minfx.line_search.test

1 #! /usr/bin/python 2 3 ############################################################################### 4 # # 5 # Copyright (C) 2003-2013 Edward d'Auvergne # 6 # # 7 # This file is part of the minfx optimisation library, # 8 # https://sourceforge.net/projects/minfx # 9 # # 10 # This program is free software: you can redistribute it and/or modify # 11 # it under the terms of the GNU General Public License as published by # 12 # the Free Software Foundation, either version 3 of the License, or # 13 # (at your option) any later version. # 14 # # 15 # This program is distributed in the hope that it will be useful, # 16 # but WITHOUT ANY WARRANTY; without even the implied warranty of # 17 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # 18 # GNU General Public License for more details. # 19 # # 20 # You should have received a copy of the GNU General Public License # 21 # along with this program. If not, see <http://www.gnu.org/licenses/>. # 22 # # 23 ############################################################################### 24 25 # Module docstring. 26 """Testing functions. 27 28 This file is part of the U{minfx optimisation library<https://sourceforge.net/projects/minfx>}. 29 """ 30 31 # Python module imports. 32 from math import cos, pi, sin, sqrt 33 from numpy import array, float64 34 35 # Minfx module imports. 36 from more_thuente import more_thuente 37 38

39 -def run():

40 print("\n\n\n\n\n\n\n\n\n\n\n\n\t\t<<< Test Functions >>>\n\n\n") 41 print("\nSelect the function to test:") 42 while True: 43 input = raw_input('> ') 44 valid_functions = ['1', '2', '3', '4', '5', '6'] 45 if input in valid_functions: 46 func = int(input) 47 break 48 else: 49 print("Choose a function number between 1 and 6.") 50 51 print("\nSelect a0:") 52 while True: 53 input = raw_input('> ') 54 valid_vals = ['1e-3', '1e-1', '1e1', '1e3'] 55 if input in valid_vals: 56 a0 = float(input) 57 break 58 else: 59 print("Choose a0 as one of ['1e-3', '1e-1', '1e1', '1e3'].") 60 61 print("Testing line minimiser using test function " + repr(func)) 62 if func == 1: 63 f, df = func1, dfunc1 64 mu, eta = 0.001, 0.1 65 elif func == 2: 66 f, df = func2, dfunc2 67 mu, eta = 0.1, 0.1 68 elif func == 3: 69 f, df = func3, dfunc3 70 mu, eta = 0.1, 0.1 71 elif func == 4: 72 f, df = func456, dfunc456 73 beta1, beta2 = 0.001, 0.001 74 mu, eta = 0.001, 0.001 75 elif func == 5: 76 f, df = func456, dfunc456 77 beta1, beta2 = 0.01, 0.001 78 mu, eta = 0.001, 0.001 79 elif func == 6: 80 f, df = func456, dfunc456 81 beta1, beta2 = 0.001, 0.01 82 mu, eta = 0.001, 0.001 83 84 xk = array([0.0], float64) 85 pk = array([1.0], float64) 86 if func >= 4: 87 args = (beta1, beta2) 88 else: 89 args = () 90 f0 = f(*(xk,)+args) 91 g0 = df(*(xk,)+args) 92 a = more_thuente(f, df, args, xk, pk, f0, g0, a_init=a0, mu=mu, eta=eta, print_flag=1) 93 print("The minimum is at " + repr(a))

94 95

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

97 """Test function 1. 98 99 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286-307. 100 101 The function is:: 102 103 alpha 104 phi(alpha) = - --------------- 105 alpha**2 + beta 106 """ 107 108 return - alpha[0]/(alpha[0]**2 + beta)

109 110

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

112 """Derivative of test function 1. 113 114 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286-307. 115 116 The gradient is:: 117 118 2*alpha**2 1 119 phi'(alpha) = -------------------- - --------------- 120 (alpha**2 + beta)**2 alpha**2 + beta 121 """ 122 123 temp = array([0.0], float64) 124 if alpha[0] > 1e90: 125 return temp 126 else: 127 a = 2.0*(alpha[0]**2)/((alpha[0]**2 + beta)**2) 128 b = 1.0/(alpha[0]**2 + beta) 129 temp[0] = a - b 130 return temp

131 132

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

134 """Test function 2. 135 136 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286-307. 137 138 The function is:: 139 140 phi(alpha) = (alpha + beta)**5 - 2(alpha + beta)**4 141 """ 142 143 return (alpha[0] + beta)**5 - 2.0*((alpha[0] + beta)**4)

