| 1 |
# ******NOTICE*************** |
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# optimize.py module by Travis E. Oliphant |
| 3 |
# |
| 4 |
# You may copy and use this module as you see fit with no |
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# guarantee implied provided you keep this notice in all copies. |
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# *****END NOTICE************ |
| 7 |
|
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# A collection of optimization algorithms. Version 0.3.1 |
| 9 |
|
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# Minimization routines |
| 11 |
"""optimize.py |
| 12 |
|
| 13 |
A collection of general-purpose optimization routines using numarrayeric |
| 14 |
|
| 15 |
fmin --- Nelder-Mead Simplex algorithm (uses only function calls) |
| 16 |
fminBFGS --- Quasi-Newton method (uses function and gradient) |
| 17 |
fminNCG --- Line-search Newton Conjugate Gradient (uses function, gradient and hessian (if it's provided)) |
| 18 |
|
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""" |
| 20 |
import numarray |
| 21 |
__version__="0.3.1" |
| 22 |
|
| 23 |
def para(x): |
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return (x[0]-2.)**2 |
| 25 |
|
| 26 |
def para_der(x): |
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return 2*(x-2.) |
| 28 |
|
| 29 |
def para_hess(x): |
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return numarray.ones((1,1))*2. |
| 31 |
|
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def para_hess_p(x,p): |
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return 2*p |
| 34 |
|
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def rosen(x): # The Rosenbrock function |
| 36 |
return numarray.sum(100.0*(x[1:]-x[:-1]**2.0)**2.0 + (1-x[:-1])**2.0) |
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|
| 38 |
def rosen_der(x): |
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xm = x[1:-1] |
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xm_m1 = x[:-2] |
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xm_p1 = x[2:] |
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der = numarray.zeros(x.shape,x.typecode()) |
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der[1:-1] = 200*(xm-xm_m1**2) - 400*(xm_p1 - xm**2)*xm - 2*(1-xm) |
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der[0] = -400*x[0]*(x[1]-x[0]**2) - 2*(1-x[0]) |
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der[-1] = 200*(x[-1]-x[-2]**2) |
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return der |
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|
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def rosen3_hess_p(x,p): |
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assert(len(x)==3) |
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assert(len(p)==3) |
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hessp = numarray.zeros((3,),x.typecode()) |
| 52 |
hessp[0] = (2 + 800*x[0]**2 - 400*(-x[0]**2 + x[1])) * p[0] \ |
| 53 |
- 400*x[0]*p[1] \ |
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+ 0 |
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hessp[1] = - 400*x[0]*p[0] \ |
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+ (202 + 800*x[1]**2 - 400*(-x[1]**2 + x[2]))*p[1] \ |
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- 400*x[1] * p[2] |
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hessp[2] = 0 \ |
| 59 |
- 400*x[1] * p[1] \ |
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+ 200 * p[2] |
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|
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return hessp |
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|
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def rosen3_hess(x): |
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assert(len(x)==3) |
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hessp = numarray.zeros((3,3),x.typecode()) |
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hessp[0,:] = [2 + 800*x[0]**2 -400*(-x[0]**2 + x[1]), -400*x[0], 0] |
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hessp[1,:] = [-400*x[0], 202+800*x[1]**2 -400*(-x[1]**2 + x[2]), -400*x[1]] |
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hessp[2,:] = [0,-400*x[1], 200] |
