def _setup_driver(self, problem):
"""
Prepare the driver for execution.
This is the final thing to run during setup.
Parameters
----------
paropt_problem : <Problem>
Pointer
"""
# TODO:
# - logic for different opt algorithms
# - treat equality constraints
super(ParOptDriver, self)._setup_driver(problem)
# Raise error if multiple objectives are provided
if len(self._objs) > 1:
msg = 'ParOpt currently does not support multiple objectives.'
raise RuntimeError(msg.format(self.__class__.__name__))
# Create the ParOptProblem from the OpenMDAO problem
self.paropt_problem = ParOptProblem(problem)
self.opt = ParOpt.Optimizer(self.paropt_problem, self.options)
return
def paropt_truss(truss, use_hessian=False):
'''
Optimize the given truss structure using ParOpt
'''
fname = os.path.join(prefix, 'truss_paropt%dx%d.out' % (N, M))
options = {
'algorithm': 'ip',
'qn_subspace_size': 10,
'abs_res_tol': 1e-6,
'barrier_strategy': 'complementarity_fraction',
'use_hvec_product': True,
'gmres_subspace_size': 25,
'nk_switch_tol': 1.0,
'eisenstat_walker_gamma': 0.01,
'eisenstat_walker_alpha': 0.0,
'max_gmres_rtol': 1.0,
'output_level': 1,
'armijo_constant': 1e-5,
'output_file': fname
}
if use_hessian is False:
options['use_hvec_product'] = False
opt = ParOpt.Optimizer(truss, options)
opt.optimize()
return opt
def plot_it_all(problem):
"""
Plot a carpet plot with the search histories for steepest descent,
conjugate gradient and BFGS from the same starting point.
"""
# Create the data for the carpet plot
n = 150
xlow = -4.0
xhigh = 4.0
x1 = np.linspace(xlow, xhigh, n)
r = np.zeros((n, n))
for j in range(n):
for i in range(n):
fail, fobj, con = problem.evalObjCon([x1[i], x1[j]])
r[j, i] = fobj
# Assign the contour levels
levels = np.min(r) + np.linspace(0, 1.0, 75)**2 * (np.max(r) - np.min(r))
# Create the plot
fig = plt.figure(facecolor='w')
plt.contour(x1, x1, r, levels)
colours = [
'-bo', '-ko', '-co', '-mo', '-yo', '-bx', '-kx', '-cx', '-mx', '-yx'
]
options = {
'algorithm': 'tr',
'tr_init_size': 0.05,
'tr_min_size': 1e-6,
'tr_max_size': 10.0,
'tr_eta': 0.1,
'tr_adaptive_gamma_update': True,
'tr_max_iterations': 200
}
for k in range(len(colours)):
# Optimize the problem
problem.x_hist = []
opt = ParOpt.Optimizer(rosen, options)
opt.optimize()
# Copy out the steepest descent points
sd = np.zeros((2, len(problem.x_hist)))
for i in range(len(problem.x_hist)):
sd[0, i] = problem.x_hist[i][0]
sd[1, i] = problem.x_hist[i][1]
plt.plot(sd[0, :], sd[1, :], colours[k], label='IP %d' % (sd.shape[1]))
plt.plot(sd[0, -1], sd[1, -1], '-ro')
plt.legend()
plt.axis([xlow, xhigh, xlow, xhigh])
plt.show()
def solve_problem(eigs, filename=None, use_stdout=False, use_tr=False):
# Get the A matrix
A = create_random_problem(eigs)
# Create the other problem data
b = np.random.uniform(size=len(eigs))
Acon = np.random.uniform(size=len(eigs))
bcon = np.random.uniform()
problem = Quadratic(A, b, Acon, bcon)
options = {
'algorithm': 'ip',
'abs_res_tol': 1e-8,
'starting_point_strategy': 'affine_step',
