def dampedNewton(self, start):
"""
Damped Newton method for analytics center [Nesterov, p. 204]
Args:
start (Matrix): the starting point of the algorithm
Returns:
Matrix: approximation of the analytic center
"""
k = 0
# starting point
y = start
if self.drawPlot:
y0All = [y[0, 0]]
y1All = [y[1, 0]]
# gradient and hessian
Fd, Fdd, _ = Utils.gradientHessian(self.AAll, y, self.boundR)
FddInv = inv(Fdd)
self.logStdout.info('AUXILIARY PATH-FOLLOWING')
while True:
k += 1
self.logStdout.info('\nk = ' + str(k))
# iteration step
y = y - dot(FddInv, Fd)/(1+Utils.LocalNorm(Fd, FddInv))
self.logStdout.info('y = ' + str(y))
if self.drawPlot:
y0All.append(y[0, 0])
y1All.append(y[1, 0])
# gradient and hessian
Fd, Fdd, A = Utils.gradientHessian(self.AAll, y, self.boundR)
FddInv = inv(Fdd)
# print eigenvalues
if self.logStdout.isEnabledFor(logging.INFO):
for i in range(0, len(A)):
eigs, _ = eig(A[i])
eigs.sort()
self.logStdout.info('EIG[' + str(i) + '] = ' + str(eigs))
# breaking condition
if Utils.LocalNorm(Fd, FddInv) <= beta:
break
# plot auxiliary path
if self.drawPlot:
self.plot.add(y1All, xvals = y0All, title = 'Damped Newton', w = 'points', pt = 1)
self.gnuplot.show(self.plot)
return y
def mainFollow(self, x):
"""
Main following algorithm [Nesterov, p. 202]
Args:
x (Matrix): good approximation of the analytic center, used as the starting point of the algorithm
Returns:
Matrix: found optimal solution of the problem
"""
if self.drawPlot:
x0All = [x[0, 0]]
x1All = [x[1, 0]]
# Main path-following scheme [Nesterov, p. 202]
self.logStdout.info('\nMAIN PATH-FOLLOWING')
# initialization of the iteration process
t = 0
eps = 10**(-3)
k = 0
# print the input condition to verify that is satisfied
if self.logStdout.isEnabledFor(logging.INFO):
Fd, Fdd, _ = Utils.gradientHessian(self.AAll, x, self.boundR)
self.logStdout.info('Input condition = ' + str(Utils.LocalNormA(Fd, Fdd)))
while True:
k += 1
self.logStdout.info('\nk = ' + str(k))
# gradient and hessian
Fd, Fdd, A = Utils.gradientHessian(self.AAll, x, self.boundR)
FddInv = inv(Fdd)
# iteration step
t = t + gamma/Utils.LocalNorm(self.c, FddInv)
x = x - dot(FddInv, (t*self.c + Fd))
if self.drawPlot:
x0All.append(x[0, 0])
x1All.append(x[1, 0])
self.logStdout.info('t = ' + str(t))
self.logStdout.info('x = ' + str(x))
# print eigenvalues
if self.logStdout.isEnabledFor(logging.INFO):
for i in range(0, len(A)):
eigs, _ = eig(A[i])
eigs.sort()
self.logStdout.info('EIG[' + str(i) + '] = ' + str(eigs))
# breaking condition
self.logStdout.info('Breaking condition = ' + str(eps*t))
if eps*t >= self.nu + (beta + sqrt(self.nu))*beta/(1 - beta):
break
self.solved = True
self.result = x
self.resultA = A
# plot main path
if self.drawPlot:
self.gnuplot.replot(gp.Data(x0All, x1All, title = 'Main path', with_ = 'points pt 1', filename = 'tmp/mainPath.dat'))
print('\nPress enter to continue')
input()
return x
def auxFollow(self, start):
"""
Auxiliary path-following algorithm [Nesterov, p. 205]
Warning: This function appears to be unstable. Please use dampedNewton instead.
Args:
start (Matrix): the starting point of the algorithm
Returns:
Matrix: approximation of the analytic center
"""
t = 1
k = 0
# starting point
y = start
if self.drawPlot:
y0All = [y[0, 0]]
y1All = [y[1, 0]]
# gradient and hessian
Fd, Fdd, A = Utils.gradientHessian(self.AAll, y, self.boundR)
Fd0 = Fd
FddInv = inv(Fdd)
# print eigenvalues
if self.logStdout.isEnabledFor(logging.INFO):
for i in range(0, len(A)):
eigs, _ = eig(A[i])
eigs.sort()
self.logStdout.info('EIG[' + str(i) + '] = ' + str(eigs))
self.logStdout.info('AUXILIARY PATH-FOLLOWING')
while True:
k += 1
self.logStdout.info('\nk = ' + str(k))
# iteration step
t = t - gamma/Utils.LocalNorm(Fd0, FddInv)
y = y - dot(FddInv, (t*Fd0 + Fd))
#self.logStdout.info('t = ' + str(t))
self.logStdout.info('y = ' + str(y))
if self.drawPlot:
y0All.append(y[0, 0])
y1All.append(y[1, 0])
# gradient and hessian
Fd, Fdd, A = Utils.gradientHessian(self.AAll, y, self.boundR)
FddInv = inv(Fdd)
# print eigenvalues
if self.logStdout.isEnabledFor(logging.INFO):
for i in range(0, len(A)):
eigs, _ = eig(A[i])
eigs.sort()
self.logStdout.info('EIG[' + str(i) + '] = ' + str(eigs))
# breaking condition
if Utils.LocalNorm(Fd, FddInv) <= sqrt(beta)/(1 + sqrt(beta)):
break
# prepare x
x = y - dot(FddInv, Fd)
# plot auxiliary path
if self.drawPlot:
self.gnuplot.replot(gp.Data(y0All, y1All, title = 'Auxiliary path', with_ = 'points pt 1', filename = 'tmp/auxiliaryPath.dat'))
return x