Spring2021/int_point.ipynb

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 "cells": [
  {
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   "metadata": {
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    }
   },
   "source": [
    "# Методы внутренней точки"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## На прошлом семинаре\n",
    "\n",
    "- Задачи оптимизации с ограничениями на простые множества\n",
    "- Метод проекции градиента как частный случай проксимального градиентного метода\n",
    "- Метод условного градента (Франка-Вольфа) и его сходимость"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "## Задача выпуклой оптимизации с ограничениями типа равенств\n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "&\\min f(x) \\\\ \n",
    "\\text{s.t. } & Ax = b,\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "где $f$ - выпукла и дважды дифференцируема, $A \\in \\mathbb{R}^{p \\times n}$ и $\\mathrm{rank} \\; A = p < n$"
   ]
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   "source": [
    "### Двойственная задача\n",
    "Двойственная функция \n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "g(\\mu) &  = -b^{\\top}\\mu + \\inf_x(f(x) + \\mu^{\\top}Ax) \\\\\n",
    "& = -b^{\\top}\\mu - \\sup_x((-A^{\\top}\\mu)^{\\top}x -f(x)) \\\\\n",
    "& = -b^{\\top}\\mu - f^*(-A^{\\top}\\mu)\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "Двойственная задача\n",
    "\n",
    "$$\n",
    "\\max_\\mu -b^{\\top}\\mu - f^*(-A^{\\top}\\mu)\n",
    "$$\n",
    "\n",
    "**Подход 1**: найти сопряжённую функцию и решить безусловную задачу оптимизации"
   ]
  },
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   "metadata": {
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   "source": [
    "**Трудности**\n",
    "- не всегда легко восстановить решение прямой задачи по решению двойственной\n",
    "- сопряжённая функция $f^*$ должна быть дважды дифференцируемое для быстрого решения двойственной задачи. Это не всегда так."
   ]
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   "source": [
    "### Условия оптимальности\n",
    "\n",
    "- $Ax^* = b$\n",
    "- $f'(x^*) + A^{\\top}\\mu^* = 0$\n",
    "\n",
    "или\n",
    "\n",
    "$$ \\begin{bmatrix} f' & A^{\\top} \\\\ A & 0 \\end{bmatrix} \\begin{bmatrix} x^{\\\\*} \\\\ \\mu^{\\\\*} \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ b \\end{bmatrix} $$\n",
    "\n",
    "**Подход 2**: решить нелинейную в общем случае систему методом Ньютона.\n",
    "\n",
    "**Вопрос**: в каком случае система окажется линейной?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "## Метод Ньютона для выпуклых задач с ограничениями типа равенств\n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "& \\min_v f(x) + f'(x)^{\\top}v + \\frac{1}{2}v^{\\top}f''(x)v\\\\\n",
    "\\text{s.t. } & A(x + v) = b\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "Из условий оптимальности имеем\n",
    "\n",
    "$$ \\begin{bmatrix} f''(x) & A^{\\top} \\\\ A & 0 \\end{bmatrix} \\begin{bmatrix} v \\\\ w \\end{bmatrix} = \\begin{bmatrix} -f'(x) \\\\ 0 \\end{bmatrix} $$\n",
    "\n",
    "**Шаг метода Ньютона определён только для невырожденной матрицы!**"
   ]
  },
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   "metadata": {
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   },
   "source": [
    "**Упражнение**. Посчитайте за сколько итераций метод Ньютона сойдётся для квадратичной функции с ограничениями типа равенств."
   ]
  },
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   },
   "source": [
    "### Линеаризация условий оптимальности\n",
    "\n",
    "- $A(x + v) = b \\rightarrow Av = 0$\n",
    "- $f'(x + v) + A^{\\top}w \\approx f'(x) + f''(x)v + A^{\\top}w = 0$\n",
    "\n",
    "или\n",
    "\n",
    "- $f''(x)v + A^{\\top}w = -f'(x)$"
   ]
  },
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   "metadata": {
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   "source": [
    "### Псевдокод\n",
    "\n",
    "**Важно:** начальная точка должна лежать в допустимом множестве!\n",
    "\n",
    "```python\n",
    "def NewtonEqualityFeasible(f, gradf, hessf, A, b, stop_crit, line_search, x0, tol):\n",
    "    \n",
    "    x = x0\n",
    "    \n",
    "    n = x.shape[0]\n",
    "    \n",
    "    while True:\n",
    "        \n",
    "        newton_matrix = [[hessf(x), A.T], [A, 0]]\n",
    "        \n",
    "        rhs = [-gradf(x), 0]\n",
    "        \n",
    "        w = solve_lin_sys(newton_matrix, rhs)\n",
    "        \n",
    "        h = w[:n]\n",
    "        \n",
    "        if stop_crit(x, h, gradf(x), **kwargs) < tol:\n",
    "            \n",
    "            break\n",
    "            \n",
    "        alpha = line_search(x, h, f, gradf(x), **kwargs)\n",
    "        \n",
    "        x = x + alpha * h\n",
    "        \n",
    "    return x\n",
    "```"
   ]
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   "metadata": {
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   "source": [
    "### Критерий остановки\n",
    "Получим выражение для значения\n",
    "\n",
    "$$\n",
    "f(x) - \\inf_v(\\hat{f}(x + v) \\; | \\; A(x+v) = b),\n",
    "$$\n",
    "\n",
    "где $\\hat{f}$ - квадратичная аппроксимация функции $f$.\n",
    "\n",
    "Для этого \n",
    "\n",
    "$$\n",
    "\\langle h^{\\top} \\rvert \\cdot \\quad f''(x)h + A^{\\top}w = -f'(x)\n",
    "$$\n",
    "\n",
    "с учётом $Ah = 0$ получаем \n",
    "\n",
    "$$\n",
    "h^{\\top}f''(x)h = -f'(x)^{\\top}h\n",
    "$$\n",
    "\n",
    "Тогда \n",
    "\n",
    "$$\n",
    "\\inf_v(\\hat{f}(x + v) \\; | \\; A(x+v) = b) = f(x) - \\frac{1}{2}h^{\\top}f''(x)h\n",
    "$$\n",
    "\n",
    "**Вывод:** величина $h^{\\top}f''(x)h$ является наиболее адекватным критерием остановки метода Ньютона."
