org-mode-lab-example

#+TITLE: ОТЧЕТ ПО ПРАКТИЧЕСКОЙ РАБОТЕ № 7
#+AUTHOR: Чащинский Н.И.
#+MACRO: GROUP      ИВБО-03-16
#+MACRO: DEPARTMENT Вычислительной Техники
#+MACRO: SUBJECT    Теория принятия решений
#+MACRO: THEMETITLE Тема практической работы:
#+MACRO: THEME      Методы безусловной оптимизации 1-го порядка: Метод наискорейшего спуска
#+MACRO: TIMETITLE  Практическая работа выполнена \today
#+MACRO: TEACHER    доцент Сорокин А.Б.
#+INCLUDE: ~/BORG/common/header.org

* Задание
Вариант №9
\begin{equation}
f(\vec{x}) = 1.8x_1^2 + 1.8x_1x_2 + 2x_2^2 - 1.1x_1 + 2.3x_2 \rightarrow \text{min}
\end{equation}
\begin{align*}
\vec{x}^{(0)} &= (2, 3)\\
\vec{x}^* &= (0.7652, -0.9194)\\
f^* &= -1.478\\
\epsilon &= 0.0001\\
\end{align*}
* Формулы
\begin{equation*}
f_k = f(\vec{x}^{(k)})
\end{equation}
\begin{equation}
\vec{x}^{(k+1)} = \vec{x}^{(k)} - h_k \nabla f_k
\end{equation}
\begin{equation}
h_k = \frac{(\nabla f_k, \nabla f_k)}{(H\vec{x}^{(k)} \nabla f_k, \nabla f_k)}
\end{equation}
* Решение
#+call: shortest_descent()

#+RESULTS:
|  k |      x1 |      x2 |       f | grad norm |
|----+---------+---------+---------+-----------|
|  0 |     2.0 |     3.0 |    40.7 |   21.2758 |
|  1 | -1.2135 | -2.0018 | 11.7682 |   12.0239 |
|  2 |  1.2592 |  0.1492 |  2.1946 |     6.353 |
|  3 |  0.2414 | -1.2705 | -0.4064 |    3.4426 |
|  4 |   0.924 | -0.6341 | -1.1884 |    1.7927 |
|  5 |  0.6276 | -1.0238 | -1.3964 |    0.9539 |
|  6 |  0.8125 | -0.8438 | -1.4563 |    0.4937 |
|  7 |  0.7292 | -0.9492 | -1.4721 |    0.2601 |
|  8 |  0.7788 | -0.8994 | -1.4765 |    0.1343 |
|  9 |  0.7558 | -0.9277 | -1.4777 |    0.0703 |
| 10 |   0.769 | -0.9141 |  -1.478 |    0.0362 |
| 11 |  0.7628 | -0.9217 | -1.4781 |    0.0189 |
| 12 |  0.7663 |  -0.918 | -1.4781 |    0.0097 |
| 13 |  0.7646 |   -0.92 | -1.4781 |    0.0051 |
| 14 |  0.7655 |  -0.919 | -1.4781 |    0.0026 |
| 15 |  0.7651 | -0.9195 | -1.4781 |    0.0014 |
| 16 |  0.7653 | -0.9193 | -1.4781 |    0.0007 |
| 17 |  0.7652 | -0.9194 | -1.4781 |    0.0004 |
| 18 |  0.7653 | -0.9193 | -1.4781 |    0.0002 |
| 19 |  0.7652 | -0.9194 | -1.4781 |    0.0001 |

* Код
#+NAME: shortest_descent
#+BEGIN_SRC python
  import numpy as np

  coefs = [1.8, 1.8, 2, -1.1, 2.3]

  def f(x):
      c = coefs
      c0 = c[0]*x[0]**2
      c1 = c[1]*x[0]*x[1]
      c2 = c[2]*x[1]**2
      c3 = c[3]*x[0]
      c4 = c[4]*x[1]
      return c0 + c1 + c2 + c3 + c4

  def grad(x):
      c = coefs
      x0 = 2*c[0]*x[0] + c[1]*x[1] + 0*c[2]      +   c[3] + 0*c[4]
      x1 = 0*c[0]      + c[1]*x[0] + 2*c[2]*x[1] + 0*c[3] +   c[4]
      return np.array([x0, x1])

  def hesse():
      c = coefs
      row0 = [c[0], c[1]]
      row1 = [c[1], c[2]]
      return np.array([row0, row1])

  def h_k(x):
      g = grad(x)
      H = hesse()
      return (np.dot(g, g) / np.dot(H.dot(g), g))

  def step(x):
      h = h_k(x)
      return x - h*grad(x)

  def optimize(x, epsilon):
      report = []
      g = grad(x)
      grad_norm = np.sqrt(g.dot(g))
      k = 0
      row = [k, x[0], x[1], f(x), grad_norm]
      row = [round(c, 4) for c in row]
      report.append(row)
      while grad_norm > epsilon:
          h = h_k(x)
          x -= h * g
          g = grad(x)
          grad_norm = np.sqrt(g.dot(g))
          k += 1
          row = [k, x[0], x[1], f(x), grad_norm]
          row = [round(c, 4) for c in row]
          report.append(row)
      header = ['k', 'x1', 'x2', 'f', 'grad norm']
      report.insert(0, header)
      report.insert(1, None)
      return report

  x = np.array([2.0, 3.0])
  epsilon = 0.0001
  report = optimize(x, epsilon)
  return report
#+END_SRC

#+RESULTS: shortest_descent
|  k |      x1 |      x2 |       f | grad norm |
|----+---------+---------+---------+-----------|
|  0 |     2.0 |     3.0 |    40.7 |   21.2758 |
|  1 | -1.2135 | -2.0018 | 11.7682 |   12.0239 |
|  2 |  1.2592 |  0.1492 |  2.1946 |     6.353 |
|  3 |  0.2414 | -1.2705 | -0.4064 |    3.4426 |
|  4 |   0.924 | -0.6341 | -1.1884 |    1.7927 |
|  5 |  0.6276 | -1.0238 | -1.3964 |    0.9539 |
|  6 |  0.8125 | -0.8438 | -1.4563 |    0.4937 |
|  7 |  0.7292 | -0.9492 | -1.4721 |    0.2601 |
|  8 |  0.7788 | -0.8994 | -1.4765 |    0.1343 |
|  9 |  0.7558 | -0.9277 | -1.4777 |    0.0703 |
| 10 |   0.769 | -0.9141 |  -1.478 |    0.0362 |
| 11 |  0.7628 | -0.9217 | -1.4781 |    0.0189 |
| 12 |  0.7663 |  -0.918 | -1.4781 |    0.0097 |
| 13 |  0.7646 |   -0.92 | -1.4781 |    0.0051 |
| 14 |  0.7655 |  -0.919 | -1.4781 |    0.0026 |
| 15 |  0.7651 | -0.9195 | -1.4781 |    0.0014 |
| 16 |  0.7653 | -0.9193 | -1.4781 |    0.0007 |
| 17 |  0.7652 | -0.9194 | -1.4781 |    0.0004 |
| 18 |  0.7653 | -0.9193 | -1.4781 |    0.0002 |
| 19 |  0.7652 | -0.9194 | -1.4781 |    0.0001 |