Cartman0/Lagrange.ipynb

## Lagrange.ipynb
{
  "cells": [
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# 不等式制約の場合のLagrange"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## 等式制約の場合"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "等式制約条件 $g(\\vec{x}) = 0$の下で，\n目的関数$f(\\vec{x})$ の最大化・または最小化する（つまり，そのときの点を求めたい）．\n \nまず解候補は，$g(\\vec{x})$と$f(\\vec{x})$の交点のどれかになる．\n交点は交差しているか接しているかの2パターンがある．\n\n$g(\\vec{x})$で表される平面（2次元の場合は直線）と\n$f(\\vec{x})$で表される平面が交差しているとき，\n$f(\\vec{x}) > 0$になる点または$f(\\vec{x}) < 0$が存在するので極大・極小値になりえない．\n（つまり，被っている領域のほうに解がある．）\n\nこの辺については，この記事 https://www.yunabe.jp/docs/lagrange_multiplier.html がわかりやすい．\n\n接している場合，$g(\\vec{x})$と$f(\\vec{x})$の2つの接線ベクトルが平行になり，また接線ベクトルと直交関係にある勾配ベクトルも平行になる．\nこの点から$f(\\vec{x})$の勾配方向にズラして最適化してしまうと，制約条件から外れてしまうので，接している点のときに，制約条件を満たした極大値または極小値になる．\n\n勾配ベクトルが平行になるを式に表すと\n\n$$\n\\vec{\\nabla} f = \\lambda \\vec{\\nabla} g\n$$\n\n(例えば2次元の時，$\\vec{\\nabla} f =(\\frac{df}{dx}, \\frac{df}{dy})$\n\n右辺0に直せば，以下の式になる．\n以下の式を満たす点が極大・極小値になる．\n\n$$\n\\vec{\\nabla} f - \\lambda \\vec{\\nabla} g = \\vec{0}\n$$\n\n勾配ベクトルを考えたときに上式になればいいので，目的関数を作り直すと，\n\n$$\nL(\\vec{x}, \\lambda) = f(\\vec{x}) - \\lambda g(\\vec{x})\n$$\n\n特にこれをラグランジュ関数（ラグランジアン）という．\nこの目的関数は，$f(\\vec{x}), g(\\vec{x})$の2つの勾配ベクトルが平行になる点があるかという最適化問題に置き換わっているので双対問題になっている．\nこの双対問題を解くことで極値を得られる．\n\n解き方：\n\n変数をn個として，偏微分すると\n\n\\begin{eqnarray}\n  \\left\\{\n    \\begin{array}{l}\n\\frac{\\partial L}{\\partial x_{1}} = \\frac{ \\partial f }{\\partial x_{1} } - \\lambda \\frac{\\partial g}{\\partial x_{1}} = 0 \\\\\\\\\n\\frac{\\partial L}{\\partial x_{2}} = \\frac{ \\partial f }{\\partial x_{2} } - \\lambda \\frac{\\partial g}{\\partial x_{2}} = 0 \\\\\\\\\n\\vdots \\\\\\\\\n\\frac{\\partial L}{\\partial x_{n}} = \\frac{ \\partial f }{\\partial x_{n} } - \\lambda \\frac{\\partial g}{\\partial x_{n}} = 0 \\\\\\\\\n\\frac{\\partial L}{\\partial \\lambda} = g(\\vec{x}) = 0 \\\\\\\\\n \\end{array}\n  \\right.\n\\end{eqnarray}\n\n最後の式は等式制約条件を表しており，この連立方程式を解いた点が極値になる．\n連立方程式の式の数は，変数の数+等式制約条件の数 になる．\n上記の例だと$n+1$個になる．\n\n連立１次方程式になれば，行列と変数ベクトルに式変形して定数ベクトルを右辺に移項して，\n係数行列が（式の数と変数の数が一致し）正方行列なので逆行列を解いて求めることができる．\n\n"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 等式制約条件が複数ある場合"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "以下のような等式制約条件が2つある場合を考える．\n\n$$\ng(\\vec{x})_{1} = 0\\\\\\\\\ng(\\vec{x})_{2} = 0\\\\\\\\\n$$\n\n目的関数の勾配ベクトルと$g(\\vec{x})_{1}, g(\\vec{x})_{2}$の勾配ベクトルはそれぞれ平行するので，\n\n$$\n\\vec{\\nabla} f = a_{1} \\vec{\\nabla} g_{1} \\\\\\\\\n\\vec{\\nabla} f = a_{2} \\vec{\\nabla} g_{2} \\\\\\\\\n$$\n\nつまり，等式制約条件の勾配ベクトル同士も平行関係にあり，\n足しても定数倍されるだけで同じ向きになる．\n両辺を足すと，\n\n$$\n2 \\vec{\\nabla} f = a_{1} \\vec{\\nabla} g_{1} + a_{2} \\vec{\\nabla} g_{2} \\\\\\\\\n\\vec{\\nabla} f = \\frac{a_{1}}{2} \\vec{\\nabla} g_{1} + \\frac{a_{2}}{2} \\vec{\\nabla} g_{2}\n$$\n\n定数倍を$\\lambda$に置き直す．\n\n$$\n\\vec{\\nabla} f = \\lambda_{1} \\vec{\\nabla} g_{1} + \\lambda_{2} \\vec{\\nabla} g_{2} \\\\\\\\\n\\vec{\\nabla} f - \\lambda_{1} \\vec{\\nabla} g_{1} - \\lambda_{2} \\vec{\\nabla} g_{2} = \\vec{0}\n$$\n\nラグランジュ関数は，\n\n$$\nL(\\vec{x}, \\vec{\\lambda}) \n= f(\\vec{x}) - \\lambda_{1} g_{1}(\\vec{x}) - \\lambda_{2} g_{2}(\\vec{x})\n= f(\\vec{x}) - \\sum_{i=1}^{2} \\lambda_{i} g_{i}(\\vec{x}) \n$$\n\n制約条件が3個以上でも同じ．n個の制約条件のラグランジュ関数は以下になる．\n\n$$\nL(\\vec{x}, \\vec{\\lambda}) \n= f(\\vec{x}) - \\sum_{i=1}^{n} \\lambda_{i} g_{i}(\\vec{x}) \n$$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## 不等式制約の場合"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "参考：https://github.com/levelfour/machine-learning-2014/wiki/%E7%AC%AC4%E5%9B%9E---%E4%B8%8D%E7%AD%89%E5%BC%8F%E5%88%B6%E7%B4%84%E6%9D%A1%E4%BB%B6%E3%81%AE%E5%A0%B4%E5%90%88%E3%81%AE%E6%9C%AA%E5%AE%9A%E4%B9%97%E6%95%B0%E6%B3%95\n\n"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "不等式制約条件$g(\\vec{x}) \\geq 0$の場合を考える．\n等式では面上（2次元では線上）のみであったが，不等式領域($\\geq 0$)により条件が緩くなっている．\n\n"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "下図のような停留点(Statinary Point)が制約条件を満たしている状況では，\nLagrange関数にg(x)を含める必要がない。 \n制約条件を含めることのない通常の目的関数の最適化で解ける．\n"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-05-29T13:35:38.413834Z",
          "end_time": "2019-05-29T13:35:38.661171Z"
        },
        "hide_input": true,
        "trusted": true,
        "scrolled": true
      },
      "cell_type": "code",
      "source": "import scipy as sp\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\nx = sp.linspace(-3, 3)\ny = -x\nplt.plot(x, y, color=\"r\")\nplt.text(3.1, -3, \"$g(\\\\vec{x}) = 0$\", color=\"r\", fontsize=18)\n\ny2 = sp.zeros_like(x) + 3\nplt.fill_between(x,y,y2,facecolor='g',alpha=0.3)\n\n# 条件内\nplt.text(-2, 2.5, \"$g(\\\\vec{x}) > 0$\", color=\"g\", fontsize=18)\n\n# 条件外\nplt.text(-4.5, 2, \"$g(\\\\vec{x}) < 0$\", color=\"k\", fontsize=18)\n\n# 停留 Stationary point\nplt.scatter(2, 1, color=\"k\")\nplt.text(0.8, 1.5, \"StationaryP\", color=\"k\", fontsize=14)\n\nplt.xlim(-3, 3)\nplt.ylim(-3, 3)\nplt.axis(\"equal\")\nplt.tight_layout()\nplt.show()",