   ]
  },
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   "metadata": {
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   "source": [
    "### Теорема сходимости\n",
    "\n",
    "Сходимость метода аналогична сходимости метода Ньютона для задачи безусловной оптимизации.\n",
    "\n",
    "**Теорема**\n",
    "Пусть выполнены следующие условия\n",
    "- множество уровней $S = \\{ x \\; | \\; x \\in D(f), \\; f(x) \\leq f(x_0), \\; Ax = b \\}$ замкнуто и $x_0 \\in D(f), \\; Ax_0 = b$\n",
    "-  для любых $x \\in S$ и $\\tilde{x} \\in S$ гессиан $f''(x)$ липшицев\n",
    "- на множестве $S$ $\\|f''(x)\\|_2 \\leq M $ и норма обратной матрицы KKT системы ограничена сверху\n",
    "\n",
    "Тогда, метод Ньютона сходится к паре $(x^*, \\mu^*)$ линейно, а при достижении достаточной близости к решению - квадратично."
   ]
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   "source": [
    "## Случай недопустимой начальной точки\n",
    "\n",
    "- Метод Ньютона требует чтобы начальная точка лежала в допустимом множестве\n",
    "- Что делать, если поиск такой точки неочевиден: например, если область определения $f$ не сопадает с $\\mathbb{R}^n$\n",
    "- Пусть начальная точка не является допустимой, в этом случае условия KKT можно записать так\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "f''(x) & A^{\\top}\\\\\n",
    "A & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "v\\\\\n",
    "w\n",
    "\\end{bmatrix}\n",
    " = -\n",
    "\\begin{bmatrix}\n",
    "f'(x)\\\\\n",
    "{\\color{red}{Ax - b}}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "- Если $x$ допустима, то система совпадает с системой для обычного метода Ньютона"
   ]
  },
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   "source": [
    "### Прямо-двойственная интерпретация\n",
    "\n",
    "- Метод *прямо-двойственный*, если на каждой итерации обновляются прямые и двойственные переменные\n",
    "- Покажем, что это значит. Для этого запишем условия оптимальности в виде\n",
    "\n",
    "$$\n",
    "r(x^*, \\mu^*) = (r_d(x^*, \\mu^*), r_p(x^*, \\mu^*)) = 0,\n",
    "$$\n",
    "\n",
    "где $r_p(x, \\mu) = Ax - b$ и $r_d(x, \\mu) = f'(x) + A^{\\top}\\mu$\n",
    "- Решим систему методом Ньютона:\n",
    "\n",
    "$$\n",
    "r(y + z) \\approx r(y) + Dr(y)z = 0\n",
    "$$\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "- Прямо-двойственный шаг в методе Ньютона определим как решение линейной системы\n",
    "\n",
    "$$\n",
    "Dr(y)z = -r(y)\n",
    "$$\n",
    "\n",
    "или более подробно\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "f''(x) & A^{\\top}\\\\\n",
    "A & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "z_p\\\\\n",
    "z_d\n",
    "\\end{bmatrix}\n",
    " = -\n",
    "\\begin{bmatrix}\n",
    "r_d(x, \\mu)\\\\\n",
    "r_p(x, \\mu)\n",
    "\\end{bmatrix}\n",
    "= - \n",
    "\\begin{bmatrix}\n",
    "f'(x) + A^{\\top}\\mu\\\\\n",
    "Ax - b\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "- Заменим $z_d^+ = \\mu + z_d$ и получим\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "f''(x) & A^{\\top}\\\\\n",
    "A & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "z_p\\\\\n",
    "z_d^+\n",
    "\\end{bmatrix}\n",
    "= - \n",
    "\\begin{bmatrix}\n",
    "f'(x)\\\\\n",
    "Ax - b\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "- Система полностью эквивалентна ранее полученной в обозначениях \n",
    "\n",
    "$$\n",
    "v = z_p \\qquad w = z_d^+ = \\mu + z_d \n",
    "$$\n",
    "\n",
    "- Метод Ньютона даёт шаг для прямой переменной и обновлённое значение для двойственной"
   ]
  },
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   "metadata": {
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   },
   "source": [
    "### Способ инициализации\n",
    "\n",
    "- Удобный способ задания начального приближения: найти точку из области определения $f$ гораздо проще, чем из пересечения области определения и допустимого множества\n",
    "- Метод Ньютона с недопустимой начальной точкой не может определить согласованность ограничений"
   ]
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   "source": [
    "### Псевдокод\n",
    "\n",
    "```python\n",
    "def NewtonEqualityInfeasible(f, gradf, hessf, A, b, stop_crit, line_search, x0, mu0, tol):\n",
    "    \n",
    "    x = x0\n",
    "    \n",
    "    mu = mu0\n",
    "    \n",
    "    n = x.shape[0]\n",
    "    \n",
    "    while True:\n",
    "        \n",
    "        z_p, z_d = ComputeNewtonStep(hessf(x), A, b)\n",
    "        \n",
    "        if stop_crit(x, z_p, z_d, gradf(x), **kwargs) < tol:\n",
    "            \n",
    "            break\n",
    "            \n",
    "        alpha = line_search(x, z_p, z_d, f, gradf(x), **kwargs)\n",
    "        \n",
    "        x = x + alpha * z_p\n",
    "        \n",
    "        mu = z_d\n",
    "        \n",
    "    return x\n",
    "```"
   ]
  },
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   "metadata": {
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   },
   "source": [
    "### Критерий остановки и линейный поиск\n",
    "\n",
    "- Изменение $r_p$ после шага $z_p$\n",
    "\n",
    "$$\n",
    "A(x + \\alpha z_p) - b = [A(x + z_p) = b] = Ax + \\alpha(b - Ax) - b = (1 - \\alpha)(Ax - b)\n",
    "$$\n",
    "\n",
    "- Итоговое изменение после $k$ шагов\n",
    "\n",
    "$$\n",
    "r^{(k)} = \\prod_{i=0}^{k-1}(1 - \\alpha^{(i)})r^{(0)}\n",
    "$$\n",
    "\n",
    "- Критерий остановки: $Ax = b$ и $\\|r(x, \\mu)\\|_2 \\leq \\varepsilon$"
   ]
  },
  {
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   "metadata": {
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    }
   },
   "source": [
    "- Линейный поиск: $c \\in (0, 1/2)$, $\\beta = (0, 1)$\n",
    "\n",
    "```python\n",
    "def linesearch(r, x, mu, z_p, z_d, c, beta):\n",
    "    \n",
    "    alpha = 1\n",
    "    \n",
    "    while np.linalg.norm(r(x + alpha * z_p, mu + alpha * z_d)) >= (1 - c * alpha) * np.linalg.norm(r(x, mu)):\n",
    "        \n",
    "        alpha *= beta\n",
    "        \n",
    "    return alpha\n",
    "\n",
    "```"
   ]
  },
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   "metadata": {
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   },
   "source": [
    "### Теорема сходимости\n",
    "\n",
    "Результат аналогичен результаты для допустимой начальной точки\n",
    "\n",
    "**Теорема.** Пусть\n",
    "- множество подуровней $S = \\{(x, \\mu) \\; | \\; x \\in D(f), \\; \\| r(x, \\mu) \\|_2 \\leq \\| r(x_0, \\mu_0)  \\|_2 \\}$ замкнуто\n",
    "- на множестве $S$ норма матрицы обратной к ККТ матрице ограничена\n",
    "- гессиан липшицев на $S$.\n",
    "\n",
    "Тогда сходимость метода линейна при удалении от решении и квадратичная при достаточном приближении к решению."
