minimise.line_search.test

1 #! /usr/bin/python 2 3 ############################################################################### 4 # # 5 # Copyright (C) 2003 Edward d'Auvergne # 6 # # 7 # This file is part of the program relax. # 8 # # 9 # relax is free software; you can redistribute it and/or modify # 10 # it under the terms of the GNU General Public License as published by # 11 # the Free Software Foundation; either version 2 of the License, or # 12 # (at your option) any later version. # 13 # # 14 # relax is distributed in the hope that it will be useful, # 15 # but WITHOUT ANY WARRANTY; without even the implied warranty of # 16 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # 17 # GNU General Public License for more details. # 18 # # 19 # You should have received a copy of the GNU General Public License # 20 # along with relax; if not, write to the Free Software # 21 # Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA # 22 # # 23 ############################################################################### 24 25 26 from math import cos, pi, sin, sqrt 27 from Numeric import Float64, array, dot 28 29 from more_thuente import more_thuente 30

31 -def run():

32 print "\n\n\n\n\n\n\n\n\n\n\n\n\t\t<<< Test Functions >>>\n\n\n" 33 print "\nSelect the function to test:" 34 while 1: 35 input = raw_input('> ') 36 valid_functions = ['1', '2', '3', '4', '5', '6'] 37 if input in valid_functions: 38 func = int(input) 39 break 40 else: 41 print "Choose a function number between 1 and 6." 42 43 print "\nSelect a0:" 44 while 1: 45 input = raw_input('> ') 46 valid_vals = ['1e-3', '1e-1', '1e1', '1e3'] 47 if input in valid_vals: 48 a0 = float(input) 49 break 50 else: 51 print "Choose a0 as one of ['1e-3', '1e-1', '1e1', '1e3']." 52 53 print "Testing line minimiser using test function " + `func` 54 if func == 1: 55 f, df = func1, dfunc1 56 mu, eta = 0.001, 0.1 57 elif func == 2: 58 f, df = func2, dfunc2 59 mu, eta = 0.1, 0.1 60 elif func == 3: 61 f, df = func3, dfunc3 62 mu, eta = 0.1, 0.1 63 elif func == 4: 64 f, df = func456, dfunc456 65 beta1, beta2 = 0.001, 0.001 66 mu, eta = 0.001, 0.001 67 elif func == 5: 68 f, df = func456, dfunc456 69 beta1, beta2 = 0.01, 0.001 70 mu, eta = 0.001, 0.001 71 elif func == 6: 72 f, df = func456, dfunc456 73 beta1, beta2 = 0.001, 0.01 74 mu, eta = 0.001, 0.001 75 76 xk = array([0.0], Float64) 77 pk = array([1.0], Float64) 78 if func >= 4: 79 args = (beta1, beta2) 80 else: 81 args = () 82 f0 = apply(f, (xk,)+args) 83 g0 = apply(df, (xk,)+args) 84 a = more_thuente(f, df, args, xk, pk, f0, g0, a_init=a0, mu=mu, eta=eta, print_flag=1) 85 print "The minimum is at " + `a`

86 87

88 -def func1(alpha, beta=2.0):

89 """Test function 1. 90 91 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient 92 decrease. ACM Trans. Math. Softw. 20, 286-307. 93 94 The function is: 95 alpha 96 phi(alpha) = - --------------- 97 alpha**2 + beta 98 """ 99 100 return - alpha[0]/(alpha[0]**2 + beta)

101 102

103 -def dfunc1(alpha, beta=2.0):

104 """Derivative of test function 1. 105 106 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient 107 decrease. ACM Trans. Math. Softw. 20, 286-307. 108 109 The gradient is: 110 2*alpha**2 1 111 phi'(alpha) = -------------------- - --------------- 112 (alpha**2 + beta)**2 alpha**2 + beta 113 """ 114 115 temp = array([0.0], Float64) 116 if alpha[0] > 1e90: 117 return temp 118 else: 119 a = 2.0*(alpha[0]**2)/((alpha[0]**2 + beta)**2) 120 b = 1.0/(alpha[0]**2 + beta) 121 temp[0] = a - b 122 return temp

123 124

125 -def func2(alpha, beta=0.004):

126 """Test function 2. 127 128 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient 129 decrease. ACM Trans. Math. Softw. 20, 286-307. 130 131 The function is: 132 133 phi(alpha) = (alpha + beta)**5 - 2(alpha + beta)**4 134 """ 135 136 return (alpha[0] + beta)**5 - 2.0*((alpha[0] + beta)**4)

137 138

139 -def dfunc2(alpha, beta=0.004):