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return hessp |
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|
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|
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def fmin(func, x0, args=(), xtol=1e-4, ftol=1e-4, maxiter=None, maxfun=None, fulloutput=0, printmessg=1): |
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"""xopt,{fval,warnflag} = fmin(function, x0, args=(), xtol=1e-4, ftol=1e-4, |
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maxiter=200*len(x0), maxfun=200*len(x0), fulloutput=0, printmessg=0) |
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|
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Uses a Nelder-Mead Simplex algorithm to find the minimum of function |
| 78 |
of one or more variables. |
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""" |
| 80 |
x0 = numarray.asarray(x0) |
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assert (len(x0.shape)==1) |
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N = len(x0) |
| 83 |
if maxiter is None: |
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maxiter = N * 200 |
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if maxfun is None: |
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maxfun = N * 200 |
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|
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rho = 1; chi = 2; psi = 0.5; sigma = 0.5; |
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one2np1 = range(1,N+1) |
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|
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sim = numarray.zeros((N+1,N),x0.typecode()) |
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fsim = numarray.zeros((N+1,),'d') |
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sim[0] = x0 |
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fsim[0] = apply(func,(x0,)+args) |
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nonzdelt = 0.05 |
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zdelt = 0.00025 |
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for k in range(0,N): |
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y = numarray.array(x0,copy=1) |
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if y[k] != 0: |
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y[k] = (1+nonzdelt)*y[k] |
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else: |
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y[k] = zdelt |
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|
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sim[k+1] = y |
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f = apply(func,(y,)+args) |
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fsim[k+1] = f |
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|
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ind = numarray.argsort(fsim) |
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fsim = numarray.take(fsim,ind) # sort so sim[0,:] has the lowest function value |
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sim = numarray.take(sim,ind,0) |
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|
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iterations = 1 |
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funcalls = N+1 |
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|
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while (funcalls < maxfun and iterations < maxiter): |
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if (max(numarray.ravel(numarray.absolute(sim[1:]-sim[0]))) <= xtol \ |
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and max(numarray.absolute(fsim[0]-fsim[1:])) <= ftol): |
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break |
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|
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xbar = numarray.add.reduce(sim[:-1],0) / N |
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xr = (1+rho)*xbar - rho*sim[-1] |
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fxr = apply(func,(xr,)+args) |
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funcalls = funcalls + 1 |
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doshrink = 0 |
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|
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if fxr < fsim[0]: |