      "execution_count": 33,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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y2fNfPO6lEhMTGTt2LACdOnXi9ddfp0qVKld+Y7mIjIzMMwRTUlKyjnGLAkpEvCDg7jkdOnSI/fv306RJkxz7rLVs3rw52wwlJsZ5jMOxY8fyHPP9999nw4YNAJQqVeqy32/IHCdz3EvFxsYyZ84c2rVrx8KFC4mLi6N169ZMmzaNpKSkK7/Bi1x99dUcOXIkx2INgH379hEdHV3wWdPFdA9KRIpYwIVTcnIyACEhOd/aokWLSExMzLqkB9CwYUOAPDs7LF68mD59+tClSxd69OjBtGnT2LJlS57n3759e7ZxLxUWFkavXr347LPP2LdvH2PHjuX48eMMGDCAihUr0rVrVz788MOsmc/lNG3alIyMDL777rts21NSUtiwYQPx8Vf8npvrFFAiUoTcDidjTFVjzNfGmC3GmE3GmCGeKKygqlatSlhYGMuWLcu2/dChQzz22GMA2WZOjRs3pnTp0qxZsybHWGvXrqVr1640b96cOXPm8PzzzxMSEsKIESPyPP+aNWuoWLEidevWvWKtFSpUYMiQIaxbt44tW7YwfPhwfvjhB7p160bFihXp168fJ0+ezPP13bt3xxjD+PHjs21/8803SU5Opnfv3lesIV8UUCJSRDwxc0oDhltr6wO3AI8aY669wmsKTbFixejfvz/r16+nffv2TJ48mdGjR9O4cWPCwpxbbBeHU2hoKF27duWrr77Kdnlsy5Yt3H333dSpU4f58+cTERFBrVq1GDBgAB9//DHffPNNjnOfOnWKlStX0q1bt3zXXa9ePf71r3/x+++/s3z5crp3787HH3/M0aNH83zNddddx6OPPsq8efPo2rUrb731FsOHD+eJJ56gZcuW9OrVK991XJECSkSKgMd76xljPgZes9YuyeuYwu6tl5yczN///nfmzZvHyZMnadq0KU899RSvvfYa33//Pfv27ct2/HfffcfNN9/Mhx9+yL333svu3btp3rw54eHhrF69mooVK2Yde+DAAWrVqkXjxo1zBNSMGTPo168fP//8c56X9fIjNTWVkJCQHF8Ovlh6ejrjx49nypQp7Ny5k+joaLp3787o0aMpWbKk2zVc5sTqxSdBS731Cs7V3noeDSdjTHVgBdDQWnvikn2DgcEAcXFxTXbt2uWx87oiMTGR2NhY+vfvz8SJE3Psb9euHadPn2blypUFPoc3Gr96lQJKgpTCqeBcDSePLSU3xpQEPgKGXhpMANbaKcAUcGZOnjrvpc6dO4cxJusSHjgLBPr27Ztrq59M//nPf7jhhhtYvHgxbdu2zfd558+fz88//5z/fnb+LPMSn7VaZi4iHuWRcDLGFMMJpjnWWq9OG7755hsGDhxIz549iYuLY+/evcyePZudO3cyceLEPBcqNGjQgLS0tAKft3Pnzpw9e7bAr/dboaEwcyag70GJiOe4HU7GmdtOBbZYa8e6X5J7SpYsSVxcHBMnTiQpKYkyZcpw6623Mn36dG677TZvlxeYFFAi4mGemDk1B/oAPxtjNpzf9rS1dpEHxs63+Ph4vvrqK2+cOrgpoETEg9wOJ2vtKkB3BkUBJSIeE3C99cTLFFAi4gEKJ/E8BZSIuEnhJIVDASUibgi4xq/iQzIDSq2OgtahvYe4p949bPs598bKInnRzEkKl54H5XOSjiXxzoR3WL98PccSjxFVOopqtatx76B7ady8MQPvHMjdve+my4Au+Rr36T5PE1c7joeefShrW3TlaGasnEHpsqU9/TYKxaG9hxjUelDWz1Glo6hWpxr3P34/DW9yvyWZuE7hJIVPAeVTXnj8Bc6eOctj/3qMynGVSTqWxMbvNnLyeN4d8AsqNDSUsjFlPT5uQaSdSyOsmGt/5f3zzX9So14Njh87zqyxsxj111FMWDCBSrGVCrlKyaRwkqKhgPIJp06cYvO6zYyeNpobmt0AQIUqFah9XW3Amf0c3n+Yt196m7dfehuAT7Z+wok/TjD5uclsXr+Zk8dPUrFqRbo82IXW97YGYPxT49mYsJGNCRtZ9I7zFcc3l74JwKDWg/jPB//JOsfGhI1Mf2k6O7buILJUJC07tKTv8L4UCy+WVUPVa6oSVSqKL97/gpCQEO7odAf9/t4v6zltX3/yNQtmLmDf7/sILx5Og6YNGPT0IMpXLA/Az2t/5pm+z/Ds5GeZ+9pcdmzdQf9/9Oetf7/FS++9lFULwBfvf8HMsTOZvmJ61rZSZUpRNqYsZWPK8ujoR3mw5YNsWLWBdj3aFcp/F8lJ4SRFRwHldSUiS1AisgTfffUd1za5lvCI7E9KHjFhBEM6D6F119bc1fOurO3nzp6jVoNa3DvoXiJLRrJh9Qbe+OcbxFwdww3NbmDQM4PYv3M/VWpW4YFhDwBQulxpjhw4km38o4eOMmrwKO645w6GvDCEg7sPMuF/J2CMYcBTA7KOW75gOR0f6MiLc19kx9YdvPzky9RqUIuWHVoCziyo12O9iK0Zy4k/TjDj5Rm8NPwlxswek+18M16eQf9/9KdytcqUiCpBwtcJLJ23NFs4Lf1oKXd0uiMrHC8VUTzCOacb7c0k/7QgQopWZkD16qVFEl4QGhbKkBeGsGzBMno27cnfu/+daf83jV9+/AVwZgwhISGUiCqRNXMAKF+xPF0HdKVm/ZpUqlqJdt3b0axNM1YsXAFAVKkowoqFEVE8Iut1oaGhOc6/6J1FlI0py0MjH6Jqrao0vaMpfZ/oy8I5C0k9c+F5alVrVaX3472pUqMKLe5qwXU3X8dPa37K2t/m3jbEt4ynUtVK1Lm+Dg//82E2r9vMkYPZw7Dn33rSuEVjKlWtxFXlrqJtt7asWLiCs6lOH8w9v+3hlx9/oc29bXL9vFKSU5g5diYhoSE0bKp7TkVJMycpelpm7lW3/vlW4lvFs2ndJn7Z8Avfr/ye+W/P5/6h93NfHv8d0tPT+ejNj1i5aCXHDh3j3LlzpJ1Ly/df2Ht+30O9RvWyLs8B1G9Sn7RzaezfvZ8adWsAUL1u9WyvKxdTjqSjSVk//7bpN+a+7lyuO3X8FBbnQQeJ+xOJrhSdddw1Da/JNs7Nf7qZSc9N4tvF39KyY0uWfrSUOtfXoVqdatmOG9FnBCEhIaSeSaVsTFmGvDAkR01SuBRO4h0KKK8KjwincfPGNG7emB6P9mDC/0zg3dffpUv/3FfozZ82n/lvz2fQ04OoVqcaxSOLM2vcrGyB4RJLns9BMhd1QQsNyz7rMsaQkZEBOLOZkQNHckOzGxj2f8MoU74MJ/44wVO9nyLtXPZLb8Uji2f7OaxYGHd2upOl85bS4q4WfP3J1/R6LOcTo598+Umq1alGVKkov1lpGGgUTuI9CiifUbVWVdLT0jl39hxhxcKygiDT5vWbadqqKXd0ugMAay37d+4nqlRU1jG5vS6386z6fBUZGRlZs6ct67cQViyMynGVXap17+97OfHHCfo80Sdr9dzqxatdfq9tu7Xl0bsfZdE7izhz+gy33317jmPKVyzvcj1SOHTPSbwrM6B0D6pInPjjBM/0fYavP/maHb/s4ODeg6z6fBXzps7j+mbXE1kykgpVKrBp3SaOHjrKiT+c54ZeXf1qflrzE5vXb2bv73uZ/NxkDu09lG3sClUqsO2nbRzae4gTf5zINaja92rPscPHmDRqEnt+20PCsgRmjJ3B3b3vJqJEhEvvIebqGIqFF2Ph7IUc3HOQhGUJzHl1jsufQZUaVah/Y33efultbm17K5ElI11+rRQdzZzE+zSDKjIlokpQ94a6LJi5gAO7D3Du7DnKVyxPy7tbct/Dzmfe6/FevDHyDQa3Gcy5s+f4ZOsndH+4O4f2HWLUoFGEFw/nzi530rJjS/Zs35M1dpf+XRj/1Hge7fAoZ1POZi0lv1j5iuUZOWUk01+azpDOQ4gqHUXLDi154IkHXH4PV5W7iqFjhjJr3CwWvbOI6nWrM+AfA/jnoH+6PEabv7Rh07pNtPlL7gshxPuMtYX2xPQ8xcfH23Xr1hX5ecXHpafDAw/AO++wZej9CigpNB+9+RFLPlzCpC8mFej1Hep0yPPemVyeMWa9tTb+Ssdp5iS+QzMoKWRnTp9h7+97WTBzAd0e6ubtcuQyFE7iWxRQUogmPzeZFQtXcNOdN9Guu7o9+DKFk/iezIAyRgElHjV0zFCGjhnq7TLEBQon8U1qdSQS1BRO4rsUUCJBS+Ekvk0BJRKUFE7i+xRQIkFH4ST+ITOgrFVAiQQBhZP4Dy0zFwkaCifxLwookaCgcBL/o4ASCXgKJ/FPCiiRgKZwEv+lgBIJWAon8W8KKJGA5JFwMsZMAzoAh621DT0xpojLFFABZ9mCZcwaN4sjB44QXTmaPsP60KpjK2+XJUXIUzOn6cBrwEwPjSeSPwqogLFswTJe/9/XSU1JBSBxfyKv/+/rAAqoIOKRx7Rba1cAxzwxlkiBZQZU79565LsfmzVuVlYwZUpNSWXWuFleqki8ocjuORljBgODAeLi4orqtBJs1OrI7x05cCRf2yUwFVk4WWunAFPAeUx7UZ1XgpACyq9FV44mcX9irtsleHjksp6Iz8kMKF3i8zt9hvUhonhEtm0RxSPoM6yPlyoSb9BScglcmkH5pcxFD1qtF9w8tZR8LtAKiDbG7AVGWmunemJsEbeom7lfatWxlcIoyHkknKy1PT0xjkih0DJzEb+jy3oSHBRQIn5F4STBQwEl4jcUThJcFFAifkHhJMFHASXi8xROEpwUUCI+TeEkwUsBJeKzFE4S3DIDyhgFlIgPUTiJqJOEiM9ROImAAkrExyicRDIpoER8hsJJ5GIKKBGfoHASuZQCSsTrFE4iuVE3cxGvUjiJ5EXfgxLxGoWTyOUooES8QuEkciUKKJEip3AScYUCSqRIKZxEXKWAEikyCieR/FBAiRQJhZNIfimgRAqdwkmkINTNXKRQKZxECkqdJEQKjcJJxB0KKJFCoXAScZcCSsTjFE4inqCAEvEohZOIpyigRDxG4STiSepmLuIRCicRT9P3oETcpnASKQwKKBG3eCScjDHtgFeAUOAta+0YT4wr4tcuDShj2P7Xbl4uSsQ/uB1OxphQ4HWgDbAXSDDGfGKt3ezu2CJ+7+KAGjcLQAEl4gJPzJxuArZba38HMMa8C3QCFE4ioIASKQBPhFMVYM9FP+8Fbr70IGPMYGAwQFxcnAdOK+JHFFAi+eKJcDK5bLM5Nlg7BZgCEB8fn2O/SMBTQIm4zBPhtBeoetHPscB+D4wrEngu7maugBLJkyfCKQGobYypAewDegC9PDCuSGC6uJOEAkokV26Hk7U2zRjzN+ALnKXk06y1m9yuTCSQKaBELssj33Oy1i4CFnliLJGgoYASyZM6RIh406UBZa06SYigcBLxPnUzF8lB4STiC9TNXCQbhZOIr1CzWJEsCicRX6KAEgEUTiK+R93MRRROIj5JrY4kyCmcRHyVAkqCmMJJxJcpoCRIKZxEfJ0CSoKQwknEH6ibuQQZhZOIv1AvPgkiCicRf6KAkiChcBLxNwooCQIKJxF/pG7mEuAUTiL+St3MJYApnET8mbqZS4BSOIn4OzWLlQCkcBIJBGoWKwFG4SQSKNRJQgKIwkkkkCigJEAonEQCjQJKAoDCSSQQKaDEzymcRAKVAkr8mMJJJJCpm7n4KYWTSKBTLz7xQwonkWCggBI/o3ASCRZqFit+xK1wMsZ0A/4J1Adustau80RRIlJI1CxW/IS7M6eNQFdgsgdqEZGioIASP+BWOFlrtwAYYzxTjYgUDXUzFx+ne04iwUrdzMWHXTGcjDFLgUq57HrGWvuxqycyxgwGBgPExcW5XKCIFCJ1MxcfdcVwsta29sSJrLVTgCkA8fHx1hNjiogHqJOE+CBd1hMRBZT4HHeXkncBJgAxwEJjzAZr7Z89UpmIFC0FlPgQd1fr/Rf4r4dqERFvU0CJj9BlPRHJTgElPkDhJCI5qZu5eJnCSURyp2ax4kUKJxHJmwJKvEThJCKXp27m4gUKJxG5MjWLlSKmcBIR16hZrBQhhZOIuE7NYqWIKJxEJH8UUFIEFE4ikn/qZi6FTOEkIgWjThJSiBROIlJwCigpJAonEXGPAkoKgcJJRNyngBIx14RsAAALjElEQVQPUziJiGcooMSDFE4i4jnqZi4eonASEc9Ss1jxAIWTiHieAkrcpHASkcKhbubiBoWTiBQedTOXAlI4iUjhUjdzKQCFk4gUPjWLlXxSOIlI0VBAST4onESk6KibubhI4SQiRUudJMQFCicRKXoKKLkChZOIeIcCSi5D4SQi3qOAkjwonETEuxRQkgu3wskY8xLQETgL/AY8aK097onCRCSIqJu5XCLEzdcvARpaa68HfgVGuF+SiASlzE4SvXtTf9wsrpn8gbcrEi9ya+ZkrV180Y9rgL+4V46IBDV1M5fzPHnPqT/wXl47jTGDgcEAcXFxHjytiAQUdTMXXAgnY8xSoFIuu56x1n58/phngDRgTl7jWGunAFMA4uPjbYGqFZHgoG7mQe+K4WStbX25/caYvkAH4E/WWoWOiHiGAiqoubtarx3wD6CltTbZMyWJiJyngApa7t5zeg2IAJYYYwDWWGsfcrsqEZFMCqig5O5qvWs8VYiISJ4uDSh1Mw946hAhIv5By8yDisJJRPyHAipoKJxExL8ooIKCwklE/I8CKuApnETEPymgAprCSUT8lwIqYCmcRMS/KaACksJJRPyfAirgKJxEJDCom3lAUTiJSOBQq6OAoXASkcCigAoICicRCTwKKL+ncBKRwKSA8msKJxEJXOpm7rcUTiIS2LTM3C8pnEQk8Cmg/I7CSUSCQyAElLXwwQfQrh2ULu3tagqVwklEgoe/B5QxcPo0dO8On37qvJ8AFeLtAkREilRmQPXuTf1xs7hm8gferWfjRggLgyVLsm+fOBGqV8/5a9Qo+PxzWLDAOW7+fAgPh23birTswqZwEpHg40sB9cQT0Lw5tGmTffvDD8POnTl/DRwI/fpB587OcZ07w3XXwT/+UZRVQ0YGjBsH9epB8eJQtSoMH+7M7DxAl/VEJDj5wiW+b791Zkzz57t2vLVw5505g2jIEOjbFzZtggYNPF9nboYNg1dfhS5dnFDassX5+YcfYOlSCHFv7qNwEpHg5e2AeuMNKF8e2rd37Xhj4NZbc27v2tWZaU2aBBMmeLbG3Gza5Jyna1f46KML22vUgMcfh3ffhV693DqFLuuJSHC79BLfpPfdHzMhwQmc0qWhbFnnUlxSEkRGQp8+zjFpac6MqU0bKFYs++vPnIHYWIiLg9TU7PsGDnRqfvfdC9tKloTbbnNW8hWFuXOdWdzQodm3DxrkvMfZs90+hWZOIiKebHX02WfOfaBq1eDJJ51wmjYN7rrLCZ3GjZ3j1q+HU6fgpptyjlGihLPwYeBAZ3Y1bJizfcQImDoVXn8devTI/ppmzeCLL2DrVuc+0KUyMuDYMdffR7lyeV+aS0hw9l1ae/Hi0KiRs99NCicREfBMQB086FzOio+HxYshKsrZ3qePs9IOLoTT5s3O77Vq5T5Wv37OgoMXXnBmJG+9BWPGOKH1yCM5j88cZ9Om3MNp927nspurduy4UPOl9u+H6GiIiMi5r0oVWL0azp51VhEWkMJJRCSTuwH14otw4oQzu8kMJoAyZZwZxfLlzu8AiYnO7+XK5V3LmDHQsaMzE/vqK3jsMXj22dyPL1/e+f3w4dz3V6qUc7n65VSqlPe+5OTcgwmc2VPmMQonEREPKWhAWQvvvQctWuQ+cwHnUl/Zss6fjbnwurx06AA33ghffulcxnvllcuf/+JxL1W8OLRuffn34KrIyLxDMCXlwjFuUDiJiFyqIAF16JBzuat795z7rHUu4zVvfmFbTIzz++XuA73/PmzY4Py5VKm8g+ficTLHvVR6+oXZmitiYvLuQHH11c77SU3NOYPat8+55OfGrAkUTiIiuctvQCUnO7/ntohg0SInGDIv6QE0bOj8nldnh8WLnXtVXbo4q/mmTXMWRtSvn/vx27dnH/dSe/Z47p5T06ZOfd9956wSzJSS4oTp7be7fp48uBVOxpjngE5ABnAY6Get3e92VSIiviA/AVW1qtOGaNmy7NsPHXLuFcGFxRCZfy5dGtasyTnW2rXOd4iaN4c5c2DvXuf7RCNG5P2F3TVroGJFqFs39/2evOfUvTv8+98wfnz2cHrzTSeke/d2/Tx5sdYW+BdQ+qI/Pw5McuV1TZo0sSIifiMtzdreva0Fu3no/TYjIyP34wYPthasvesuaydNsnbUKGsrV7a2dm1n++7d2Y/v18/asmWtTUm5sG3zZmvLl7e2cWNrk5IubH/oIWeMVatynvfkSWsjI63929/cf6+u+tvfnHq6dLH2zTetfeIJa8PCrG3Z0tr09DxfBqyzruSLKwe5NBCMACa6cqzCSUT8zkUBlfH887kfc/q0tY88Ym2lStZGRVnbqpW1n39ubYcO1l59dc7j1651/hr+8EPn5127rI2NtbZmTWsPHsx+7P791pYoYe2tt+YcZ/p0Z5yff3bvPeZHWpq1L79sbZ061oaHO+9v2DAnKC/D1XAy9nIrRVxgjPkX8ACQBNxhrc31jpsxZjAwGCAuLq7Jrl273DqviEiRS093etilpcE777jWPy4x0en20L+/02n8Uu3aOc1SV64seF1NmjgrAefNK/gYRcQYs95aG3/F464UTsaYpUBuFx+fsdZ+fNFxI4Di1tqRVzppfHy8Xbdu3ZUOExHxPenpzuq7sEtu2Z8756ymu3h7Sopz72jpUvj559zvB23aBDfc4CyaaNs2//XMnw/33eeMU7t2/l9fxFwNpysuiLDWurow/h1gIXDFcBIR8Vt5La/+5hun3VDPnk5PvL17nR5zO3c6M6a8Fio0aODMxAqqc2enG0OAcXe1Xm1rbeY6yHuAre6XJCLih0qWdEJp4kSnyWuZMk4H8enTs69oE5e4+z2nMcaYujhLyXcBD7lfkoiIH4qPd1oMiUe4FU7W2ns9VYiIiEgmPc9JRER8jsJJRER8jsJJRER8jsJJRER8jtsdIgp0UmMScVb3+bJo4Ii3i/AT+qzyR5+X6/RZ5Y8/fF7VrLV5PNfjAq+Ekz8wxqxz5VvMos8qv/R5uU6fVf4E0uely3oiIuJzFE4iIuJzFE55m+LtAvyIPqv80eflOn1W+RMwn5fuOYmIiM/RzElERHyOwklERHyOwskFxpgnjTHWGBPt7Vp8lTHmJWPMVmPMT8aY/xpjyni7Jl9jjGlnjPnFGLPdGPOUt+vxZcaYqsaYr40xW4wxm4wxQ7xdk68zxoQaY34wxnzq7Vo8QeF0BcaYqkAbYLe3a/FxS4CG1trrgV+BEV6ux6cYY0KB14G7gGuBnsaYa71blU9LA4Zba+sDtwCP6vO6oiHAFm8X4SkKpysbB/w/QCtHLsNau9ham/k4zzVArDfr8UE3Aduttb9ba88C7wKdvFyTz7LWHrDWfn/+zydx/tKt4t2qfJcxJha4G3jL27V4isLpMowx9wD7rLU/ersWP9Mf+MzbRfiYKsCei37ei/6ydYkxpjrQGFjr3Up82nicf0RneLsQT3H3Sbh+zxizFKiUy65ngKeBtkVbke+63Gdlrf34/DHP4FySmVOUtfkBk8s2zcavwBhTEvgIGGqtPeHtenyRMaYDcNhau94Y08rb9XhK0IeTtbZ1btuNMdcBNYAfjTHgXKb63hhzk7X2YBGW6DPy+qwyGWP6Ah2AP1l9ge5Se4GqF/0cC+z3Ui1+wRhTDCeY5lhr53m7Hh/WHLjHGNMeKA6UNsbMttbe7+W63KIv4brIGLMTiLfW+nrHX68wxrQDxgItrbWJ3q7H1xhjwnAWivwJ2AckAL2stZu8WpiPMs6/CGcAx6y1Q71dj784P3N60lrbwdu1uEv3nMRTXgNKAUuMMRuMMZO8XZAvOb9Y5G/AFzg3999XMF1Wc6APcOf5/582nJ8ZSJDQzElERHyOZk4iIuJzFE4iIuJzFE4iIuJzFE4iIuJzFE4iIuJzFE4iIuJzFE4iIuJz/j+Mnxa6kHjOpQAAAABJRU5ErkJggg==\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "問題は下図のように、\n停留点が制約の境界上にあるときである。\nこのとき$\\nabla f$の向きによっては、最適化しようとすると制約を外れる向き\n（つまり緑色の領域→白色の領域の向き）に動く可能性がある。\nそのため、この状況では等式制約条件下で解く必要がある．\n"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-05-29T13:48:57.469278Z",
          "end_time": "2019-05-29T13:48:57.624572Z"
        },
        "hide_input": true,
        "trusted": true,
        "scrolled": true
      },
      "cell_type": "code",
      "source": "import scipy as sp\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\nx = sp.linspace(-3, 3)\ny = -x\nplt.plot(x, y, color=\"r\")\nplt.text(3.1, -3, \"$g(\\\\vec{x}) = 0$\", color=\"r\", fontsize=18)\n\ny2 = sp.zeros_like(x) + 3\nplt.fill_between(x,y,y2,facecolor='g',alpha=0.3)\n\n# 条件内\nplt.text(-2, 2.5, \"$g(\\\\vec{x}) > 0$\", color=\"g\", fontsize=18)\n\n# 条件外\nplt.text(-4.5, 2, \"$g(\\\\vec{x}) < 0$\", color=\"k\", fontsize=18)\n\n# 停留 Stationary point\nplt.scatter(0, 0, color=\"k\")\nplt.text(-0.5, 0.6, \"StationaryP\", color=\"k\", fontsize=14)\n\n# 勾配 f\nplt.quiver(0,0, -1,-1, scale_units='xy', scale=2, linestyle=\"--\", color=\"b\") \nplt.text(-1, -1, \"$\\\\vec{\\\\nabla} f$\", color=\"b\", fontsize=14)\n\nplt.xlim(-3, 3)\nplt.ylim(-3, 3)\nplt.axis(\"equal\")\nplt.tight_layout()\nplt.show()",
      "execution_count": 45,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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XL3v+S8fNKTExkUmTJgHQvXt3XnnlFWrVqlXwC8tDeHh4viGYkpKSdYxbFFAi4gV+d8/pyJEjHDx4kBYtWuTaZ61ly5Yt2WYo0dHO1zgcP3483zHff/99Nm3aBECFChUu+/mGzHEyx80pJiaGefPm0aVLFxYvXkxsbCydOnVi1qxZJCUlFfwCL3HFFVdw7NixXIs1AA4cOEBUVFTRZ02X0j0oESlhfhdOycnJAAQF5X5pS5YsITExMeuSHkCzZs0A8u3ssHTpUgYOHEjPnj3p27cvs2bNYuvWrfmef+fOndnGzSkkJIT+/fvz6aefcuDAASZNmsSJEycYOnQo1atXp1evXnz44YdZM5/LadmyJRkZGXz33XfZtqekpLBp0ybi4wv8nJvrFFAiUoLcDidjTG1jzFfGmK3GmARjzEhPFFZUtWvXJiQkhBUrVmTbfuTIER566CGAbDOnuLg4IiMjWbt2ba6x1q1bR69evWjXrh3z5s3j6aefJigoiLFjx+Z7/rVr11K9enUaN25cYK3VqlVj5MiRbNiwga1btzJmzBi+//57evfuTfXq1Rk8eDCnTp3K9/l9+vTBGMPkyZOzbX/ttddITk5mwIABBdZQKAooESkhnpg5pQFjrLVNgdbAg8aYqwp4TrEJDQ1lyJAhbNy4ka5duzJ9+nQmTJhAXFwcISHOLbZLwyk4OJhevXrx5ZdfZrs8tnXrVm6//XYaNWrEggULKFOmDA0aNGDo0KEsXLiQb775Jte5T58+zddff03v3r0LXXeTJk34xz/+wa+//srKlSvp06cPCxcu5Lfffsv3Oddccw0PPvgg8+fPp1evXrz++uuMGTOGRx55hI4dO9K/f/9C11EgBZSIlACP99YzxiwEXrbWLsvvmOLurZecnMxf//pX5s+fz6lTp2jZsiWPPfYYL7/8Mv/97385cOBAtuO/++47brjhBj788EPuvPNO9u7dS7t27QgLC2PNmjVUr14969hDhw7RoEED4uLicgXUm2++yeDBg/npp5/yvaxXGKmpqQQFBeX6cPCl0tPTmTx5MjNmzGD37t1ERUXRp08fJkyYQPny5d2u4TInVi8+CVjqrVd0rvbW82g4GWPqAquAZtbakzn2jQBGAMTGxrbYs2ePx87risTERGJiYhgyZAhTp07Ntb9Lly6cOXOGr7/+usjn8EbjV69SQEmAUjgVnavh5LGl5MaY8sBHwKicwQRgrZ0BzABn5uSp8+Z0/vx5jDFZl/DAWSAwaNCgPFv9ZPr3v//Nddddx9KlS+ncuXOhz7tgwQJ++umnwvezK80yL/FZq2XmIuJRHgknY0woTjDNs9Z6ddrwzTffMGzYMPr160dsbCz79+9n7ty57N69m6lTp+a7UOHqq68mLS2tyOft0aMH586dK/LzS63gYJgzB9DnoETEc9wOJ+PMbWcCW621k9wvyT3ly5cnNjaWqVOnkpSURKVKlWjbti2zZ8+mQ4cO3i7PPymgRMTDPDFzagcMBH4yxmy6sO1xa+0SD4xdaPHx8Xz55ZfeOHVgU0CJiAe5HU7W2tWA7gyKAkpEPMbveuuJlymgRMQDFE7ieQooEXGTwkmKhwJKRNygcJLikxlQxiigRKRQFE5SvPR9UCJSBAonKX4KKBEpJIWTlAwFlIgUgsJJSo4CSkRcpHCSkqVmsSLiAoWTlDwtMxeRAiicxDsUUCJyGQon8R4FlIjkQ+Ek3qWAEpE8KJzE+xRQIpKDwkl8gwJKRC6hcBLfoYASkQsUTuJbFFAigsJJfJG6mYsEPIWT+Ca1OhIJaAon8V0KKJGApXAS36aAEglICifxfQookYCjcJLSQd3MRQKKwklKDy0zFwkYCicpXRRQIgFB4SSljwJKxO8pnKR0UkCJ+DWFk5ReCigRv6VwktJNASXilzwSTsaYWUA34Ki1tpknxhRxmQJKxO94auY0G3gZmOOh8UQKRwEl4lc8Ek7W2lXGmLqeGEukyNTNXMRvlNg9J2PMCGAEQGxsbEmdVgKNWh2J+IUSCydr7QxgBkB8fLwtqfNKAFJAiZR6Wq0n/kkBJVKqKZzEfymgREotTy0lfwe4CYgyxuwHxllrZ3pibBG3qJu5SKkU5IlBrLX9rLU1rbWh1toYBZP4lMxVfP3703TyXK6c9r63K3LJkf1H+GOTP7Ljpx3eLkWkxOmyngQGD30OKul4Em9PeZuNKzdyPPE4EZER1GlYhzuH30lcuziG3TKM2wfcTs+hPQs17uMDHye2YSz3P3l/1raomlG8+fWbRFaOLHSd3nBk/xGGdxqe9TgiMoI6jepw98N306yVPpsvhaNwksDhgYB65uFnOHf2HA/94yFqxtYk6XgSm7/bzKkTp4qh3GAqR1f2+LhFkXY+jZBQ1/66+Ptrf6dek3qcOH6Ctya9xfj7xjNl0RRqxNQo5irFnyicJLC4EVCnT55my4YtTJg1gevaXAdAtVrVaHhNQ8CZ/Rw9eJQ3nnuDN557A4CPt33Myd9PMv2p6WzZuIVTJ05RvXZ1et7bk053dgJg8mOT2bx+M5vXb2bJ20sAeG35awAM7zScf3/w76xzbF6/mdnPzWbXtl2EVwinY7eODBoziNCw0Kwaal9Zm4gKEXz+/ucEBQVxc/ebGfzXwQQFOVfxv/r4KxbNWcSBXw8QVjaMq1tezfDHh1O1elUAflr3E08MeoInpz/JOy+/w65tuxjytyG8/s/Xee6957JqAfj8/c+ZM2kOs1fNztpWoVIFKkdXpnJ0ZR6c8CD3dryXTas30aVvl8L8LyUBTuEkgaeIAVUuvBzlwsvx3ZffcVWLqwgrE5Zt/9gpYxnZYySdenXitn63ZW0/f+48Da5uwJ3D7yS8fDib1mzi1b+/SvQV0VzX5jqGPzGcg7sPUqt+Le4ZfQ8AkVUiOXboWLbxfzvyG+NHjOfmP97MyGdGcnjvYab83xSMMQx9bGjWcSsXreSOe+7g2XeeZde2XTz/6PM0uLoBHbt1BJxZUP+H+hNTP4aTv5/kzeff5LkxzzFx7sRs53vz+TcZ8rch1KxTk3IR5Vj/1XqWz1+eLZyWf7Scm7vfnBWOOZUpW8Y5Z1page+vyKU8siBCpNQpwiKJ4JBgRj4zkhWLVtCvZT/+2uevzPrXLLb/sB1wZgxBQUGUiyiXNXMAqFq9Kr2G9qJ+0/rUqF2DLn260ObWNqxavAqAiAoRhISGUKZsmaznBQcH5zr/kreXUDm6MvePu5/aDWrT8uaWDHpkEIvnLSb1bGrWcbUb1GbAwwOoVa8W7W9rzzU3XMOPa3/M2n/rnbcS3zGeGrVr0OjaRjzw9wfYsmELxw5nD8N+f+lHXPs4atSuQcUqFencuzOrFq/iXOo5APb9so/tP2zn1jtvzfP9SklOYc6kOQQFB9Gspe45SeFo5iSBqwgzqLb/05b4m+JJ2JDA9k3b+e/X/2XBGwu4e9Td3JXPc9PT0/notY/4esnXHD9ynPPnz5N2Pq3Qf2Hv+3UfTZo3ybo8B9C0RVPSzqdxcO9B6jWuB0DdxnWzPa9KdBWSfkvKevxLwi+884pzue70idNYnIYtiQcTiaoRlXXclc2uzDbODX+4gWlPTePbpd/S8Y6OLP9oOY2ubUSdRnWyHTd24FiCgoJIPZtK5ejKjHxmZK6aRAqicJLAVoRmsWFlwohrF0dcuzj6PtiXKf9vCu++8i49h+S9Qm/BrAUseGMBwx8fTp1GdSgbXpa3XngrW2C4xIIxJs9dhovbg0Oyz7qMMWRkZADObGbcsHFc1+Y6Rv9rNJWqVuLk7yd5bMBjpJ3PfumtbHjZbI9DQkO4pfstLJ+/nPa3teerj7+i/0P9c9Xy6POPUqdRHSIqRJSalYbiexROIm52kqjdoDbpaemcP3eekNCQrCDItGXjFlre1JKbu98MgLWWg7sPElEhIuuYvJ6X13lWf7aajIyMrNnT1o1bCQkNoWZsTZdq3f/rfk7+fpKBjwzMWj23Zukal19r596defD2B1ny9hLOnjnLjbffmOuYqtWrulyPSH50z0kELgbUgAH53oM6+ftJnhj0BF99/BW7tu/i8P7DrP5sNfNnzufaNtcSXj6carWqkbAhgd+O/MbJ308CcEXdK/hx7Y9s2biF/b/uZ/pT0zmy/0i2savVqsaOH3dwZP8RTv5+Ms+g6tq/K8ePHmfa+Gns+2Uf61es581Jb3L7gNspU66MSy8z+opoQsNCWTx3MYf3HWb9ivXMe2mey29TrXq1aHp9U9547g3adm5LePlwl58rUhiaOYlkKmAGVS6iHI2va8yiOYs4tPcQ58+dp2r1qnS8vSN3PeAc1//h/rw67lVG3DqC8+fO8/G2j+nzQB+OHDjC+OHjCSsbxi09b6HjHR3Zt3Nf1tg9h/Rk8mOTebDbg5xLOZe1lPxSVatXZdyMccx+bjYje4wkIjKCjt06cs8j97j8EitWqcioiaN464W3WPL2Euo2rsvQvw3l78P/7vIYt/7pVhI2JHDrn/JeCCHiCcbakv/2ivj4eLthw4YSP6+IS9LTYdAgmDePraPuVi++HD567SOWfbiMaZ9P83YpXtOtUbd87//J5RljNlpr4ws6TjMnkZzUzTxPZ8+cZf+v+1k0ZxG97+/t7XLEzymcRPKibua5TH9qOqsWr6LVLa3o0kfdHqR4KZxE8uOhZrH+YtTEUYyaOMrbZUiAUDiJXI4CSsQrFE4iBVFAiZQ4hZOIKxRQIiVK4STiKgWUSIlROIkUhgJKpEQonEQKSwElUuwUTiJFUYRu5iLiOoWTSFGpk4RIsVE4ibhDASVSLBROIu5SQIl4nMJJxBMUUCIepXAS8RQFlIjHKJxEPEndzEU8QuEk4mn6HJSI2xROIsVBASXiFo+EkzGmC/AiEAy8bq2d6IlxRUq1HAH16/bd3PvDdo4dOkZUzSgGjh7ITXfc5N0aRXxUkLsDGGOCgVeA24CrgH7GmKvcHVfEL1wIqF1t23L7p6sZejARay2JBxN55f9eYcWiFd6uUMQnuR1OQCtgp7X2V2vtOeBdoLsHxhXxD8HB3LQnlXnczjPAYxc2p6ak8tYLb3mzMhGf5YlwqgXsu+Tx/gvbsjHGjDDGbDDGbEhMTPTAaUVKh8OHYe+BOdzDQubRP1tAHTt0zJulifgsT4STyWObzbXB2hnW2nhrbXx0dLQHTivi+w4fhltuAbiKDIK5h5eYR5WsgIqqGeXdAkV8lCcWROwHal/yOAY46IFxRUq1zGDaujVzywkyuI17OI4FngHaX9eYDO+VKOKzPDFzWg80NMbUM8aEAX2Bjz0wrkiplTOYKlaE8ePXUqfOUawxPBkbm7VI4srpH3i3WBEf5PbMyVqbZoz5C/A5zlLyWdbaBLcrEyml8gqmpUuhVasuPPnk7osHpqfDoEE0vbAoYud9vUu+WBEf5ZHPOVlrlwBLPDGWSGny++8QGgrlyzuP8w+mPJ58aS8+BZRINuoQIeKGf/0L4uKgT59CBlOmnAFlrTpJiKBwEimy/fvhxRfh9tuhY8ciBFMmdTMXyUXhJFJEf/87pKTA4sWQkADbtjnbCxVMmdTNXCQbhZNIEWzZAm+84fyekuJmMGVSs1iRLAonkSJ4/HHIyOMDSv/7v3D99W4MrIASATzzOSeRgPLNN7BwYd77nngCataE8ePzDi+XZAZU//40nTxXn4OSgKSZk0ghWAuPPZb//ooVYcwYePhhCHLnn36XzqC0zFwCkMJJpBA++QRWr869PSwMHnoIxo6FqlU9dDIFlAQwhZOIi9LTc8+ajIF77nEu49WpUwwnVUBJgFI4ibhozhxnlV6m22+HZ56Ba64p5hMroCQAKZxEXHD2LDz5pPN769ZOZ4gbbyzBAjIDyhgFlAQEhZOIC15+GSIiYP586NHDuZxX4tSLTwKIwkmkAOnpUL8+bN4MId7+f4wCSgKEt/+vJuLzgoPhzju9XcUlFFASABROIqWRupmLn1OHCAk4Bw/CAw+40cHBV2QG1IABTieJae97uyIRj9HMSfzeCy/ARx9l37ZjBzRrBg8+6J2aPEbdzMVPKZzE740e7fxkOnYMBg+Gu+/2WkmepWax4ocUThJwIiOdmVSZMt6uxIOqa5JtAAAOi0lEQVRyBpQxWiQhpZrCSQJOWJi3Kygm6iQhfkQLIsSvpaQU/JOefvH4115zPtMUEgL33ee9uovs0q/beOEtfd2GlFqaOYlfu/562Lr18sds3Ogct22bs4rvgw+cFkUVKpRMjR6nGZT4AYWT+LXhw+H//s8JoJz3mEaPhn37Ln5z7ccfOyv4evYs+To9TgElpZzCSfzaPfc437G0aRP06XNx++nTsHw5PP+887hRI2d5OTh983r0gP/8p+Tr9SgFlJRiCifxa1WrQq9eMGtW9nB6/33nW2379XMer14NHTo4YTZ0KISHe6dej1M3cymlFE7i94YPh06dYO9eiI11ts2c6YRVZKTzODISfv0V2rWDGjW8V2uxUC8+KYW0Wk/83s03Q4MG8MYbzuPt22HNGie0Mm3eDGlp0Ly5d2osdpe2OtIqPikFFE4SEIYNg9mznUt5M2c6Cx9at764f9Mm52vWK1XyWonFL2dAqRef+DC3wskY09sYk2CMyTDGxHuqKBFPGzwYDhyAzz5zbsFcOmsCJ5z8dtZ0KTWLlVLC3XtOm4FewHQP1CJSbKpVg+7dnQ/WJiXl7qu3aZNzXyogXHoPSr34xEe5NXOy1m611m73VDEixem++5yvy+jdG6pUubjdWvjxxwCZOWXKDKj+/TWDEp+k1XoSMDp1chY95GQMnDxZ8vV4nbqZiw8rMJyMMcuBvBbXPmGtXejqiYwxI4ARALGZ63lFxLvUzVx8VIHhZK31yJV4a+0MYAZAfHy89cSYIuIB6iQhPkiX9UREASU+x61wMsb0BKYA0cBiY8wma+3/eKQyESlZCijxIW6Fk7X2P0Bpb48pIpkUUOIjdFlPRLJTQIkPUDiJSG7qZi5epnASkbypm7l4kcJJRPKngBIvUTiJyOXlDChr1UlCip3CSUQKpmaxUsIUTiLimsyAslYBJcVO4SQirlOzWCkhCicRKRwFlJQAhZOIFJ66mUsxUziJSNGok4QUI4WTiBSdAkqKicJJRNyjgJJioHASEfcpoMTDFE4i4hkKKPEghZOIeI66mYuHKJxExLPULFY8QOEkIp6ngBI3KZxEpHiom7m4QeEkIsVH3cyliBROIlK81M1cikDhJCLFT81ipZAUTiJSMhRQUggKJxEpOepmLi5SOIlIyVInCXGBwklESp4CSgqgcBIR71BAyWUonETEexRQkg+Fk4h4lwJK8uBWOBljngPuAM4BvwD3WmtPeKIwEQkg6mYuOQS5+fxlQDNr7bXAz8BY90sSkYCU2UliwACavvAWV07/wNsViRe5NXOy1i695OFa4E/ulSMiAU3dzOUCT95zGgK8l99OY8wIYARAbGysB08rIn5F3cwFF8LJGLMcqJHHriestQsvHPMEkAbMy28ca+0MYAZAfHy8LVK1IhIY1M084BUYTtbaTpfbb4wZBHQD/mCtVeiIiGcooAKau6v1ugB/Azpaa5M9U5KIyAUKqIDl7j2nl4EywDJjDMBaa+39blclIpJJARWQ3F2td6WnChERyVfOgFI3c7+nDhEiUjpomXlAUTiJSOmhgAoYCicRKV0UUAFB4SQipY8Cyu8pnESkdFJA+TWFk4iUXgoov6VwEpHSTQHllxROIlL6KaD8jsJJRPyDupn7FYWTiPgPtTryGwonEfEvCii/oHASEf+jgCr1FE4i4p8UUKWawklE/Je6mZdaCicR8W9aZl4qKZxExP8poEodhZOIBAZ/CChr4YMPoEsXiIz0djXFSuEkIoGjtAeUMXDmDPTpA5984rwePxXk7QJEREpUZkANGEDTF97iyukfeLeezZshJASWLcu+fepUqFs398/48fDZZ7BokXPcggUQFgY7dpRo2cVN4SQigceXAuqRR6BdO7j11uzbH3gAdu/O/TNsGAweDD16OMf16AHXXAN/+1tJVg0ZGfDCC9CkCZQtC7Vrw5gxzszOA3RZT0QCky9c4vv2W2fGtGCBa8dbC7fckjuIRo6EQYMgIQGuvtrzdeZl9Gh46SXo2dMJpa1bncfffw/Ll0OQe3MfhZOIBC5vB9Srr0LVqtC1q2vHGwNt2+be3quXM9OaNg2mTPFsjXlJSHDO06sXfPTRxe316sHDD8O770L//m6dQpf1RCSw5bzEN+1998dcv94JnMhIqFzZuRSXlATh4TBwoHNMWpozY7r1VggNzf78s2chJgZiYyE1Nfu+YcOcmt999+K28uWhQwdnJV9JeOcdZxY3alT27cOHO69x7ly3T6GZk4iIJ1sdffqpcx+oTh149FEnnGbNgttuc0InLs45buNGOH0aWrXKPUa5cs7Ch2HDnNnV6NHO9rFjYeZMeOUV6Ns3+3PatIHPP4dt25z7QDllZMDx466/jipV8r80t369sy9n7WXLQvPmzn43KZxERMAzAXX4sHM5Kz4eli6FiAhn+8CBzko7uBhOW7Y4fzZokPdYgwc7Cw6eecaZkbz+Okyc6ITWn/+c+/jMcRIS8g6nvXudy26u2rXrYs05HTwIUVFQpkzufbVqwZo1cO6cs4qwiBROIiKZ3A2oZ5+Fkyed2U1mMAFUquTMKFaudP4ESEx0/qxSJf9aJk6EO+5wZmJffgkPPQRPPpn38VWrOn8ePZr3/ho1ci9Xv5waNfLfl5ycdzCBM3vKPEbhJCLiIUUNKGvhvfegffu8Zy7gXOqrXNn53ZiLz8tPt25w/fXwxRfOZbwXX7z8+S8dN6eyZaFTp8u/BleFh+cfgikpF49xg8JJRCSnogTUkSPO5a4+fXLvs9a5jNeu3cVt0dHOn5e7D/T++7Bpk/N7hQr5B8+l42SOm1N6+sXZmiuio/PvQHHFFc7rSU3NPYM6cMC55OfGrAkUTiIieStsQCUnO3/mtYhgyRInGDIv6QE0a+b8mV9nh6VLnXtVPXs6q/lmzXIWRjRtmvfxO3dmHzenffs8d8+pZUunvu++c1YJZkpJccL0xhtdP08+3AonY8xTQHcgAzgKDLbWHnS7KhERX1CYgKpd22lDtGJF9u1Hjjj3iuDiYojM3yMjYe3a3GOtW+d8hqhdO5g3D/bvdz5PNHZs/h/YXbsWqleHxo3z3u/Je059+sA//wmTJ2cPp9dec0J6wADXz5Mfa22Rf4DIS35/GJjmyvNatGhhRURKjbQ0awcMsBbsllF324yMjLyPGzHCWrD2ttusnTbN2vHjra1Z09qGDZ3te/dmP37wYGsrV7Y2JeXiti1brK1a1dq4OGuTki5uv/9+Z4zVq3Of99Qpa8PDrf3LX9x/ra76y1+cenr2tPa116x95BFrQ0Ks7djR2vT0fJ8GbLCu5IsrB7k0EIwFprpyrMJJREqdSwIq4+mn8z7mzBlr//xna2vUsDYiwtqbbrL2s8+s7dbN2iuuyH38unXOX8Mffug83rPH2pgYa+vXt/bw4ezHHjxobbly1rZtm3uc2bOdcX76yb3XWBhpadY+/7y1jRpZGxbmvL7Ro52gvAxXw8nYy60UcYEx5h/APUAScLO1Ns87bsaYEcAIgNjY2BZ79uxx67wiIiUuPd3pYZeWBm+/7Vr/uMREp9vDkCFOp/GcunRxmqV+/XXR62rRwlkJOH9+0ccoIcaYjdba+AKPKyicjDHLgbwuPj5hrV14yXFjgbLW2nEFnTQ+Pt5u2LChoMNERHxPerqz+i4kxy378+ed1XSXbk9Jce4dLV8OP/2U9/2ghAS47jpn0UTnzoWvZ8ECuOsuZ5yGDQv//BLmajgVuCDCWuvqwvi3gcVAgeEkIlJq5be8+ptvnHZD/fo5PfH273d6zO3e7cyY8luocPXVzkysqHr0cLox+Bl3V+s1tNZmroP8I7DN/ZJEREqh8uWdUJo61WnyWqmS00F89uzsK9rEJe5+zmmiMaYxzlLyPcD97pckIlIKxcc7LYbEI9wKJ2vtnZ4qREREJJO+z0lERHyOwklERHyOwklERHyOwklERHyO2x0iinRSYxJxVvf5sijgmLeLKCX0XhWO3i/X6b0qnNLwftWx1ubzvR4XeSWcSgNjzAZXPsUseq8KS++X6/ReFY4/vV+6rCciIj5H4SQiIj5H4ZS/Gd4uoBTRe1U4er9cp/eqcPzm/dI9JxER8TmaOYmIiM9ROImIiM9ROLnAGPOoMcYaY6K8XYuvMsY8Z4zZZoz50RjzH2NMJW/X5GuMMV2MMduNMTuNMY95ux5fZoypbYz5yhiz1RiTYIwZ6e2afJ0xJtgY870x5hNv1+IJCqcCGGNqA7cCe71di49bBjSz1l4L/AyM9XI9PsUYEwy8AtwGXAX0M8Zc5d2qfFoaMMZa2xRoDTyo96tAI4Gt3i7CUxROBXsB+F9AK0cuw1q71Fqb+XWea4EYb9bjg1oBO621v1przwHvAt29XJPPstYestb+98Lvp3D+0q3l3ap8lzEmBrgdeN3btXiKwukyjDF/BA5Ya3/wdi2lzBDgU28X4WNqAfsuebwf/WXrEmNMXSAOWOfdSnzaZJx/RGd4uxBPcfebcEs9Y8xyoEYeu54AHgc6l2xFvuty75W1duGFY57AuSQzryRrKwVMHts0Gy+AMaY88BEwylp70tv1+CJjTDfgqLV2ozHmJm/X4ykBH07W2k55bTfGXAPUA34wxoBzmeq/xphW1trDJViiz8jvvcpkjBkEdAP+YPUBupz2A7UveRwDHPRSLaWCMSYUJ5jmWWvne7seH9YO+KMxpitQFog0xsy11t7t5brcog/husgYsxuIt9b6esdfrzDGdAEmAR2ttYnersfXGGNCcBaK/AE4AKwH+ltrE7xamI8yzr8I3wSOW2tHebue0uLCzOlRa203b9fiLt1zEk95GagALDPGbDLGTPN2Qb7kwmKRvwCf49zcf1/BdFntgIHALRf+e9p0YWYgAUIzJxER8TmaOYmIiM9ROImIiM9ROImIiM9ROImIiM9ROImIiM9ROImIiM9ROImIiM/5/8DDCMZAeK8qAAAAAElFTkSuQmCC\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "最大化，最小化と目的関数と制約条件の勾配ベクトルの向きによってラグランジュ関数がどうなるかみる．\n"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 最大化 and 目的関数と制約条件の勾配ベクトルの向きが同じ"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "下図の場合，最大化するには，$\\nabla f$の向きにそのまま動かせばよく，\nこれは制約条件を満たしている．\n制約条件なしの目的関数$f$の最大化$\\lambda =0$に等しい．\n"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-05-29T14:40:41.394304Z",
