def fminBFGS(f, x0, fprime=None, args=(), avegtol=1e-5, maxiter=None, fulloutput=0, printmessg=1):
"""xopt = fminBFGS(f, x0, fprime=None, args=(), avegtol=1e-5,
maxiter=None, fulloutput=0, printmessg=1)
Optimize the function, f, whose gradient is given by fprime using the
quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)
See Wright, and Nocedal 'Numerical Optimization', 1999, pg. 198.
"""
app_fprime = 0
if fprime is None:
app_fprime = 1
x0 = Num.asarray(x0)
if maxiter is None:
maxiter = len(x0)*200
func_calls = 0
grad_calls = 0
k = 0
N = len(x0)
gtol = N*avegtol
I = MLab.eye(N)
Hk = I
if app_fprime:
gfk = apply(approx_fprime,(x0,f)+args)
func_calls = func_calls + len(x0) + 1
else:
gfk = apply(fprime,(x0,)+args)
grad_calls = grad_calls + 1
xk = x0
sk = [2*gtol]
while (Num.add.reduce(abs(gfk)) > gtol) and (k < maxiter):
pk = -Num.dot(Hk,gfk)
alpha_k, fc, gc = line_search_BFGS(f,xk,pk,gfk,args)
func_calls = func_calls + fc
xkp1 = xk + alpha_k * pk
sk = xkp1 - xk
xk = xkp1
if app_fprime:
gfkp1 = apply(approx_fprime,(xkp1,f)+args)
func_calls = func_calls + gc + len(x0) + 1
else:
gfkp1 = apply(fprime,(xkp1,)+args)
grad_calls = grad_calls + gc + 1
yk = gfkp1 - gfk
k = k + 1
rhok = 1 / Num.dot(yk,sk)
A1 = I - sk[:,Num.NewAxis] * yk[Num.NewAxis,:] * rhok
A2 = I - yk[:,Num.NewAxis] * sk[Num.NewAxis,:] * rhok
Hk = Num.dot(A1,Num.dot(Hk,A2)) + rhok * sk[:,Num.NewAxis] * sk[Num.NewAxis,:]
gfk = gfkp1
if printmessg or fulloutput:
fval = apply(f,(xk,)+args)
if k >= maxiter:
warnflag = 1
if printmessg:
print "Warning: Maximum number of iterations has been exceeded"
print " Current function value: %f" % fval
print " Iterations: %d" % k
print " Function evaluations: %d" % func_calls
print " Gradient evaluations: %d" % grad_calls
else:
warnflag = 0
if printmessg:
print "Optimization terminated successfully."
print " Current function value: %f" % fval
print " Iterations: %d" % k
print " Function evaluations: %d" % func_calls
print " Gradient evaluations: %d" % grad_calls
if fulloutput:
return xk, fval, func_calls, grad_calls, warnflag
else:
return xk
def fminBFGS(f,
x0,
fprime=None,
args=(),
avegtol=1e-5,
maxiter=None,
fulloutput=0,
printmessg=1):
"""xopt = fminBFGS(f, x0, fprime=None, args=(), avegtol=1e-5,
maxiter=None, fulloutput=0, printmessg=1)
Optimize the function, f, whose gradient is given by fprime using the
quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS)
See Wright, and Nocedal 'Numerical Optimization', 1999, pg. 198.
"""
app_fprime = 0
if fprime is None:
app_fprime = 1
x0 = Num.asarray(x0)
if maxiter is None:
maxiter = len(x0) * 200
func_calls = 0
grad_calls = 0
k = 0
N = len(x0)
gtol = N * avegtol
I = MLab.eye(N)
Hk = I
if app_fprime:
gfk = apply(approx_fprime, (x0, f) + args)
func_calls = func_calls + len(x0) + 1
else:
gfk = apply(fprime, (x0, ) + args)
grad_calls = grad_calls + 1
xk = x0
sk = [2 * gtol]
while (Num.add.reduce(abs(gfk)) > gtol) and (k < maxiter):
pk = -Num.dot(Hk, gfk)
alpha_k, fc, gc = line_search_BFGS(f, xk, pk, gfk, args)
func_calls = func_calls + fc
xkp1 = xk + alpha_k * pk
sk = xkp1 - xk
xk = xkp1
if app_fprime:
gfkp1 = apply(approx_fprime, (xkp1, f) + args)
func_calls = func_calls + gc + len(x0) + 1
else:
gfkp1 = apply(fprime, (xkp1, ) + args)
grad_calls = grad_calls + gc + 1
yk = gfkp1 - gfk
k = k + 1
rhok = 1 / Num.dot(yk, sk)
A1 = I - sk[:, Num.NewAxis] * yk[Num.NewAxis, :] * rhok
A2 = I - yk[:, Num.NewAxis] * sk[Num.NewAxis, :] * rhok
Hk = Num.dot(A1, Num.dot(
Hk, A2)) + rhok * sk[:, Num.NewAxis] * sk[Num.NewAxis, :]
gfk = gfkp1
if printmessg or fulloutput:
fval = apply(f, (xk, ) + args)
if k >= maxiter:
warnflag = 1
if printmessg:
print "Warning: Maximum number of iterations has been exceeded"
print " Current function value: %f" % fval
print " Iterations: %d" % k
print " Function evaluations: %d" % func_calls
print " Gradient evaluations: %d" % grad_calls
else:
warnflag = 0
if printmessg:
print "Optimization terminated successfully."
print " Current function value: %f" % fval
print " Iterations: %d" % k
print " Function evaluations: %d" % func_calls
print " Gradient evaluations: %d" % grad_calls
if fulloutput:
return xk, fval, func_calls, grad_calls, warnflag
else:
return xk