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xe = (1+rho*chi)*xbar - rho*chi*sim[-1] |
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fxe = apply(func,(xe,)+args) |
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funcalls = funcalls + 1 |
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|
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if fxe < fxr: |
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sim[-1] = xe |
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fsim[-1] = fxe |
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else: |
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sim[-1] = xr |
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fsim[-1] = fxr |
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else: # fsim[0] <= fxr |
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if fxr < fsim[-2]: |
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sim[-1] = xr |
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fsim[-1] = fxr |
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else: # fxr >= fsim[-2] |
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# Perform contraction |
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if fxr < fsim[-1]: |
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xc = (1+psi*rho)*xbar - psi*rho*sim[-1] |
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fxc = apply(func,(xc,)+args) |
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funcalls = funcalls + 1 |
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|
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if fxc <= fxr: |
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sim[-1] = xc |
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fsim[-1] = fxc |
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else: |
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doshrink=1 |
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else: |
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# Perform an inside contraction |
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xcc = (1-psi)*xbar + psi*sim[-1] |
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fxcc = apply(func,(xcc,)+args) |
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funcalls = funcalls + 1 |
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|
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if fxcc < fsim[-1]: |
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sim[-1] = xcc |
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fsim[-1] = fxcc |
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else: |
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doshrink = 1 |
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|
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if doshrink: |
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for j in one2np1: |
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sim[j] = sim[0] + sigma*(sim[j] - sim[0]) |
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fsim[j] = apply(func,(sim[j],)+args) |
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funcalls = funcalls + N |
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|
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ind = numarray.argsort(fsim) |
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sim = numarray.take(sim,ind,0) |
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fsim = numarray.take(fsim,ind) |
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iterations = iterations + 1 |
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|
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x = sim[0] |
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fval = min(fsim) |
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warnflag = 0 |
| 179 |
|
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if funcalls >= maxfun: |
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warnflag = 1 |
| 182 |
if printmessg: |
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print "Warning: Maximum number of function evaluations has been exceeded." |
| 184 |
elif iterations >= maxiter: |
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warnflag = 2 |
| 186 |
if printmessg: |