          "end_time": "2019-05-29T14:40:41.558352Z"
        },
        "hide_input": true,
        "trusted": true
      },
      "cell_type": "code",
      "source": "import scipy as sp\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\nx = sp.linspace(-3, 3)\ny = -x\nplt.plot(x, y, color=\"r\")\nplt.text(3.1, -3, \"$g(\\\\vec{x}) = 0$\", color=\"r\", fontsize=18)\n\ny2 = sp.zeros_like(x) + 3\nplt.fill_between(x,y,y2,facecolor='g',alpha=0.3)\n\n# 条件内\nplt.text(-2, 2.5, \"$g(\\\\vec{x}) > 0$\", color=\"g\", fontsize=18)\n\n# 条件外\nplt.text(-4.5, 2, \"$g(\\\\vec{x}) < 0$\", color=\"k\", fontsize=18)\n\n# 停留 Stationary point\nplt.scatter(0, 0, color=\"k\")\n# plt.text(-0.5, 0.6, \"StationaryP\", color=\"k\", fontsize=14)\n\n# 勾配g\nplt.quiver(0,0, 1,1, scale_units='xy', scale=1, linestyle=\"--\", color=\"r\") \nplt.text(1, 1, \"$\\\\vec{\\\\nabla} g$\", color=\"r\", fontsize=14)\n\n# 勾配 f\nplt.quiver(0,0, 1,1, scale_units='xy', scale=2, linestyle=\"--\", color=\"b\") \nplt.text(1/2, 0, \"$\\\\vec{\\\\nabla} f$\", color=\"b\", fontsize=14)\n\nplt.xlim(-3, 3)\nplt.ylim(-3, 3)\nplt.axis(\"equal\")\nplt.tight_layout()\nplt.show()",
      "execution_count": 55,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 最大化 and 目的関数と制約条件の勾配ベクトルの向きが逆"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "$\\vec{\\nabla} f$の向きに最大化すると，制約条件を外れてしまうので等式制約条件を入れなければならない．\n\n勾配ベクトルは平行し，向きが逆なので\n\n$$\n\\vec{\\nabla} f = \\lambda (-\\vec{\\nabla} g) \\\\\\\\\n\\vec{\\nabla} f + \\lambda \\vec{\\nabla} g = \\vec{0}, (\\lambda > 0)\n$$\n\nラグランジュ関数は，\n\n$$\nL(\\vec{x}, \\lambda) = f(\\vec{x}) + \\lambda g(\\vec{x}), (\\lambda > 0 )\n$$"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-05-29T14:44:26.922457Z",
          "end_time": "2019-05-29T14:44:27.097604Z"
        },
        "hide_input": true,
        "trusted": true,
        "scrolled": true
      },
      "cell_type": "code",
      "source": "import scipy as sp\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\nx = sp.linspace(-3, 3)\ny = -x\nplt.plot(x, y, color=\"r\")\nplt.text(3.1, -3, \"$g(\\\\vec{x}) = 0$\", color=\"r\", fontsize=18)\n\ny2 = sp.zeros_like(x) + 3\nplt.fill_between(x,y,y2,facecolor='g',alpha=0.3)\n\n# 条件内\nplt.text(-2, 2.5, \"$g(\\\\vec{x}) > 0$\", color=\"g\", fontsize=18)\n\n# 条件外\nplt.text(-4.5, 2, \"$g(\\\\vec{x}) < 0$\", color=\"k\", fontsize=18)\n\n# 停留 Stationary point\nplt.scatter(0, 0, color=\"k\")\n# plt.text(-0.5, 0.6, \"StationaryP\", color=\"k\", fontsize=14)\n\n# 勾配g\nplt.quiver(0,0, 1,1, scale_units='xy', scale=1, linestyle=\"--\", color=\"r\") \nplt.text(1, 1, \"$\\\\vec{\\\\nabla} g$\", color=\"r\", fontsize=14)\n\n# 勾配 f\nplt.quiver(0,0, -1, -1, scale_units='xy', scale=2, linestyle=\"--\", color=\"b\") \nplt.text(-1, -1, \"$\\\\vec{\\\\nabla} f$\", color=\"b\", fontsize=14)\n\nplt.xlim(-3, 3)\nplt.ylim(-3, 3)\nplt.axis(\"equal\")\nplt.tight_layout()\nplt.show()",
      "execution_count": 59,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 最小化 and 目的関数と制約条件の勾配ベクトルの向きが同じ"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "最小化により，$- \\vec{\\nabla} f$の方向に動かしていく．\n$- \\vec{\\nabla} f$は，$\\vec{\\nabla} g$の向きと逆向きになる．\n$- \\vec{\\nabla} f$の向きに動かしていくと，制約条件から外れてしまうので等式制約を含ませる必要がある．\n\n$$\n- \\vec{\\nabla} f = \\lambda (- \\vec{\\nabla}g )\\\\\\\\\n\\vec{\\nabla} f = \\lambda \\vec{\\nabla}g \\\\\\\\\n\\vec{\\nabla} f - \\lambda \\vec{\\nabla}g = \\vec{0}, (\\lambda > 0) \\\\\\\\\n\\vec{\\nabla} f + \\lambda \\vec{\\nabla}g = \\vec{0}, (\\lambda < 0) \\\\\\\\\n$$\n\nラグランジュ関数は，\n\n$$\nL(\\vec{x}, \\lambda) = f(\\vec{x}) - \\lambda g(\\vec{x}), (\\lambda > 0) \\\\\\\\\nL(\\vec{x}, \\lambda) = f(\\vec{x}) + \\lambda g(\\vec{x}), (\\lambda < 0) \\\\\\\\\n$$"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-05-29T15:15:22.956773Z",
          "end_time": "2019-05-29T15:15:23.132389Z"
        },
        "hide_input": true,
        "trusted": true
      },
      "cell_type": "code",
