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24 """A line search algorithm based on interpolation.
25
26 This file is part of the U{minfx optimisation library<https://gna.org/projects/minfx>}.
27 """
28
29
30 from copy import deepcopy
31 from numpy import dot, sqrt
32
33
34 from minfx.line_search.interpolate import cubic_ext, quadratic_fafbga
35
36
37 quadratic = quadratic_fafbga
38
39
41 """A line search algorithm based on interpolation.
42
43 Pages 56-57, from 'Numerical Optimization' by Jorge Nocedal and Stephen J. Wright, 1999, 2nd ed.
44
45 Requires the gradient function.
46
47
48 @param func: The function to minimise.
49 @type func: func
50 @param args: The tuple of arguments to supply to the functions func.
51 @type args: tuple
52 @param x: The parameter vector at minimisation step k.
53 @type x: numpy array
54 @param f: The function value at the point x.
55 @type f: float
56 @param g: The function gradient vector at the point x.
57 @type g: numpy array
58 @param p: The descent direction.
59 @type p: numpy array
60 @keyword a_init: Initial step length.
61 @type a_init: flaot
62 @keyword mu: Constant determining the slope for the sufficient decrease condition (0 < mu < 1).
63 @type mu: float
64 @keyword print_flag: The higher the value, the greater the amount of info printed out.
65 @type print_flag: int
66 """
67
68
69 i = 1
70 f_count = 0
71 a0 = {}
72 a0['a'] = 0.0
73 a0['phi'] = f
74 a0['phi_prime'] = dot(g, p)
75
76
77 a = {}
78 a['a'] = a_init
79 a['phi'] = func(*(x + a['a']*p,)+args)
80 f_count = f_count + 1
81
82 if print_flag:
83 print("\n<Line search initial values>")
84 print_data("Pre (a0)", i-1, a0)
85
86
87 if a0['phi_prime'] >= 0.0:
88 raise NameError("The gradient at point 0 of this line search is positive, ie p is not a descent direction and the line search will not work.")
89
90
91 if a['phi'] <= a0['phi'] + mu * a['a'] * a0['phi_prime']:
92 return a['a'], f_count
93
94
95 a_last = deepcopy(a)
96
97
98 a_new = - 0.5 * a0['phi_prime'] * a['a']**2 / (a['phi'] - a0['phi'] - a0['phi_prime']*a['a'])
99 a['a'] = a_new
100 a['phi'] = func(*(x + a['a']*p,)+args)
101 f_count = f_count + 1
102
103
104 if a['phi'] <= a0['phi'] + mu * a['a'] * a0['phi_prime']:
105 return a['a'], f_count
106
107 while True:
108 if print_flag:
109 print("<Line search iteration i = " + repr(i) + " >")
110 print_data("Initial (a)", i, a)
111 print_data("Initial (a_last)", i, a_last)
112
113 beta1 = 1.0 / (a_last['a']**2 * a['a']**2 * (a['a'] - a_last['a']))
114 beta2 = a['phi'] - a0['phi'] - a0['phi_prime'] * a['a']
115 beta3 = a_last['phi'] - a0['phi'] - a0['phi_prime'] * a_last['a']
116
117 fact_a = beta1 * (a_last['a']**2 * beta2 - a['a']**2 * beta3)
118 fact_b = beta1 * (-a_last['a']**3 * beta2 + a['a']**3 * beta3)
119
120 a_new = {}
121 a_new['a'] = (-fact_b + sqrt(fact_b**2 - 3.0 * fact_a * a0['phi_prime'])) / (3.0 * fact_a)
122 a_new['phi'] = func(*(x + a_new['a']*p,)+args)
123 f_count = f_count + 1
124
125
126 if a_new['phi'] <= a0['phi'] + mu * a_new['a'] * a0['phi_prime']:
127 return a_new['a'], f_count
128
129
130 if a['a'] - a_new['a'] > 0.5 * a['a'] or 1.0 - a_new['a']/a['a'] < 0.9:
131 a_new['a'] = 0.5 * a['a']
132 a_new['phi'] = func(*(x + a_new['a']*p,)+args)
133 f_count = f_count + 1
134
135
136 a_last = deepcopy(a)
137 a = deepcopy(a_new)
138
139
141 """Temp func for debugging."""
142
143 print(text + " data printout:")
144 print(" Iteration: " + repr(k))
145 print(" a: " + repr(a['a']))
146 print(" phi: " + repr(a['phi']))
147 print(" phi_prime: " + repr(a['phi_prime']))
148