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print "Warning: Maximum number of iterations has been exceeded" |
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else: |
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if printmessg: |
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print "Optimization terminated successfully." |
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print " Current function value: %f" % fval |
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print " Iterations: %d" % iterations |
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print " Function evaluations: %d" % funcalls |
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|
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if fulloutput: |
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return x, fval, warnflag |
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else: |
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return x |
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|
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|
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def zoom(a_lo, a_hi): |
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pass |
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|
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|
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|
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def line_search(f, fprime, xk, pk, gfk, args=(), c1=1e-4, c2=0.9, amax=50): |
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"""alpha, fc, gc = line_search(f, xk, pk, gfk, args=(), c1=1e-4, c2=0.9, amax=1) |
| 208 |
|
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minimize the function f(xk+alpha pk) using the line search algorithm of |
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Wright and Nocedal in 'numarrayerical Optimization', 1999, pg. 59-60 |
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""" |
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|
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fc = 0 |
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gc = 0 |
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alpha0 = 1.0 |
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phi0 = apply(f,(xk,)+args) |
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phi_a0 = apply(f,(xk+alpha0*pk,)+args) |
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fc = fc + 2 |
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derphi0 = numarray.dot(gfk,pk) |
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derphi_a0 = numarray.dot(apply(fprime,(xk+alpha0*pk,)+args),pk) |
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gc = gc + 1 |
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|
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# check to see if alpha0 = 1 satisfies Strong Wolfe conditions. |
| 224 |
if (phi_a0 <= phi0 + c1*alpha0*derphi0) \ |
| 225 |
and (numarray.absolute(derphi_a0) <= c2*numarray.absolute(derphi0)): |
| 226 |
return alpha0, fc, gc |
| 227 |
|
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alpha0 = 0 |
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alpha1 = 1 |
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phi_a1 = phi_a0 |
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phi_a0 = phi0 |
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|
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i = 1 |
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while 1: |
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if (phi_a1 > phi0 + c1*alpha1*derphi0) or \ |
| 236 |
((phi_a1 >= phi_a0) and (i > 1)): |
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return zoom(alpha0, alpha1) |
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|
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derphi_a1 = numarray.dot(apply(fprime,(xk+alpha1*pk,)+args),pk) |
| 240 |
gc = gc + 1 |
| 241 |
if (numarray.absolute(derphi_a1) <= -c2*derphi0): |
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return alpha1 |
| 243 |
|
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if (derphi_a1 >= 0): |
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return zoom(alpha1, alpha0) |
| 246 |
|
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alpha2 = (amax-alpha1)*0.25 + alpha1 |
| 248 |
i = i + 1 |