   ]
  },
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   "metadata": {
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    }
   },
   "source": [
    "## Общая задача выпуклой оптимизации\n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "& \\min_{x \\in \\mathbb{R}^n} f_0(x)\\\\\n",
    "\\text{s.t. } & f_i (x) \\leq 0 \\qquad i=1,\\ldots,m\\\\\n",
    "& Ax = b,\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "где $f_i$ - выпуклые и дважды непрерывно дифференцируемы, $A \\in \\mathbb{R}^{p \\times n}$ и $\\mathrm{rank} \\; A = p < n$. \n",
    "\n",
    "Предполагаем, что задача строго разрешима, то есть выполняется условие Слейтера."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "## Условия оптимальности\n",
    "\n",
    "- Разрешимость прямой задачи\n",
    "\n",
    "$$\n",
    "Ax^* = b, \\; f_i(x^*) \\leq 0, \\; i = 1,\\ldots,m\n",
    "$$\n",
    "\n",
    "- Разрешимость двойственной задачи\n",
    "\n",
    "$$\n",
    "\\lambda^* \\geq 0\n",
    "$$\n",
    "\n",
    "- Стационарность лагранжиана\n",
    "\n",
    "$$\n",
    "f'_0(x^*) + \\sum_{i=1}^m \\lambda^*_if'_i(x^*) + A^{\\top}\\mu^* = 0\n",
    "$$\n",
    "\n",
    "- Условие дополняющей нежёсткости\n",
    "\n",
    "$$\n",
    "\\lambda^*_i f_i(x^*) = 0, \\qquad i = 1,\\ldots, m\n",
    "$$"
   ]
  },
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   "metadata": {
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   },
   "source": [
    "## Идея\n",
    "\n",
    "- Свести задачу с ограничениями типа **неравенств** к последовательности задач с ограничениями типа **равенств**\n",
    "- Использовать методы для решения задачи с ограничениями типа равенств"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "& \\min f_0(x) + \\sum_{i=1}^m I_-(f_i(x))\\\\\n",
    "\\text{s.t. } & Ax = b,\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "где $I_-$ - индикаторная функция\n",
    "\n",
    "$$\n",
    "I_-(u) = \n",
    "\\begin{cases}\n",
    "0, & u \\leq 0\\\\\n",
    "\\infty, & u > 0\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**Проблема.** Теперь целевая функция - **недифференцируема**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "## Логарифмический барьер\n",
    "\n",
    "**Идея.** Приблизить функцию $I_-(u)$ функцией\n",
    "\n",
    "$$\n",
    "\\hat{I}_-(u) = -t\\log(-u),\n",
    "$$\n",
    "\n",
    "где $t > 0$ - параметр.\n",
    "\n",
    "- Функции $I_-(u)$ и $\\hat{I}_-(u)$ выпуклые и неубывающие\n",
    "- Однако $\\hat{I}_-(u)$ **дифференцируема** и приближается к $I_-(u)$ при $t \\to 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 0, 'u')"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "x = np.linspace(-2, 0, 100000, endpoint=False)\n",
    "plt.figure(figsize=(10, 6))\n",
    "for t in [0.1, 0.5, 1, 1.5, 2]:\n",
    "    plt.plot(x, -t * np.log(-x), label=r\"$t = \" + str(t) + \"$\")\n",
    "plt.legend(fontsize=20)\n",
    "plt.xticks(fontsize=20)\n",
    "plt.yticks(fontsize=20)\n",
    "plt.xlabel(\"u\", fontsize=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### \"Ограниченная\" задача\n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "& \\min f_0(x) + \\sum_{i=1}^m -t \\log(-f_i(x))\\\\\n",
    "\\text{s.t. } & Ax = b\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "- Задача по-прежнему **выпуклая**\n",
    "- Функция \n",
    "\n",
    "$$\n",
    "\\phi(x) = -\\sum\\limits_{i=1}^m \\log(-f_i(x))\n",
    "$$ \n",
    "называется *логарифмическим барьером*. Её область определения - множество точек, для котороых ограничения типа неравенств выполняются строго.\n",
    "\n",
    "**Упражнение.** Найдите градиент и гессиан $\\phi(x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Центральный путь\n",
    "\n",
    "Для каждого $t > 0$ \"ограниченная\" задача имеет единственное решение $x^*(t)$.\n",
    "\n",
    "**Определение.** Последовательность $x^*(t)$ для $t > 0$ образует *центральный путь*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Условия оптимальности для \"ограниченной\" задачи\n",
    "\n",
    "- Разрешимость прямой задачи\n",
    "\n",
    "$$\n",
    "Ax^*(t) = b, \\; f_i(x^*) < 0, \\; i = 1,\\ldots,m\n",
    "$$\n",
    "\n",
    "- Стационарность лагранжиана\n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "& f'_0(x^*(t)) + \\phi'(x^*(t)) + A^{\\top}\\hat{\\mu} = \\\\\n",
    "& = f'_0(x^*(t)) - t\\sum_{i=1}^m \\frac{f_i'(x^*(t))}{f_i(x^*(t))} + A^{\\top}\\hat{\\mu} = 0\n",
    "\\end{split}\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- Обозначим \n",
    "\n",
    "$$\n",
    "\\lambda^*_i(t) = -\\frac{t}{f_i(x^*(t))} \\; i=1,\\ldots,m \\text{ и } \\mu^* = \\hat{\\mu}\n",
    "$$\n",
    "\n",
    "- Тогда условие оптимальности можно записать как\n",
    "\n",
    "$$\n",
    "f'_0(x^*(t)) + \\sum_{i=1}^m \\lambda^*_i(t)f_i'(x^*(t)) + A^{\\top}\\mu^* = 0\n",
    "$$\n",
    "\n",
    "- Тогда $x^*(t)$ минимизирует лагранжиан \n",
    "\n",
    "$$\n",
    "L = f_0(x) + \\sum_{i=1}^m \\lambda_if_i(x) + \\mu^{\\top}(Ax - b)\n",
    "$$\n",
    "\n",
    "для $\\lambda = \\lambda^*(t)$ и $\\mu = \\mu^*$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Зазор двойственности\n",
    "\n",
    "- Двойственная функция $g(\\lambda^*(t), \\mu^*)$ конечна и представима в виде\n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "g(\\lambda^*(t), \\mu^*) & = f_0(x^*(t)) + \\sum_{i=1}^m \\lambda^*_i(t)f_i(x^*(t)) + (\\mu^*)^{\\top}(Ax^*(t) - b)\\\\\n",
    "& = f_0(x^*(t)) - mt\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "- Зазор двойственности\n",
    "\n",
    "$$\n",
    "f_0(x^*(t)) - p^* \\leq mt\n",