      "source": "import scipy as sp\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\nx = sp.linspace(-3, 3)\ny = -x\nplt.plot(x, y, color=\"r\")\nplt.text(3.1, -3, \"$g(\\\\vec{x}) = 0$\", color=\"r\", fontsize=18)\n\ny2 = sp.zeros_like(x) + 3\nplt.fill_between(x,y,y2,facecolor='g',alpha=0.3)\n\n# 条件内\nplt.text(-2, 2.5, \"$g(\\\\vec{x}) > 0$\", color=\"g\", fontsize=18)\n\n# 条件外\nplt.text(-4.5, 2, \"$g(\\\\vec{x}) < 0$\", color=\"k\", fontsize=18)\n\n# 停留 Stationary point\nplt.scatter(0, 0, color=\"k\")\n# plt.text(-0.5, 0.6, \"StationaryP\", color=\"k\", fontsize=14)\n\n# 勾配g\nplt.quiver(0,0, 1,1, scale_units='xy', scale=1, linestyle=\"--\", color=\"r\") \nplt.text(1, 1, \"$\\\\vec{\\\\nabla} g$\", color=\"r\", fontsize=14)\n\n# 勾配 f\nplt.quiver(0,0, 1, 1, scale_units='xy', scale=2, linestyle=\"--\", color=\"b\") \nplt.text(0, 0.5, \"$\\\\vec{\\\\nabla} f$\", color=\"b\", fontsize=14)\n\nplt.quiver(0,0, -1, -1, scale_units='xy', scale=2, linestyle=\"--\", color=\"b\") \nplt.text(-1, -1, \"$-\\\\vec{\\\\nabla} f$\", color=\"b\", fontsize=14)\n\nplt.xlim(-3, 3)\nplt.ylim(-3, 3)\nplt.axis(\"equal\")\nplt.tight_layout()\nplt.show()",
      "execution_count": 63,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 最小化 and 目的関数と制約条件の勾配ベクトルの向きが逆向き"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "不等式領域に動かしていくので,制約条件なしの最適化に等しい．\nつまり，$\\lambda = 0$\n"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-05-29T15:42:22.718232Z",
          "end_time": "2019-05-29T15:42:22.892063Z"
        },
        "hide_input": true,
        "trusted": true,
        "scrolled": true
      },
      "cell_type": "code",
      "source": "import scipy as sp\nimport matplotlib.pyplot as plt\n\n%matplotlib inline\n\nx = sp.linspace(-3, 3)\ny = -x\nplt.plot(x, y, color=\"r\")\nplt.text(3.1, -3, \"$g(\\\\vec{x}) = 0$\", color=\"r\", fontsize=18)\n\ny2 = sp.zeros_like(x) + 3\nplt.fill_between(x,y,y2,facecolor='g',alpha=0.3)\n\n# 条件内\nplt.text(-2, 2.5, \"$g(\\\\vec{x}) > 0$\", color=\"g\", fontsize=18)\n\n# 条件外\nplt.text(-4.5, 2, \"$g(\\\\vec{x}) < 0$\", color=\"k\", fontsize=18)\n\n# 停留 Stationary point\nplt.scatter(0, 0, color=\"k\")\n# plt.text(-0.5, 0.6, \"StationaryP\", color=\"k\", fontsize=14)\n\n# 勾配g\nplt.quiver(0,0, 1,1, scale_units='xy', scale=1, linestyle=\"--\", color=\"r\") \nplt.text(1, 1, \"$\\\\vec{\\\\nabla} g$\", color=\"r\", fontsize=14)\n\n# 勾配 f\nplt.quiver(0,0, 1, 1, scale_units='xy', scale=2, linestyle=\"--\", color=\"b\") \nplt.text(-0.3, 0.5, \"$-\\\\vec{\\\\nabla} f$\", color=\"b\", fontsize=14)\n\nplt.quiver(0,0, -1, -1, scale_units='xy', scale=2, linestyle=\"--\", color=\"b\") \nplt.text(-1, -1, \"$\\\\vec{\\\\nabla} f$\", color=\"b\", fontsize=14)\n\nplt.xlim(-3, 3)\nplt.ylim(-3, 3)\nplt.axis(\"equal\")\nplt.tight_layout()\nplt.show()",
      "execution_count": 67,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "<Figure size 432x288 with 1 Axes>",
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### まとめる"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "以上をまとめる．\n\n$$\n\\text{subject to } g(\\vec{x}) \\geq 0 \\\\\\\\\n\\text{max} f(\\vec{x}) \\\\\\\\\n\\Rightarrow \nL(\\vec{x}, \\lambda) = f(\\vec{x}) + \\lambda g(\\vec{x}), (\\lambda \\geq 0)\n$$\n\n$$\n\\text{subject to } g(\\vec{x}) \\geq 0 \\\\\\\\\n\\text{min} f(\\vec{x}) \\\\\\\\\n\\Rightarrow \nL(\\vec{x}, \\lambda) = f(\\vec{x}) - \\lambda g(\\vec{x}), (\\lambda \\geq 0)\n$$\n"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 相補条件"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "- 等式条件（境界）上に停留点があれば，$g(\\vec{x}) = 0$\n\n- 不等式領域$g(\\vec{x}) > 0$に停留点があれば，制約を考える必要がない通常の最適化より$\\lambda = 0$\n\n以上をまとめると，以下の条件を満たす．\n\n$$\n\\lambda g(\\vec{x}) = 0\n$$\n\nこれを特にKKT条件の中では相補条件という．\n\n"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "",
      "execution_count": null,
      "outputs": []
    }
  ],
  "metadata": {
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3",
      "language": "python"
    },
    "hide_input": false,
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      "number_sections": true,
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      "title_cell": "Table of Contents",
      "title_sidebar": "Contents",
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      "toc_position": {
        "height": "calc(100% - 180px)",
        "width": "165px",
        "left": "10px",
        "top": "150px"
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      "toc_section_display": true,
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    },
    "language_info": {
      "name": "python",
      "version": "3.7.0",
      "mimetype": "text/x-python",
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
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      "pygments_lexer": "ipython3",
      "nbconvert_exporter": "python",
      "file_extension": ".py"
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    "gist": {
      "id": "",
      "data": {
        "description": "Lagrange",
        "public": true
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  "nbformat": 4,
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