| 249 |
alpha0 = alpha1 |
| 250 |
alpha1 = alpha2 |
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phi_a0 = phi_a1 |
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phi_a1 = apply(f,(xk+alpha1*pk,)+args) |
| 253 |
|
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|
| 255 |
|
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def line_search_BFGS(f, xk, pk, gfk, args=(), c1=1e-4, alpha0=1): |
| 257 |
"""alpha, fc, gc = line_search(f, xk, pk, gfk, args=(), c1=1e-4, alpha0=1) |
| 258 |
|
| 259 |
minimize over alpha, the function f(xk+alpha pk) using the interpolation |
| 260 |
algorithm (Armiijo backtracking) as suggested by |
| 261 |
Wright and Nocedal in 'numarrayerical Optimization', 1999, pg. 56-57 |
| 262 |
""" |
| 263 |
|
| 264 |
fc = 0 |
| 265 |
phi0 = apply(f,(xk,)+args) # compute f(xk) |
| 266 |
phi_a0 = apply(f,(xk+alpha0*pk,)+args) # compute f |
| 267 |
fc = fc + 2 |
| 268 |
derphi0 = numarray.dot(gfk,pk) |
| 269 |
|
| 270 |
if (phi_a0 <= phi0 + c1*alpha0*derphi0): |
| 271 |
return alpha0, fc, 0 |
| 272 |
|
| 273 |
# Otherwise compute the minimizer of a quadratic interpolant: |
| 274 |
|
| 275 |
alpha1 = -(derphi0) * alpha0**2 / 2.0 / (phi_a0 - phi0 - derphi0 * alpha0) |
| 276 |
phi_a1 = apply(f,(xk+alpha1*pk,)+args) |
| 277 |
fc = fc + 1 |
| 278 |
|
| 279 |
if (phi_a1 <= phi0 + c1*alpha1*derphi0): |
| 280 |
return alpha1, fc, 0 |
| 281 |
|
| 282 |
# Otherwise loop with cubic interpolation until we find an alpha which satifies |
| 283 |
# the first Wolfe condition (since we are backtracking, we will assume that |
| 284 |
# the value of alpha is not too small and satisfies the second condition. |
| 285 |
|
| 286 |
while 1: # we are assuming pk is a descent direction |
| 287 |
factor = alpha0**2 * alpha1**2 * (alpha1-alpha0) |
| 288 |
a = alpha0**2 * (phi_a1 - phi0 - derphi0*alpha1) - \ |
| 289 |
alpha1**2 * (phi_a0 - phi0 - derphi0*alpha0) |
| 290 |
a = a / factor |
| 291 |
b = -alpha0**3 * (phi_a1 - phi0 - derphi0*alpha1) + \ |
| 292 |
alpha1**3 * (phi_a0 - phi0 - derphi0*alpha0) |
| 293 |
b = b / factor |
| 294 |
|
| 295 |
alpha2 = (-b + numarray.sqrt(numarray.absolute(b**2 - 3 * a * derphi0))) / (3.0*a) |
| 296 |
phi_a2 = apply(f,(xk+alpha2*pk,)+args) |
| 297 |
fc = fc + 1 |
| 298 |
|
| 299 |
if (phi_a2 <= phi0 + c1*alpha2*derphi0): |
| 300 |
return alpha2, fc, 0 |
| 301 |
|
| 302 |
if (alpha1 - alpha2) > alpha1 / 2.0 or (1 - alpha2/alpha1) < 0.96: |
| 303 |
alpha2 = alpha1 / 2.0 |
| 304 |
|
| 305 |
alpha0 = alpha1 |
| 306 |
alpha1 = alpha2 |
| 307 |
phi_a0 = phi_a1 |
| 308 |
phi_a1 = phi_a2 |
| 309 |
|
| 310 |
epsilon = 1e-8 |
| 311 |
|
| 312 |
def approx_fprime(xk,f,*args): |
| 313 |
f0 = apply(f,(xk,)+args) |
| 314 |
grad = numarray.zeros((len(xk),),'d') |
| 315 |
ei = numarray.zeros((len(xk),),'d') |
| 316 |
for k in range(len(xk)): |
| 317 |
ei[k] = 1.0 |
| 318 |
grad[k] = (apply(f,(xk+epsilon*ei,)+args) - f0)/epsilon |
| 319 |
ei[k] = 0.0 |
| 320 |
return grad |
| 321 |
|
| 322 |
def approx_fhess_p(x0,p,fprime,*args): |
| 323 |
f2 = apply(fprime,(x0+epsilon*p,)+args) |
| 324 |
f1 = apply(fprime,(x0,)+args) |
| 325 |
return (f2 - f1)/epsilon |
| 326 |
|
| 327 |
|
| 328 |
def fminBFGS(f, x0, fprime=None, args=(), avegtol=1e-5, maxiter=None, fulloutput=0, printmessg=1): |
| 329 |
"""xopt = fminBFGS(f, x0, fprime=None, args=(), avegtol=1e-5, maxiter=None, fulloutput=0, printmessg=1) |
| 330 |
|
| 331 |
Optimize the function, f, whose gradient is given by fprime using the |
| 332 |
quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) |
| 333 |
See Wright, and Nocedal 'numarrayerical Optimization', 1999, pg. 198. |
| 334 |
""" |
| 335 |
|
| 336 |
app_fprime = 0 |
| 337 |
if fprime is None: |
| 338 |
app_fprime = 1 |
| 339 |
|
| 340 |
x0 = numarray.asarray(x0) |
| 341 |
if maxiter is None: |
| 342 |
maxiter = len(x0)*200 |
| 343 |
func_calls = 0 |
| 344 |
grad_calls = 0 |
| 345 |
k = 0 |
| 346 |
N = len(x0) |
| 347 |
gtol = N*avegtol |
| 348 |
I = numarray.identity(N) |
| 349 |
Hk = I |
| 350 |
|
| 351 |
if app_fprime: |
| 352 |
gfk = apply(approx_fprime,(x0,f)+args) |
| 353 |
func_calls = func_calls + len(x0) + 1 |
| 354 |
else: |
| 355 |
gfk = apply(fprime,(x0,)+args) |
| 356 |
grad_calls = grad_calls + 1 |
| 357 |
xk = x0 |
| 358 |
sk = [2*gtol] |
| 359 |
while (numarray.add.reduce(numarray.absolute(gfk)) > gtol) and (k < maxiter): |
