{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Optimization of a two-parameter function\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\n\n\n# Define the function that we are interested in\ndef sixhump(x):\n    return (\n        (4 - 2.1 * x[0] ** 2 + x[0] ** 4 / 3) * x[0] ** 2\n        + x[0] * x[1]\n        + (-4 + 4 * x[1] ** 2) * x[1] ** 2\n    )\n\n\n# Make a grid to evaluate the function (for plotting)\nxlim = [-2, 2]\nylim = [-1, 1]\nx = np.linspace(*xlim)  # type: ignore[call-overload]\ny = np.linspace(*ylim)  # type: ignore[call-overload]\nxg, yg = np.meshgrid(x, y)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## A 2D image plot of the function\n Simple visualization in 2D\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import matplotlib.pyplot as plt\n\nplt.figure()\nplt.imshow(sixhump([xg, yg]), extent=xlim + ylim, origin=\"lower\")  # type: ignore[arg-type]\nplt.colorbar()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## A 3D surface plot of the function\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "from mpl_toolkits.mplot3d import Axes3D\n\nfig = plt.figure()\nax: Axes3D = fig.add_subplot(111, projection=\"3d\")\nsurf = ax.plot_surface(\n    xg,\n    yg,\n    sixhump([xg, yg]),\n    rstride=1,\n    cstride=1,\n    cmap=\"viridis\",\n    linewidth=0,\n    antialiased=False,\n)\n\nax.set_xlabel(\"x\")\nax.set_ylabel(\"y\")\nax.set_zlabel(\"f(x, y)\")\nax.set_title(\"Six-hump Camelback function\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Find minima\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import scipy as sp\n\n# local minimization\nres_local = sp.optimize.minimize(sixhump, x0=[0, 0])\n\n# global minimization\nres_global = sp.optimize.differential_evolution(sixhump, bounds=[xlim, ylim])\n\nplt.figure()\n# Show the function in 2D\nplt.imshow(sixhump([xg, yg]), extent=xlim + ylim, origin=\"lower\")  # type: ignore[arg-type]\nplt.colorbar()\n# Mark the minima\nplt.scatter(res_local.x[0], res_local.x[1], label=\"local minimizer\")\nplt.scatter(res_global.x[0], res_global.x[1], label=\"global minimizer\")\nplt.legend()\nplt.show()"
      ]
    }
  ],
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    "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": "3.12.11"
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