    "$$\n",
    "\n",
    "- При $t \\to 0$ зазор двойственности равен 0 и центральный путь сходится к решению исходной задачи."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## ККТ интерпретация\n",
    "\n",
    "Условия оптимальности для \"ограниченной\" задачи эквивалентны условиям оптимальности для исходной задачи если\n",
    "\n",
    "$$\n",
    "-\\lambda_i f_i(x) = 0 \\to - \\lambda_i f_i(x) = t \\quad i = 1,\\ldots, m\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Физическая интерпретация\n",
    "- Предположим, что ограничений типа равенства нет\n",
    "- Рассмотрим неквантовую частицу в поле сил\n",
    "- Каждому ограничению $f_i(x) \\leq 0$ поставим в соответствие силу\n",
    "$$\n",
    "F_i(x) = -\\nabla(-\\log(-f_i(x))) = \\frac{f'_i(x)}{f_i(x)}\n",
    "$$\n",
    "- Целевой функции также поставим в соответствие силу \n",
    "$$\n",
    "F_0(x) = -\\frac{f'_0(x)}{t}\n",
    "$$\n",
    "- Каждая точка из центрального пути $x^*(t)$ - это положение частицы, в котором выполняется баланс сил ограничений и целевой функции\n",
    "- С уменьшением $t$ сила для целевой функции доминирует, и частица стремится занять положение, расположенное ближе к оптимальному\n",
    "- Поскольку сила ограничений стремится к бесконечности при приближении частицы к границе, частица никогда не вылетит из допустимого множества"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Барьерный метод\n",
    "\n",
    "- $x_0$ должна быть допустимой\n",
    "- $t_0 > 0$ - начальное значение параметра\n",
    "- $\\alpha \\in (0, 1)$ - множитель для уменьшения $t_0$\n",
    "\n",
    "```python\n",
    "def BarrierMethod(f, x0, t0, tol, alpha, **kwargs):\n",
    "    \n",
    "    x = x0\n",
    "    \n",
    "    t = t0\n",
    "    \n",
    "    while True:\n",
    "        \n",
    "        x = SolveBarrierProblem(f, t, x, **kwargs)\n",
    "        \n",
    "        if m * t < tol:\n",
    "            \n",
    "            break\n",
    "            \n",
    "        t *= alpha\n",
    "        \n",
    "    return x\n",
    "\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Точность решения \"ограниченной\" задачи\n",
    "\n",
    "- Точное решение \"ограниченной\" задачи не требуется, так как приближённый центральный путь всё равно сойдётся к решению исходной задачи\n",
    "- Двойственные переменные перестают быть двойственными при неточном решении, но это поправимо введением поправочных слагаемых\n",
    "- Разница в стоимости точного и неточного центрального пути - несколько итераций метода Ньютона, поэтому существенного ускорения добиться нельзя"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Выбор параметров\n",
    "- Множитель $\\alpha$\n",
    "    - При $\\alpha \\sim 1$, **мало** итераций нужно для решения \"ограниченной\" задачи, но **много** для нахождения точного решения исходной задачи\n",
    "    - При $\\alpha \\sim 10^{-5}$ **много** итераций нужно для решения \"ограниченной\" задачи, но **мало** для нахождения точного решения исходной задачи\n",
    "- Начальный параметр $t_0$\n",
    "    - Аналогичная альтернатива как и для параметра $\\alpha$\n",
    "    - Параметр $t_0$ задаёт начальную точку для центрального пути"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Почти теорема сходимости\n",
    "\n",
    "- Как было показано выше при $t \\to 0$ барьерный метод сходится к решению исходной задачи\n",
    "- Скорость сходимости напрямую связана с параметрами $\\alpha$ и $t_0$, как показано ранее\n",
    "- Основная сложность - быстрое решение вспомогательных задач методом Ньютона"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Задача поиска допустимого начального приближения\n",
    "\n",
    "- Барьерный метод требует допустимого начального приближения\n",
    "- Метод разбивается на две фазы\n",
    "    - Первая фаза метода ищет допустимое начальное приближение\n",
    "    - Вторая фаза использует найденное начальное приближение для запуска барьерного метода"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Первая фаза метода\n",
    "\n",
    "Простой поиск допустимой точки\n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "& \\min s\\\\\n",
    "\\text{s.t. } & f_i(x) \\leq s\\\\\n",
    "& Ax = b\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "- эта задача всегда имеет строго допустимое начальное приближение\n",
    "- если $s^* < 0$, то $x^*$ строго допустима и может быть использована в барьерном методе\n",
    "- если $s^* > 0$, то задача не разрешима и допустимое множество пусто"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Сумма несогласованностей\n",
    "\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "& \\min s_1 + \\ldots + s_m\\\\\n",
    "\\text{s.t. } & f_i(x) \\leq s_i\\\\\n",
    "& Ax = b\\\\\n",
    "& s \\geq 0\n",
    "\\end{split}\n",
    "\\end{equation*}\n",
    "\n",
    "- оптимальное значене равно нулю и достигается тогда и только тогда когда система ограничений совместна\n",
    "- если задача неразрешима, то можно определить какие ограничения к этому приводят, то есть какие $s_i > 0$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Вторая фаза метода\n",
    "\n",
    "- После получения допустимой начальной точки $x_0$ выполняется обычный метод Ньютона для задачи с ограничениями равенствами"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Прямо-двойственный метод\n",
    "\n",
    "Похож на барьерный метод, но\n",
    "- нет разделения на внешние итерации и внутренние: на каждой итерации обновляются прямые и двойственные переменные\n",
    "- направление определяется методом Ньютона, применённого к модифицированной системе ККТ\n",
