minimise.cauchy

30 """Cauchy Point trust region algorithm. 31 32 Page 69 from 'Numerical Optimization' by Jorge Nocedal and Stephen J. Wright, 1999, 2nd ed. 33 The Cauchy point is defined by: 34 35 delta 36 pCk = - tau_k ------- dfk 37 ||dfk|| 38 39 where: 40 delta_k is the trust region radius, 41 dfk is the gradient vector, 42 43 / 1 if dfk . Bk . dfk <= 0 44 tau_k = < 45 \ min(||dfk||**2/(delta . dfk . Bk . dfk), 1) otherwise. 46 """ 47 48 if print_flag: 49 if print_flag >= 2: 50 print print_prefix 51 print print_prefix 52 print print_prefix + "Cauchy point minimisation" 53 print print_prefix + "~~~~~~~~~~~~~~~~~~~~~~~~~" 54 min = Cauchy_point(func, dfunc, d2func, args, x0, func_tol, grad_tol, maxiter, delta_max, delta0, eta, full_output, print_flag, print_prefix) 55 results = min.minimise() 56 return results

60 - def __init__(self, func, dfunc, d2func, args, x0, func_tol, grad_tol, maxiter, delta_max, delta0, eta, full_output, print_flag, print_prefix):

61 """Class for Cauchy Point trust region minimisation specific functions. 62 63 Unless you know what you are doing, you should call the function 'cauchy_point' rather than 64 using this class. 65 """ 66 67 # Function arguments. 68 self.func = func 69 self.dfunc = dfunc 70 self.d2func = d2func 71 self.args = args 72 self.xk = x0 73 self.func_tol = func_tol 74 self.grad_tol = grad_tol 75 self.maxiter = maxiter 76 self.full_output = full_output 77 self.print_flag = print_flag 78 self.print_prefix = print_prefix 79 self.delta_max = delta_max 80 self.delta = delta0 81 self.eta = eta 82 83 # Initialise the function, gradient, and Hessian evaluation counters. 84 self.f_count = 0 85 self.g_count = 0 86 self.h_count = 0 87 88 # Initialise the warning string. 89 self.warning = None 90 91 # Set the convergence test function. 92 self.setup_conv_tests() 93 94 # Initial values before the first iteration. 95 self.fk, self.f_count = apply(self.func, (self.xk,)+self.args), self.f_count + 1 96 self.dfk, self.g_count = apply(self.dfunc, (self.xk,)+self.args), self.g_count + 1 97 self.d2fk, self.h_count = apply(self.d2func, (self.xk,)+self.args), self.h_count + 1

98 99

100 - def new_param_func(self):

101 """Find the Cauchy point.""" 102 103 # Calculate the curvature and norm. 104 curv = dot(self.dfk, dot(self.d2fk, self.dfk)) 105 norm_dfk = sqrt(dot(self.dfk, self.dfk)) 106 107 # tau_k. 108 if curv <= 0.0: 109 self.tau_k = 1.0 110 else: 111 self.tau_k = min(norm_dfk ** 3 / (self.delta * curv), 1.0) 112 113 if self.print_flag >= 2: 114 print self.print_prefix + "dfk . Bk . dfk: " + `curv` 115 print self.print_prefix + "tau_k: " + `self.tau_k` 116 117 # Cauchy point. 118 self.pk = - self.tau_k * self.delta * self.dfk / norm_dfk 119 120 # Find the new parameter vector and function value at that point. 121 self.xk_new = self.xk + self.pk 122 self.fk_new, self.f_count = apply(self.func, (self.xk_new,)+self.args), self.f_count + 1 123 self.dfk_new, self.g_count = apply(self.dfunc, (self.xk_new,)+self.args), self.g_count + 1

124 125

126 - def update(self):

127 """Update function. 128 129 Function to update the function value, gradient vector, and Hessian matrix. 130 """ 131 132 self.xk = self.xk_new * 1.0 133 self.fk = self.fk_new 134 self.dfk = self.dfk_new * 1.0 135 self.d2fk, self.h_count = apply(self.d2func, (self.xk,)+self.args), self.h_count + 1

Source Code for Module minimise.cauchy_point