144 145

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

147 """Derivative of test function 2. 148 149 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286-307. 150 151 The gradient is:: 152 153 phi'(alpha) = 5(alpha + beta)**4 - 8(alpha + beta)**3 154 """ 155 156 temp = array([0.0], float64) 157 temp[0] = 5.0*((alpha[0] + beta)**4) - 8.0*((alpha[0] + beta)**3) 158 return temp

159 160

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

162 """Test function 3. 163 164 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286-307. 165 166 The function is:: 167 168 2(1 - beta) / l*pi \ 169 phi(alpha) = phi0(alpha) + ----------- . sin | ---- . alpha | 170 l*pi \ 2 / 171 172 where:: 173 174 / 1 - alpha, if alpha <= 1 - beta, 175 | 176 | alpha - 1, if alpha >= 1 + beta, 177 phi0(alpha) = < 178 | 1 1 179 | ------(alpha - 1)**2 + - beta, if alpha in [1 - beta, 1 + beta]. 180 \ 2*beta 2 181 """ 182 183 # Calculate phi0(alpha). 184 if alpha[0] <= 1.0 - beta: 185 phi0 = 1.0 - alpha[0] 186 elif alpha[0] >= 1.0 + beta: 187 phi0 = alpha[0] - 1.0 188 else: 189 phi0 = 0.5/beta * (alpha[0] - 1.0)**2 + 0.5*beta 190 191 return phi0 + 2.0*(1.0 - beta)/(l*pi) * sin(0.5 * l * pi * alpha[0])

192 193

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

195 """Derivative of test function 3. 196 197 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286-307. 198 199 The gradient is:: 200 / l*pi \ 201 phi(alpha) = phi0'(alpha) + (1 - beta) . cos | ---- . alpha | 202 \ 2 / 203 204 where:: 205 206 / -1, if alpha <= 1 - beta, 207 | 208 | 1, if alpha >= 1 + beta, 209 phi0'(alpha) = < 210 | alpha - 1 211 | ---------, if alpha in [1 - beta, 1 + beta]. 212 \ beta 213 """ 214 215 # Calculate phi0'(alpha). 216 if alpha[0] <= 1.0 - beta: 217 phi0_prime = -1.0 218 elif alpha[0] >= 1.0 + beta: 219 phi0_prime = 1.0 220 else: 221 phi0_prime = (alpha[0] - 1.0)/beta 222 223 temp = array([0.0], float64) 224 temp[0] = phi0_prime + (1.0 - beta) * cos(0.5 * l * pi * alpha[0]) 225 return temp

226 227

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

229 """Test functions 4, 5, and 6. 230 231 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286-307. 232 233 The function is:: 234 235 phi(alpha) = gamma(beta1) * sqrt((1 - alpha)**2 + beta2**2) 236 + gamma(beta2) * sqrt(alpha**2 + beta1**2) 237 238 where:: 239 240 gamma(beta) = sqrt(1 + beta**2) - beta 241 """ 242 243 g1 = sqrt(1.0 + beta1**2) - beta1 244 g2 = sqrt(1.0 + beta2**2) - beta2 245 return g1 * sqrt((1.0 - alpha[0])**2 + beta2**2) + g2 * sqrt(alpha[0]**2 + beta1**2)

246 247

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

249 """Test functions 4, 5, and 6. 250 251 From More, J. J., and Thuente, D. J. 1994, Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20, 286-307. 252 253 The function is:: 254 255 (1 - alpha) 256 phi'(alpha) = - gamma(beta1) * ------------------------------- 257 sqrt((1 - alpha)**2 + beta2**2) 258 259 a 260 + gamma(beta2) * ------------------------- 261 sqrt(alpha**2 + beta1**2) 262 263 where:: 264 265 gamma(beta) = sqrt(1 + beta**2) - beta 266 """ 267 268 temp = array([0.0], float64) 269 g1 = sqrt(1.0 + beta1**2) - beta1 270 g2 = sqrt(1.0 + beta2**2) - beta2 271 a = -g1 * (1.0 - alpha[0]) / sqrt((1.0 - alpha[0])**2 + beta2**2) 272 b = g2 * alpha[0] / sqrt(alpha[0]**2 + beta1**2) 273 temp[0] = a + b 274 return temp

275 276 277 if __name__ == '__main__': 278 run() 279

Source Code for Module minfx.line_search.test_functions