    "- последовательность точек в прямо-двойственном методе не обязательно допустимы \n",
    "- работает даже когда задача не строго допустима"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Сходимость для квадратичной целевой функции\n",
    "\n",
    "При некоторых предположениях о начальной точке и начальном значении $\\mu$, можно показать, что для достижения $\\mu_k \\leq \\varepsilon$ потребуется \n",
    "\n",
    "$$\n",
    "\\mathcal{O}\\left(\\sqrt{n}\\log \\left( \\frac{1}{\\varepsilon}\\right)\\right)\n",
    "$$ \n",
    "итераций\n",
    "\n",
    "\n",
    "Доказательство и все детали можно найти [тут](https://epubs.siam.org/doi/book/10.1137/1.9781611971453?mobileUi=0) или [тут](https://www.maths.ed.ac.uk/~gondzio/reports/ipmXXV.pdf)\n",
    "\n",
    "- Сравните с методами типа градиентного спуска, которые дают сходимость типа $\\mathcal{O}\\left( \\frac{1}{\\varepsilon} \\right)$\n",
    "- Зависит от размерности как $\\sqrt{n}$\n",
    "- На практике зависимость от размерности ещё слабее"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Резюме\n",
    "\n",
    "- Метод Ньютона для выпуклой задачи с оганичениями типа равенств\n",
    "- Случай недопустимой начальной точки\n",
    "- Прямой барьерный метод\n",
    "- Прямо-двойственный метод"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Применение методов внутренней точки к задаче линейного программирования\n",
    "\n",
    "Исходная задача\n",
    "\\begin{align*}\n",
    "&\\min_x c^{\\top}x \\\\\n",
    "\\text{s.t. } & Ax = b\\\\\n",
    "& x_i \\geq 0, \\; i = 1,\\dots, n\n",
    "\\end{align*}\n",
    "\n",
    "Аппроксимированная задача\n",
    "\\begin{align*}\n",
    "&\\min_x c^{\\top}x {\\color{red}{- \\mu \\sum\\limits_{i=1}^n \\ln x_i}} \\\\\n",
    "\\text{s.t. } & Ax = b\\\\\n",
    "\\end{align*}\n",
    "для некоторого $\\mu > 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Барьерная функция\n",
    "\n",
    "**Определение.** Функция $B(x, \\mu) = -\\mu\\ln x$ называется *барьерной* для задачи с ограничением $x \\geq 0$.\n",
    "\n",
    "Более подробно о таких функциях будет рассказано в контексте нелинейной условной оптимизации..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Что произошло?\n",
    "\n",
    "- Сделали из линейной задачу нелинейную\n",
    "- Перенесли ограничение типа неравенства в целевую функцию\n",
    "- Ввели дополнительный параметр $\\mu$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Почему это хорошо?\n",
    "\n",
    "Переход к задаче с ограничениями типа равенств $\\to$ упрощение условий оптимальности, в частности\n",
    "\n",
    "- Исключено требование дополняющей нежёсткости\n",
    "- Исключено условие неотрицательности множителя Лагранжа для ограничения типа неравенства"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Условия оптимальности\n",
    "\n",
    "- Лагранжиан: $L = c^{\\top}x - \\mu\\sum\\limits_{i=1}^n \\ln x_i + \\lambda^{\\top}(Ax - b)$\n",
    "- Стационарная точка $L$: \n",
    "\n",
    "$$\n",
    "c - \\mu X^{-1}e + A^{\\top}\\lambda = 0,\n",
    "$$\n",
    "\n",
    "где $X = \\mathrm{diag}(x_1, \\dots, x_n)$ и $e = [1, \\dots, 1]$\n",
    "- Ограничение типа равенства: $Ax = b$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Пусть $s = \\mu X^{-1}e$, тогда условия оптимальности можно переписать так:\n",
    "- $A^{\\top}\\lambda + c - s = 0 $\n",
    "- $Xs = {\\color{red}{\\mu e}}$\n",
    "- $Ax = b$\n",
    "\n",
    "Также $x > 0 \\Rightarrow s > 0$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сравнение с условиями оптимальности для исходной задачи\n",
    "\n",
    "- Лагранжиан: $L = c^{\\top}x + \\lambda^{\\top}(Ax - b) - s^{\\top}x$\n",
    "- Условие стационарности: $c + A^{\\top}\\lambda - s = 0$\n",
    "- Допустимость прямой задачи: $Ax = b, \\; x \\geq 0$\n",
    "- Допустимость двойственной: $s \\geq 0$\n",
    "- Условие дополняющей нежёсткости: $s_ix_i = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### После упрощения\n",
    "\n",
    "- $A^{\\top}\\lambda + c - s = 0$\n",
    "- $Ax = b$\n",
    "- $Xs = {\\color{red}{0}}$\n",
    "- $x \\geq 0, \\; s \\geq 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Вывод\n",
    "\n",
    "- Введение барьерной функции c множителем $\\mu$ эквивалентно релаксации условий дополняющей нежёсткости на параметр $\\mu$\n",
    "- При $\\mu \\to 0$ решения задач совпадают!\n",
    "- Итеративное решение задачи с барьерной функцией вместе с уменьшением $\\mu$. Последовательность решений сойдётся к вершине симплекса по траектории из точек, лежащих внутри симплекса."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Общая схема\n",
    "```python\n",
    "def GeneralInteriorPointLP(c, A, b, x0, mu0, rho, tol):\n",
    "    \n",
    "    x = x0\n",
    "    \n",
    "    mu = mu0\n",
    "    \n",
    "    e = np.ones(c.shape[0])\n",
    "    \n",
    "    while True:\n",
    "        \n",
    "        primal_var, dual_var = StepInsideFeasibleSet(c, A, b, x, mu)\n",
    "        \n",
    "        mu *= rho\n",
    "        \n",
    "        if converge(primal_var, dual_var, c, A, b, tol) and mu < tol:\n",
    "            \n",
    "            break\n",
    "            \n",
    "    return x\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Как решать задачу с барьерной функцией?\n",
    "\n",
    "- Прямой метод \n",
    "- Прямо-двойственный метод"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Прямой метод\n",
    "\n",
    "Вспомним исходную задачу:\n",
    "\\begin{align*}\n",
    "&\\min_x c^{\\top}x - \\mu \\sum\\limits_{i=1}^n \\ln x_i \\\\\n",