| 360 |
pk = -numarray.dot(Hk,gfk) |
| 361 |
alpha_k, fc, gc = line_search_BFGS(f,xk,pk,gfk,args) |
| 362 |
func_calls = func_calls + fc |
| 363 |
xkp1 = xk + alpha_k * pk |
| 364 |
sk = xkp1 - xk |
| 365 |
xk = xkp1 |
| 366 |
if app_fprime: |
| 367 |
gfkp1 = apply(approx_fprime,(xkp1,f)+args) |
| 368 |
func_calls = func_calls + gc + len(x0) + 1 |
| 369 |
else: |
| 370 |
gfkp1 = apply(fprime,(xkp1,)+args) |
| 371 |
grad_calls = grad_calls + gc + 1 |
| 372 |
|
| 373 |
yk = gfkp1 - gfk |
| 374 |
k = k + 1 |
| 375 |
|
| 376 |
rhok = 1 / numarray.dot(yk,sk) |
| 377 |
A1 = I - sk[:,numarray.NewAxis] * yk[numarray.NewAxis,:] * rhok |
| 378 |
A2 = I - yk[:,numarray.NewAxis] * sk[numarray.NewAxis,:] * rhok |
| 379 |
Hk = numarray.dot(A1,numarray.dot(Hk,A2)) + rhok * sk[:,numarray.NewAxis] * sk[numarray.NewAxis,:] |
| 380 |
gfk = gfkp1 |
| 381 |
|
| 382 |
|
| 383 |
if printmessg or fulloutput: |
| 384 |
fval = apply(f,(xk,)+args) |
| 385 |
if k >= maxiter: |
| 386 |
warnflag = 1 |
| 387 |
if printmessg: |
| 388 |
print "Warning: Maximum number of iterations has been exceeded" |
| 389 |
print " Current function value: %f" % fval |
| 390 |
print " Iterations: %d" % k |
| 391 |
print " Function evaluations: %d" % func_calls |
| 392 |
print " Gradient evaluations: %d" % grad_calls |
| 393 |
else: |
| 394 |
warnflag = 0 |
| 395 |
if printmessg: |
| 396 |
print "Optimization terminated successfully." |
| 397 |
print " Current function value: %f" % fval |
| 398 |
print " Iterations: %d" % k |
| 399 |
print " Function evaluations: %d" % func_calls |
| 400 |
print " Gradient evaluations: %d" % grad_calls |
| 401 |
|
| 402 |
if fulloutput: |
| 403 |
return xk, fval, func_calls, grad_calls, warnflag |
| 404 |
else: |
| 405 |
return xk |
| 406 |
|
| 407 |
|
| 408 |
def fminNCG(f, x0, fprime, fhess_p=None, fhess=None, args=(), avextol=1e-5, maxiter=None, fulloutput=0, printmessg=1): |
| 409 |
"""xopt = fminNCG(f, x0, fprime, fhess_p=None, fhess=None, args=(), avextol=1e-5, maxiter=None, fulloutput=0, printmessg=1) |
| 410 |
|
| 411 |
Optimize the function, f, whose gradient is given by fprime using the |
| 412 |
Newton-CG method. fhess_p must compute the hessian times an arbitrary |
| 413 |
vector. If it is not given, finite-differences on fprime are used to |
| 414 |
compute it. See Wright, and Nocedal 'numarrayerical Optimization', 1999, |
| 415 |
pg. 140. |
| 416 |
""" |
| 417 |
|
| 418 |
x0 = numarray.asarray(x0) |
| 419 |
fcalls = 0 |
| 420 |
gcalls = 0 |
| 421 |
hcalls = 0 |
| 422 |
approx_hessp = 0 |
| 423 |
if fhess_p is None and fhess is None: # Define hessian product |
| 424 |
approx_hessp = 1 |
| 425 |
|
| 426 |
xtol = len(x0)*avextol |
| 427 |
update = [2*xtol] |
| 428 |
xk = x0 |
| 429 |
k = 0 |
| 430 |
while (numarray.add.reduce(numarray.absolute(update)) > xtol) and (k < maxiter): |
| 431 |
# Compute a search direction pk by applying the CG method to |
| 432 |
# del2 f(xk) p = - grad f(xk) starting from 0. |
| 433 |
b = -apply(fprime,(xk,)+args) |
| 434 |
gcalls = gcalls + 1 |
| 435 |
maggrad = numarray.add.reduce(numarray.absolute(b)) |
| 436 |
eta = min([0.5,numarray.sqrt(maggrad)]) |
| 437 |
termcond = eta * maggrad |
| 438 |
xsupi = numarray.zeros((len(x0),)) |
| 439 |
ri = -b |
| 440 |
psupi = -ri |
| 441 |
i = 0 |
| 442 |
dri0 = numarray.dot(ri,ri) |
| 443 |
|
| 444 |
if fhess is not None: # you want to compute hessian once. |
| 445 |
A = apply(fhess,(xk,)+args) |
| 446 |
hcalls = hcalls + 1 |
| 447 |
|
| 448 |
while numarray.add.reduce(numarray.absolute(ri)) > termcond: |
| 449 |
if fhess is None: |
| 450 |
if approx_hessp: |
| 451 |
Ap = apply(approx_fhess_p,(xk,psupi,fprime)+args) |
| 452 |
gcalls = gcalls + 2 |
| 453 |
else: |
| 454 |
Ap = apply(fhess_p,(xk,psupi)+args) |
| 455 |
hcalls = hcalls + 1 |
| 456 |
else: |
| 457 |
# Ap = numarray.dot(A,psupi) |
| 458 |
Ap = numarray.matrixmultiply(A,psupi) |
| 459 |
# check curvature |
| 460 |
curv = numarray.dot(psupi,Ap) |
| 461 |
if (curv <= 0): |
| 462 |
if (i > 0): |
| 463 |
break |
| 464 |
else: |
| 465 |
xsupi = xsupi + dri0/curv * psupi |
| 466 |
break |
| 467 |
alphai = dri0 / curv |
| 468 |
xsupi = xsupi + alphai * psupi |
| 469 |
ri = ri + alphai * Ap |
| 470 |
dri1 = numarray.dot(ri,ri) |
| 471 |
betai = dri1 / dri0 |
| 472 |
psupi = -ri + betai * psupi |
| 473 |
i = i + 1 |
