t\t:=  4*i*(N + 1 - i)/(N+1)**2; # initial guess\n",
        "\n",
        "var u{2..N-1} >= 0; # triangle area\n",
        "\n",
        "var area >= area_lb; # objective\n",
        "\n",
        "# Constraints\n",
        "# Distance between vertices\n",
        "s.t. vertices_distance{i in 1..N, j in i+1 .. N}:\n",
        "\t(x[i]-x[j])**2 + (y[i]-y[j])**2 <= max_diam**2;\n",
        "\n",
        "# Orientation + Area\n",
        "s.t. orientation_triangle_area{i in 2..N-1}:\n",
        "    x[i]*y[i+1] - y[i]*x[i+1] >= 2*u[i];\n",
        "\n",
        "# Fixed vertex\n",
        "s.t. fix_x: x[1] = 0;\n",
        "s.t. fix_y: y[1] = 0;\n",
        "\n",
        "s.t. area_formula: area = sum{i in 2..N-1} u[i];\n",
        "\n",
        "maximize total_area: area;"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "rRTZ9buavA5_"
      },
      "outputs": [],
      "source": [
        "# solve the problem for n sides, return solution (cartesian coords) and area\n",
        "def solve_cartesian(n, prev_sol=0):\n",
        "    ampl.reset()\n",
        "    ampl.read(\"cartesian_polygon.mod\")\n",
        "    ampl.param[\"N\"] = n\n",
        "    ampl.param[\"area_lb\"] = prev_sol\n",
        "    ampl.solve()\n",
        "\n",
        "    # Get result from AMPL\n",
        "    x = [ampl.var[\"x\"][i].value() for i in range(1, n + 1)]\n",
        "    y = [ampl.var[\"y\"][i].value() for i in range(1, n + 1)]\n",
        "\n",
        "    # To plot easier, repeat the first point in the last position\n",
        "    # (to close the polygon)\n",
        "    x.append(x[0])\n",
        "    y.append(y[0])\n",
        "\n",
        "    return x, y, ampl.obj[\"total_area\"].value()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "GioSJwomvA5_",
        "outputId": "398e013b-9ea5-470d-df61-1d46325f61d0"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Problem size:  10\n",
            "Ipopt 3.12.13: outlev=0\n",
            "\n",
            "\n",
            "******************************************************************************\n",
            "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
            " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
            "         For more information visit http://projects.coin-or.org/Ipopt\n",
            "******************************************************************************\n",
            "\n",
            " \n",
            "Ipopt 3.12.13: Optimal Solution Found\n",
            "\n",
            "suffix ipopt_zU_out OUT;\n",
            "suffix ipopt_zL_out OUT;\n",
            "Problem size:  11\n",
            "Ipopt 3.12.13: outlev=0\n",
            "\n",
            "\n",
            "******************************************************************************\n",
            "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
            " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
            "         For more information visit http://projects.coin-or.org/Ipopt\n",
            "******************************************************************************\n",
            "\n",
            " \n",
            "Ipopt 3.12.13: Optimal Solution Found\n",
            "\n",
            "suffix ipopt_zU_out OUT;\n",
            "suffix ipopt_zL_out OUT;\n",
            "Problem size:  12\n",
            "Ipopt 3.12.13: outlev=0\n",
            "\n",
            "\n",
            "******************************************************************************\n",
            "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
            " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
            "         For more information visit http://projects.coin-or.org/Ipopt\n",
            "******************************************************************************\n",
            "\n",
            " \n",
            "Ipopt 3.12.13: Optimal Solution Found\n",
            "\n",
            "suffix ipopt_zU_out OUT;\n",
            "suffix ipopt_zL_out OUT;\n",
            "\n",
            "Sides:  10\n",
            "Area:  0.7335979126944925\n"
          ]
        },
        {
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        },
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\n",
            "Sides:  11\n",
            "Area:  0.7521931144677444\n"
          ]
        },
        {
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        },
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\n",
            "Sides:  12\n",
            "Area:  0.7542668674915548\n"
          ]
        },
        {
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "min_sides = 10\n",
        "max_sides = 12\n",
        "\n",
        "# save previous solutions to plot\n",
        "solutions = {}\n",
        "# save last optimal value to be used as lower bound\n",
        "prev_sol = 0\n",
        "\n",
        "for sides in range(min_sides, max_sides + 1):\n",
        "    print(\"Problem size: \", sides)\n",
        "    solutions[sides] = solve_cartesian(sides, prev_sol)\n",
        "    prev_sol = solutions[sides][2]\n",
        "\n",
        "for sides in range(min_sides, max_sides + 1):\n",
        "    xs, ys, area = solutions[sides]\n",
        "    plot_polygon(sides, xs, ys, area)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FeWsbxoPvA6A"
      },
      "source": [
        "## Conclusion\n",
        "\n",
        "We have presented two different non-linear models to solve the Largest Small Polygon problem.\n",
        "\n",
        "1. The first one uses **polar coordinates** to sort the vertices by the angle. It uses $2(n-1)$ variables and $n(n-1)/2+(n-1)$ constraints. However, there are sines, cosines, and quadratic terms in most of the constraints and objective, so it might be less efficient.\n",
        "\n",
        "2. The second model uses **cartesian coordinates** and sort the vertices by another method. It gets rid of sines and cosines in constraints, and objective becomes linear by introducing $n-2$ new variables. The number of constraints is the same. This model is used in **state-of-the-art research**, and solves the problem for any $n$ without assuming symmetry, so it solves the general problem.\n",
        "\n",
        "There are other models that might be more efficient, these assume *Graham Conjecture*, related to symmetry properties in the diameter graph.\n",
        "\n",
        "* Multiple non-linear solvers could be used to tackle this problem when the number of sides become big: knitro, ipopt, octeract. For this particular problem, **knitro** (13.2.0) gave the best performance in terms of speed and optimality of solutions."
      ]
    }
  ],
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.6.9"
    },
    "vscode": {
      "interpreter": {
        "hash": "aee8b7b246df8f9039afb4144a1f6fd8d2ca17a180786b69acc140d282b71a49"
      }
    }
  },
  "nbformat": 4,
  "nbformat_minor": 1
}