    "\\text{s.t. } & Ax = b\\\\\n",
    "\\end{align*}\n",
    "\n",
    "Идея: приблизим целевую функцию до второго порядка, как в методе Ньютона."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Реализация\n",
    "На $(k+1)$-ой итерации необходимо решить следующую задачу: \n",
    "\n",
    "\\begin{align*}\n",
    "&\\min_p \\frac{1}{2}p^{\\top}Hp + g^{\\top}p\\\\\n",
    "\\text{s.t. } & A(x_k + p) = b,\\\\\n",
    "\\end{align*}\n",
    "\n",
    "где $H = \\mu X^{-2}$ - гессиан, и $g = c - \\mu X^{-1}e$ - градиент."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Снова KKT\n",
    "\n",
    "Выпишем условия ККТ для этой задачи\n",
    "- $Hp + g + A^{\\top}\\lambda = 0$\n",
    "- $Ap = 0$\n",
    "\n",
    "или\n",
    "$$\\begin{bmatrix} H & A^{\\top}\\\\ A & 0 \\end{bmatrix} \\begin{bmatrix} p\\\\ \\lambda \\end{bmatrix} = \\begin{bmatrix} -g \\\\ 0 \\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Из первой строки:\n",
    "\n",
    "$$\n",
    "-\\mu X^{-2}p + A^{\\top}\\lambda = c - \\mu X^{-1}e\n",
    "$$\n",
    "\n",
    "$$\n",
    "-\\mu Ap + AX^{2}A^{\\top}\\lambda = AX^2c - \\mu AXe\n",
    "$$\n",
    "\n",
    "$$\n",
    "AX^{2}A^{\\top}\\lambda = AX^2c - \\mu AXe\n",
    "$$\n",
    "\n",
    "Так как $X \\in \\mathbb{S}^n_{++}$ и $A$ полного ранга, то уравнение имеет единственное решение $\\lambda^*$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Найдём направление $p$\n",
    "\n",
    "$$\n",
    "-\\mu p + X^2A^{\\top}\\lambda^* = X^2c - \\mu Xe = X^2c - \\mu x\n",
    "$$\n",
    "\n",
    "$$\n",
    "p = x + \\frac{1}{\\mu}X^2(A^{\\top}\\lambda^* - c)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Способы решения системы из ККТ\n",
    "\n",
    "1. Прямой способ: формирование матрицы $(n + m) \\times (n + m)$ и явное решение системы - $\\frac{1}{3}(n + m)^3$\n",
    "2. Последовательное исключение переменных:\n",
    "    - $Hp + A^{\\top}\\lambda = -g$, $p = -H^{-1}(g + A^{\\top}\\lambda)$\n",
    "    - $Ap = -AH^{-1}(g + A^{\\top}\\lambda) = -AH^{-1}A^{\\top}\\lambda - AH^{-1}g = 0$\n",
    "    \n",
    "    Здесь матрица $-AH^{-1}A^{\\top}$ есть *дополнение по Шуру* матрицы $H$.\n",
    "3. Алгоритм вычисления решения при последовательном исключении переменных\n",
    "    - Вычислить $H^{-1}g$ и $H^{-1}A^{\\top}$ - $f_H + (m+1)s_H$ операций\n",
    "    - Вычислить дополнение по Шуру $-AH^{-1}A^{\\top}$ - $\\mathcal{O}(m^2n)$\n",
    "    - Найти $\\lambda$ - $\\frac{1}{3}m^3$ операций\n",
    "    - Найти $p$ - $s_H + \\mathcal{O}(mn)$ операций\n",
    "4. Итого: $f_H + ms_H + \\frac{m^3}{3} + \\mathcal{O}(m^2n)$ уже гораздо быстрее прямого способа"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Используем структуру матрицы $H$\n",
    "\n",
    "- В нашем случае $H = \\mu X^{-2}$ - диагональная матрица!\n",
    "- $f_H$ - $n$ операций\n",
    "- $s_H$ - $n$ операций\n",
    "- Итоговая сложность $\\frac{m^3}{3} + \\mathcal{O}(m^2n)$ операций, где $m \\ll n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Поиск шага $\\alpha$\n",
    "\n",
    "- Обычный линейный поиск с условиями достаточного убывания\n",
    "- Условие $A(x_k + \\alpha p) = b$ выполняется автоматически"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Псевдокод прямого барьерного метода\n",
    "\n",
    "```python\n",
    "def PrimalBarrierLP(c, A, b, x0, mu0, rho, tol):\n",
    "    \n",
    "    x = x0\n",
    "    \n",
    "    mu = mu0\n",
    "    \n",
    "    e = np.ones(x.shape[0])\n",
    "    \n",
    "    while True:\n",
    "        \n",
    "        p, lam = ComputeNewtonDirection(c, x, A, mu)\n",
    "        \n",
    "        alpha = line_search(p, mu, c, x)\n",
    "        \n",
    "        x = x + alpha * p\n",
    "        \n",
    "        mu = rho * mu\n",
    "        \n",
    "        if mu < tol and np.linalg.norm(x.dot(c - A.T.dot(lam)) - mu * e) < tol:\n",
    "            \n",
    "            break\n",
    "            \n",
    "    return x\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сравнение барьерного метода и прямого метода внутренней точки\n",
    "\n",
    "- Пример Klee-Minty c прошлого семинара\n",
    "\\begin{align*}\n",
    "& \\max_{x \\in \\mathbb{R}^n} 2^{n-1}x_1 + 2^{n-2}x_2 + \\dots + 2x_{n-1} + x_n\\\\\n",
    "\\text{s.t. } & x_1 \\leq 5\\\\\n",
    "& 4x_1 + x_2 \\leq 25\\\\\n",
    "& 8x_1 + 4x_2 + x_3 \\leq 125\\\\\n",
    "& \\ldots\\\\\n",
    "& 2^n x_1 + 2^{n-1}x_2 + 2^{n-2}x_3 + \\ldots + x_n \\leq 5^n\\\\\n",
    "& x \\geq 0\n",
    "\\end{align*}\n",
    "- Какая сложность работы симплекс метода?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- Сведение к стандартной форме\n",
    "\n",
    "\\begin{align*}\n",
    "& \\min_{x, \\; z} -c^{\\top}x \\\\\n",
    "\\text{s.t. } & Ax + z = b\\\\\n",
    "& x \\geq 0, \\quad z \\geq 0\n",
    "\\end{align*}\n",
    "\n",
    "- Сравним скорость работы прямого барьерного метода и симплекс-метода"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.optimize as scopt\n",
    "import scipy.linalg as sclin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "def NewtonLinConstraintsFeasible(f, gradf, hessf, A, x0, line_search, linsys_solver, args=(), \n",
    "                                 disp=False, disp_conv=False, callback=None, tol=1e-6, max_iter=100, **kwargs):\n",
    "    x = x0.copy()\n",
    "    n = x0.shape[0]\n",
    "    iteration = 0\n",
    "    lam = np.random.randn(A.shape[0])\n",
    "    while True:\n",
    "        gradient, hess = gradf(x, *args), hessf(x, *args)\n",
    "        h = linsys_solver(hess, A, gradient)\n",
    "        descent_dir = h[:n]\n",