| 474 |
dri0 = dri1 # update numarray.dot(ri,ri) for next time. |
| 475 |
|
| 476 |
pk = xsupi # search direction is solution to system. |
| 477 |
gfk = -b # gradient at xk |
| 478 |
alphak, fc, gc = line_search_BFGS(f,xk,pk,gfk,args) |
| 479 |
fcalls = fcalls + fc |
| 480 |
gcalls = gcalls + gc |
| 481 |
|
| 482 |
update = alphak * pk |
| 483 |
xk = xk + update |
| 484 |
k = k + 1 |
| 485 |
|
| 486 |
if printmessg or fulloutput: |
| 487 |
fval = apply(f,(xk,)+args) |
| 488 |
if k >= maxiter: |
| 489 |
warnflag = 1 |
| 490 |
if printmessg: |
| 491 |
print "Warning: Maximum number of iterations has been exceeded" |
| 492 |
print " Current function value: %f" % fval |
| 493 |
print " Iterations: %d" % k |
| 494 |
print " Function evaluations: %d" % fcalls |
| 495 |
print " Gradient evaluations: %d" % gcalls |
| 496 |
print " Hessian evaluations: %d" % hcalls |
| 497 |
else: |
| 498 |
warnflag = 0 |
| 499 |
if printmessg: |
| 500 |
print "Optimization terminated successfully." |
| 501 |
print " Current function value: %f" % fval |
| 502 |
print " Iterations: %d" % k |
| 503 |
print " Function evaluations: %d" % fcalls |
| 504 |
print " Gradient evaluations: %d" % gcalls |
| 505 |
print " Hessian evaluations: %d" % hcalls |
| 506 |
|
| 507 |
if fulloutput: |
| 508 |
return xk, fval, fcalls, gcalls, hcalls, warnflag |
| 509 |
else: |
| 510 |
return xk |
| 511 |
|
| 512 |
|
| 513 |
|
| 514 |
if __name__ == "__main__": |
| 515 |
import string |
| 516 |
import time |
| 517 |
|
| 518 |
|
| 519 |
times = [] |
| 520 |
algor = [] |
| 521 |
x0 = [0.8,1.2,0.7] |
| 522 |
start = time.time() |
| 523 |
x = fmin(rosen,x0) |
| 524 |
print x |
| 525 |
times.append(time.time() - start) |
| 526 |
algor.append('Nelder-Mead Simplex\t') |
| 527 |
|
| 528 |
start = time.time() |
| 529 |
x = fminBFGS(rosen, x0, fprime=rosen_der, maxiter=80) |
| 530 |
print x |
| 531 |
times.append(time.time() - start) |
| 532 |
algor.append('BFGS Quasi-Newton\t') |
| 533 |
|
| 534 |
start = time.time() |
| 535 |
x = fminBFGS(rosen, x0, avegtol=1e-4, maxiter=100) |
| 536 |
print x |
| 537 |
times.append(time.time() - start) |
| 538 |
algor.append('BFGS without gradient\t') |
| 539 |
|
| 540 |
|
| 541 |
start = time.time() |
| 542 |
x = fminNCG(rosen, x0, rosen_der, fhess_p=rosen3_hess_p, maxiter=80) |
| 543 |
print x |
| 544 |
times.append(time.time() - start) |
| 545 |
algor.append('Newton-CG with hessian product') |
| 546 |
|
| 547 |
|
| 548 |
start = time.time() |
| 549 |
x = fminNCG(rosen, x0, rosen_der, fhess=rosen3_hess, maxiter=80) |
| 550 |
print x |
| 551 |
times.append(time.time() - start) |
| 552 |
algor.append('Newton-CG with full hessian') |
| 553 |
|
| 554 |
print "\nMinimizing the Rosenbrock function of order 3\n" |
| 555 |
print " Algorithm \t\t\t Seconds" |
| 556 |
print "===========\t\t\t =========" |
| 557 |
for k in range(len(algor)): |
| 558 |
print algor[k], "\t -- ", times[k] |
| 559 |
|
| 560 |
times = [] |
| 561 |
algor=[] |
| 562 |
x0 = [1.,] |
| 563 |
start = time.time() |
| 564 |
x = fmin(para,x0) |
| 565 |
print x |
| 566 |
times.append(time.time() - start) |
| 567 |
algor.append('Nelder-Mead Simplex\t') |
| 568 |
|
| 569 |
start = time.time() |
| 570 |
x = fminBFGS(para, x0, fprime=para_der, maxiter=80) |
| 571 |
print x |
| 572 |
times.append(time.time() - start) |
| 573 |
algor.append('BFGS Quasi-Newton\t') |
| 574 |
|
| 575 |
start = time.time() |
| 576 |
x = fminBFGS(para, x0, avegtol=1e-4, maxiter=100) |
| 577 |
print x |
| 578 |
times.append(time.time() - start) |
| 579 |
algor.append('BFGS without gradient\t') |
| 580 |
|
| 581 |
|
| 582 |
start = time.time() |
| 583 |
x = fminNCG(para, x0, para_der, fhess_p=para_hess_p, maxiter=80) |
| 584 |
print x |
| 585 |
times.append(time.time() - start) |
| 586 |
algor.append('Newton-CG with hessian product') |
| 587 |
|
| 588 |
|
| 589 |
start = time.time() |
| 590 |
x = fminNCG(para, x0, para_der, fhess=para_hess, maxiter=80) |
| 591 |
print x |
| 592 |
times.append(time.time() - start) |
| 593 |
algor.append('Newton-CG with full hessian') |
| 594 |
|
| 595 |
print "\nMinimizing x^2\n" |
| 596 |
print " Algorithm \t\t\t Seconds" |
| 597 |
print "===========\t\t\t =========" |
| 598 |
for k in range(len(algor)): |
| 599 |
print algor[k], "\t -- ", times[k] |
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