140 """Derivative of test function 2. 141 142 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient 143 decrease. ACM Trans. Math. Softw. 20, 286-307. 144 145 The gradient is: 146 147 phi'(alpha) = 5(alpha + beta)**4 - 8(alpha + beta)**3 148 """ 149 150 temp = array([0.0], Float64) 151 temp[0] = 5.0*((alpha[0] + beta)**4) - 8.0*((alpha[0] + beta)**3) 152 return temp

153 154

155 -def func3(alpha, beta=0.01, l=39.0):

156 """Test function 3. 157 158 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient 159 decrease. ACM Trans. Math. Softw. 20, 286-307. 160 161 The function is: 162 163 2(1 - beta) / l*pi \ 164 phi(alpha) = phi0(alpha) + ----------- . sin | ---- . alpha | 165 l*pi \ 2 / 166 167 where: 168 / 1 - alpha, if alpha <= 1 - beta, 169 | 170 | alpha - 1, if alpha >= 1 + beta, 171 phi0(alpha) = < 172 | 1 1 173 | ------(alpha - 1)**2 + - beta, if alpha in [1 - beta, 1 + beta]. 174 \ 2*beta 2 175 """ 176 177 # Calculate phi0(alpha). 178 if alpha[0] <= 1.0 - beta: 179 phi0 = 1.0 - alpha[0] 180 elif alpha[0] >= 1.0 + beta: 181 phi0 = alpha[0] - 1.0 182 else: 183 phi0 = 0.5/beta * (alpha[0] - 1.0)**2 + 0.5*beta 184 185 return phi0 + 2.0*(1.0 - beta)/(l*pi) * sin(0.5 * l * pi * alpha[0])

186 187

188 -def dfunc3(alpha, beta=0.01, l=39.0):

189 """Derivative of test function 3. 190 191 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient 192 decrease. ACM Trans. Math. Softw. 20, 286-307. 193 194 The gradient is: 195 / l*pi \ 196 phi(alpha) = phi0'(alpha) + (1 - beta) . cos | ---- . alpha | 197 \ 2 / 198 199 where: 200 / -1, if alpha <= 1 - beta, 201 | 202 | 1, if alpha >= 1 + beta, 203 phi0'(alpha) = < 204 | alpha - 1 205 | ---------, if alpha in [1 - beta, 1 + beta]. 206 \ beta 207 """ 208 209 # Calculate phi0'(alpha). 210 if alpha[0] <= 1.0 - beta: 211 phi0_prime = -1.0 212 elif alpha[0] >= 1.0 + beta: 213 phi0_prime = 1.0 214 else: 215 phi0_prime = (alpha[0] - 1.0)/beta 216 217 temp = array([0.0], Float64) 218 temp[0] = phi0_prime + (1.0 - beta) * cos(0.5 * l * pi * alpha[0]) 219 return temp

220 221

222 -def func456(alpha, beta1, beta2):

223 """Test functions 4, 5, and 6. 224 225 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient 226 decrease. ACM Trans. Math. Softw. 20, 286-307. 227 228 The function is: 229 230 phi(alpha) = gamma(beta1) * sqrt((1 - alpha)**2 + beta2**2) 231 + gamma(beta2) * sqrt(alpha**2 + beta1**2) 232 233 where: 234 gamma(beta) = sqrt(1 + beta**2) - beta 235 """ 236 237 g1 = sqrt(1.0 + beta1**2) - beta1 238 g2 = sqrt(1.0 + beta2**2) - beta2 239 return g1 * sqrt((1.0 - alpha[0])**2 + beta2**2) + g2 * sqrt(alpha[0]**2 + beta1**2)

240 241

242 -def dfunc456(alpha, beta1, beta2):

243 """Test functions 4, 5, and 6. 244 245 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient 246 decrease. ACM Trans. Math. Softw. 20, 286-307. 247 248 The function is: 249 250 (1 - alpha) 251 phi'(alpha) = - gamma(beta1) * ------------------------------- 252 sqrt((1 - alpha)**2 + beta2**2) 253 254 a 255 + gamma(beta2) * ------------------------- 256 sqrt(alpha**2 + beta1**2) 257 258 where: 259 gamma(beta) = sqrt(1 + beta**2) - beta 260 """ 261 262 temp = array([0.0], Float64) 263 g1 = sqrt(1.0 + beta1**2) - beta1 264 g2 = sqrt(1.0 + beta2**2) - beta2 265 a = -g1 * (1.0 - alpha[0]) / sqrt((1.0 - alpha[0])**2 + beta2**2) 266 b = g2 * alpha[0] / sqrt(alpha[0]**2 + beta1**2) 267 temp[0] = a + b 268 return temp

269 270 271 run() 272

Source Code for Module minimise.line_search.test_functions