    "        decrement = descent_dir.dot(hessf(x, *args).dot(descent_dir))\n",
    "        if decrement < tol:\n",
    "            if disp_conv:\n",
    "                print(\"Tolerance achieved! Decrement = {}\".format(decrement))\n",
    "            break\n",
    "        alpha = line_search(x, descent_dir, f, gradf, args, **kwargs)\n",
    "        if alpha < 1e-16:\n",
    "            if disp_conv:\n",
    "                print(\"Step is too small!\")\n",
    "        x = x + alpha * descent_dir\n",
    "        if callback is not None:\n",
    "            callback((descent_dir, x))\n",
    "        iteration += 1\n",
    "        if disp:\n",
    "            print(\"Current function val = {}\".format(f(x, *args)))\n",
    "            print(\"Newton decrement = {}\".format(decrement))\n",
    "        if iteration >= max_iter:\n",
    "            if disp_conv:\n",
    "                print(\"Maxiter exceeds!\")\n",
    "            break\n",
    "    res = {\"x\": x, \"num_iter\": iteration, \"tol\": decrement}\n",
    "    return res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "def simple_solver(hess, A, gradient):\n",
    "    n = hess.shape[0]\n",
    "    n_lin_row, n_lin_col = A.shape\n",
    "    modified_hess = np.zeros((n + n_lin_row, n + n_lin_row))\n",
    "    modified_hess[:n, :n] = hess\n",
    "    modified_hess[n:n + n_lin_row, :n_lin_col] = A\n",
    "    modified_hess[:n_lin_col, n:n + n_lin_row] = A.T\n",
    "    rhs = np.zeros(n + n_lin_row)\n",
    "    rhs[:n] = -gradient\n",
    "    h = np.linalg.solve(modified_hess, rhs)\n",
    "    return h\n",
    "\n",
    "def elimination_solver(hess, A, gradient):\n",
    "    inv_hess_diag = np.divide(1.0, np.diag(hess))\n",
    "    inv_hess_grad = np.multiply(-inv_hess_diag, gradient)\n",
    "    rhs = A.dot(inv_hess_grad)\n",
    "    L_inv_hess = np.sqrt(inv_hess_diag)\n",
    "    AL_inv_hess = A * L_inv_hess\n",
    "    # print(AL_inv_hess.shape)\n",
    "    S = AL_inv_hess.dot(AL_inv_hess.T)\n",
    "#     cho_S = sclin.cho_factor(S)\n",
    "#     w = sclin.cho_solve(cho_S, rhs)\n",
    "    w = np.linalg.solve(S, rhs)\n",
    "    v = np.subtract(inv_hess_grad, np.multiply(inv_hess_diag, A.T.dot(w)))\n",
    "#     h = np.zeros(hess.shape[1] + A.shape[0])\n",
    "#     h[:hess.shape[1]] = v\n",
    "#     h[hess.shape[1]:hess.shape[1] + A.shape[0]] = w\n",
    "    return v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "def backtracking(x, descent_dir, f, grad_f, args, **kwargs):\n",
    "    beta1 = kwargs[\"beta1\"]\n",
    "    rho = kwargs[\"rho\"]\n",
    "    alpha = 1\n",
    "    while f(x + alpha * descent_dir, *args) >= f(x, *args) + beta1 * alpha * grad_f(x, *args).dot(descent_dir) \\\n",
    "            or np.isnan(f(x + alpha * descent_dir, *args)):\n",
    "        alpha *= rho\n",
    "        if alpha < 1e-16:\n",
    "            break\n",
    "    return alpha"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "def generate_KleeMinty_test_problem(n):\n",
    "    c = np.array([2**i for i in range(n)])\n",
    "    c = -c[::-1]\n",
    "    bounds = [(0, None) for i in range(n)]\n",
    "    b = np.array([5**(i+1) for i in range(n)])\n",
    "    a = np.array([1] + [2**(i+1) for i in range(1, n)])\n",
    "    A = np.zeros((n, n))\n",
    "    for i in range(n):\n",
    "        A[i:, i] = a[:n-i]\n",
    "    return c, A, b, bounds"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "n = 7\n",
    "c, A, b, _ = generate_KleeMinty_test_problem(n)\n",
    "eps = 1e-10\n",
    "def f(x, c, mu):\n",
    "    n = c.shape[0]\n",
    "    return c.dot(x[:n]) - mu * np.sum(np.log(eps + x))\n",
    "\n",
    "def gradf(x, c, mu):\n",
    "    grad = np.zeros(len(x))\n",
    "    n = c.shape[0]\n",
    "    grad[:n] = c - mu / (eps + x[:n])\n",
    "    grad[n:] = -mu / (eps + x[n:])\n",
    "    return grad\n",
    "\n",
    "def hessf(x, c, mu):\n",
    "    return mu * np.diag(1. / (eps + x)**2)\n",
    "\n",
    "A_lin = np.zeros((n, n + A.shape[0]))\n",
    "A_lin[:n, :n] = A\n",
    "A_lin[:n, n:n + A.shape[0]] = np.eye(A.shape[0])\n",
    "mu = 0.1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Проверим верно ли вычисляется градиент"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00025457253168965194"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scopt.check_grad(f, gradf, np.random.rand(n), c, mu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Выбор начального приближения допустимого по ограничениям и области определения целевой функции"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.1457157353758233e-13\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "x0 = np.zeros(2*n)\n",
    "x0[:n] = np.random.rand(n)\n",
    "x0[n:2*n] = b - A.dot(x0[:n])\n",
    "print(np.linalg.norm(A_lin.dot(x0) - b))\n",
    "print(np.sum(x0 <= 1e-6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Проверим сходимость"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Decrement value = 1.2632467880386624e-11\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist_conv = []\n",
    "def cl(x):\n",
    "    hist_conv.append(x)\n",
    "res = NewtonLinConstraintsFeasible(f, gradf, hessf, A_lin, x0, backtracking, elimination_solver, (c, mu), callback=cl,\n",
    "                           max_iter=2000, beta1=0.1, rho=0.7)\n",
    "print(\"Decrement value = {}\".format(res[\"tol\"]))\n",
    "fstar = f(res[\"x\"], c, mu)\n",
    "hist_conv_f = [np.abs(fstar - f(descdir_x[1], c, mu)) for descdir_x in hist_conv]\n",
    "plt.figure(figsize=(12, 5))\n",
    "plt.subplot(1,2,1)\n",
