Source Code for Module minfx.hessian_mods.cholesky_mod

 1  ############################################################################### 
 2  #                                                                             # 
 3  # Copyright (C) 2003-2013 Edward d'Auvergne                                   # 
 4  #                                                                             # 
 5  # This file is part of the minfx optimisation library,                        # 
 6  # https://sourceforge.net/projects/minfx                                      # 
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 8  # This program is free software: you can redistribute it and/or modify        # 
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16  # GNU General Public License for more details.                                # 
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19  # along with this program.  If not, see <http://www.gnu.org/licenses/>.       # 
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21  ############################################################################### 
22   
23  # Module docstring. 
24  """Cholesky Hessian modification with added multiple of the identity. 
25   
26  This file is part of the U{minfx optimisation library<https://sourceforge.net/projects/minfx>}. 
27  """ 
28   
29  # Python module imports. 
30  from numpy import dot, sqrt, trace, transpose 
31  from numpy.linalg import LinAlgError, cholesky, solve 
32   
33   
34 -def cholesky_mod(dfk, d2fk, I, n, print_prefix, print_flag, return_matrix=0): 
35      """Cholesky with added multiple of the identity. 
36   
37      Algorithm 6.3 from page 145 of 'Numerical Optimization' by Jorge Nocedal and Stephen J. Wright, 1999, 2nd ed. 
38   
39      Returns the modified Newton step. 
40      """ 
41   
42      # Find the minimum diagonal value of the Hessian. 
43      min_aii = 1e99 
44      for i in range(n): 
45          min_aii = min(d2fk[i, i], min_aii) 
46   
47      # Calculate the Frobenius norm of the Hessian. 
48      norm = sqrt(trace(dot(d2fk, d2fk))) 
49      half_norm = norm / 2.0 
50   
51      # Choose the initial tk value. 
52      if min_aii > 0.0: 
53          tk = 0.0 
54      else: 
55          tk = half_norm 
56   
57      # Debugging. 
58      if print_flag >= 3: 
59          print(print_prefix + "Frobenius norm: " + repr(norm)) 
60          print(print_prefix + "min aii: " + repr(min_aii)) 
61          print(print_prefix + "tk: " + repr(tk)) 
62   
63      # Loop until the matrix is positive definite. 
64      while True: 
65          if print_flag >= 3: 
66              print(print_prefix + "Iteration") 
67   
68          # Calculate the matrix A + tk.I 
69          matrix = d2fk + tk * I 
70   
71          # Attempt the Cholesky decomposition. 
72          try: 
73              L = cholesky(matrix) 
74              if print_flag >= 3: 
75                  print(print_prefix + "\tCholesky matrix L:") 
76                  for i in range(n): 
77                      print(print_prefix + "\t\t" + repr(L[i])) 
78          except LinAlgError: 
79              if print_flag >= 3: 
80                  print(print_prefix + "\tLinearAlgebraError, matrix is not positive definite.") 
81   
82              # Update of tk. 
83              tk = max(2.0*tk, half_norm) 
84          else: 
85              # Success, therefore break out of the while loop. 
86              break 
87   
88      # Calculate the Newton direction. 
89      y = solve(L, dfk) 
90      if return_matrix: 
91          return -solve(transpose(L), y), matrix 
92      else: 
93          return -solve(transpose(L), y) 
94