    "plt.semilogy(hist_conv_f)\n",
    "plt.xlabel(\"Number of iteration, $k$\", fontsize=18)\n",
    "plt.ylabel(\"$f^* - f_k$\", fontsize=18)\n",
    "plt.xticks(fontsize=18)\n",
    "_ = plt.yticks(fontsize=18)\n",
    "\n",
    "hist_conv_x = [np.linalg.norm(res[\"x\"] - x[1]) for x in hist_conv]\n",
    "plt.subplot(1,2,2)\n",
    "plt.semilogy(hist_conv_x)\n",
    "plt.xlabel(\"Number of iteration, $k$\", fontsize=18)\n",
    "plt.ylabel(\"$\\| x_k - x^*\\|_2$\", fontsize=18)\n",
    "plt.xticks(fontsize=18)\n",
    "_ = plt.yticks(fontsize=18)\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Реализация барьерного метода"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def BarrierPrimalLinConstr(f, gradf, hessf, A, c, x0, mu0, rho_mu, linesearch, linsys_solver, \n",
    "                           tol=1e-8, max_iter=500, disp_conv=False, **kwargs):\n",
    "    x = x0.copy()\n",
    "    n = x0.shape[0]\n",
    "    mu = mu0\n",
    "    while True:\n",
    "        res = NewtonLinConstraintsFeasible(f, gradf, hessf, A, x, linesearch, linsys_solver, (c, mu), \n",
    "                                           disp_conv=disp_conv, max_iter=max_iter, beta1=0.01, rho=0.5)\n",
    "        x = res[\"x\"].copy()\n",
    "        if n * mu < tol:\n",
    "            break\n",
    "        mu *= rho_mu\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n",
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 s ± 5.82 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "29.1 ms ± 847 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n",
      "[2.97923182e-07 5.95946415e-07 1.19199287e-06 2.38408577e-06\n",
      " 4.76827156e-06 9.53588096e-06 7.81250705e+04]\n"
     ]
    }
   ],
   "source": [
    "mu0 = 5\n",
    "rho_mu = 0.5\n",
    "x = BarrierPrimalLinConstr(f, gradf, hessf, A_lin, c, x0, mu0, rho_mu, backtracking, elimination_solver, max_iter=100)\n",
    "%timeit BarrierPrimalLinConstr(f, gradf, hessf, A_lin, c, x0, mu0, rho_mu, backtracking, elimination_solver, max_iter=100)\n",
    "%timeit BarrierPrimalLinConstr(f, gradf, hessf, A_lin, c, x0, mu0, rho_mu, backtracking, simple_solver, max_iter=100)\n",
    "print(x[:n])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Сравнение времени работы"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current dimension = 3\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current dimension = 4\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current dimension = 5\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current dimension = 6\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current dimension = 7\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current dimension = 8\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current dimension = 9\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/alex/anaconda3/envs/cvxpy/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in log\n",
      "  \n"
     ]
    }
   ],
   "source": [
    "mu0 = 2\n",
    "rho_mu = 0.5\n",
    "n_list = range(3, 10)\n",
    "n_iters = np.zeros(len(n_list))\n",
    "times_simplex = np.zeros(len(n_list))\n",
    "times_barrier_simple = np.zeros(len(n_list))\n",
    "for i, n in enumerate(n_list):\n",
    "    print(\"Current dimension = {}\".format(n))\n",
    "    c, A, b, bounds = generate_KleeMinty_test_problem(n)\n",
    "    time = %timeit -o -q scopt.linprog(c, A, b, bounds=bounds, options={\"maxiter\": 2**max(n_list) + 1}, method=\"simplex\")\n",
    "    times_simplex[i] = time.best\n",
    "    A_lin = np.zeros((n, n + A.shape[0]))\n",
    "    A_lin[:n, :n] = A\n",
    "    A_lin[:n, n:n + A.shape[0]] = np.eye(A.shape[0])\n",
    "    x0 = np.zeros(2*n)\n",
    "    x0[:n] = np.random.rand(n)\n",
    "    x0[n:2*n] = b - A.dot(x0[:n])\n",
    "    time = %timeit -o -q BarrierPrimalLinConstr(f, gradf, hessf, A_lin, c, x0, mu0, rho_mu, backtracking, simple_solver)\n",
    "    times_barrier_simple[i] = time.best"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 5))\n",
    "plt.semilogy(n_list, times_simplex, label=\"Simplex\")\n",
    "plt.semilogy(n_list, times_barrier_simple, label=\"Primal barrier\")\n",
    "plt.legend(fontsize=18)\n",
    "plt.xlabel(\"Dimension, $n$\", fontsize=18)\n",
    "plt.ylabel(\"Computation time, sec.\", fontsize=18)\n",
    "plt.xticks(fontsize=18)\n",
    "_ = plt.yticks(fontsize=18)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Комментарии\n",
    "\n",
    "- Было показано, что прямой метод эквивалентен методу Кармаркара\n",
    "- Использует информацию только о прямой задаче\n",
    "- Начальное приближение должно лежать в допустимом множестве - отдельная задача"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Барьеры для других задач\n",
    "\n",
    "- Для задач полуопределённой опримизации (SDP) - $-\\log\\det X$\n",
    "- Для задач с конусом второго порядка (SOCP) - $-\\log (t^2 - \\|x\\|^2_2)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    " ## Конические задачи\n",
    " \n",
    " - То, что барьеры находятся для конусов - это не совпадение.\n",
    " - В [работе](https://epubs.siam.org/doi/book/10.1137/1.9781611970791?mobileUi=0) Ю. Нестерова и А. Немировского показано, что наиболее важные для практики конусы имеют **самосогласованные** барьеры, которые приводят к полиномиальности соответствующих методов\n",
    " - Не все выпуклые конусы обладают такими барьерами.\n",
    " - Поэтому существуют NP-трудные задачи выпуклой оптимизации."
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3 (cvxpy)",
   "language": "python",
   "name": "cvxpy"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}