'barrier_strategy': 'monotone',
'start_affine_multiplier_min': 0.01,
'penalty_gamma': 1000.0,
'qn_subspace_size': 10,
'qn_type': 'bfgs',
'output_file': filename}
if use_tr:
options = {
'algorithm': 'tr',
'tr_init_size': 0.05,
'tr_min_size': 1e-6,
'tr_max_size': 10.0,
'tr_eta': 0.25,
'tr_adaptive_gamma_update': True,
'tr_max_iterations': 200,
'penalty_gamma': 10.0,
'qn_subspace_size': 10,
'qn_type': 'bfgs',
'abs_res_tol': 1e-8,
'output_file': filename,
'tr_output_file': os.path.splitext(filename)[0] + '.tr',
'starting_point_strategy': 'affine_step',
'barrier_strategy': 'monotone',
'start_affine_multiplier_min': 0.01}
opt = ParOpt.Optimizer(problem, options)
# Set a new starting point
opt.optimize()
x, z, zw, zl, zu = opt.getOptimizedPoint()
return
def paropt_truss(truss, use_hessian=False, use_tr=False, prefix='results'):
"""
Optimize the given truss structure using ParOpt
"""
fname = os.path.join(prefix, 'truss_paropt%dx%d.out' % (N, M))
options = {
'algorithm': 'ip',
'qn_subspace_size': 10,
'abs_res_tol': 1e-5,
'norm_type': 'l1',
'init_barrier_param': 10.0,
'monotone_barrier_fraction': 0.75,
'barrier_strategy': 'complementarity_fraction',
'starting_point_strategy': 'least_squares_multipliers',
'use_hvec_product': True,
'gmres_subspace_size': 50,
'nk_switch_tol': 1e3,
'eisenstat_walker_gamma': 0.01,
'eisenstat_walker_alpha': 0.0,
'max_gmres_rtol': 1.0,
'output_level': 1,
'armijo_constant': 1e-5,
'output_file': fname
}
if use_tr:
options['algorithm'] = 'tr'
options['abs_res_tol'] = 1e-8
options['barrier_strategy'] = 'monotone'
options['tr_max_size'] = 100.0
options['tr_linfty_tol'] = 1e-5
options['tr_l1_tol'] = 0.0
options['tr_max_iterations'] = 2000
options['tr_penalty_gamma_max'] = 1e6
options['tr_adaptive_gamma_update'] = True
options['tr_output_file'] = fname.split('.')[0] + '.tr'
if use_hessian is False:
options['use_hvec_product'] = False
opt = ParOpt.Optimizer(truss, options)
opt.optimize()
return opt
options = {
'algorithm': 'tr',
'tr_init_size': 0.05,
'tr_min_size': 1e-6,
'tr_max_size': 10.0,
'tr_eta': 0.25,
'tr_infeas_tol': 1e-6,
'tr_l1_tol': 1e-3,
'tr_linfty_tol': 0.0,
'tr_adaptive_gamma_update': True,
'tr_max_iterations': 1000,
'penalty_gamma': 10.0,
'qn_subspace_size': 10,
'qn_type': 'bfgs',
'abs_res_tol': 1e-8,
'starting_point_strategy': 'affine_step',
'barrier_strategy': 'mehrotra_predictor_corrector',
'tr_steering_barrier_strategy':
'mehrotra_predictor_corrector',
'tr_steering_starting_point_strategy': 'affine_step',
'use_line_search': False}
# Set up the optimizer
opt = ParOpt.Optimizer(problem, options)
# Set a new starting point
opt.optimize()
x, z, zw, zl, zu = opt.getOptimizedPoint()
def __call__(self,
optProb,
sens=None,
sensStep=None,
sensMode=None,
storeHistory=None,
hotStart=None,
storeSens=True):
"""
This is the main routine used to solve the optimization
problem.
Parameters
----------
optProb : Optimization or Solution class instance
This is the complete description of the optimization problem
to be solved by the optimizer
sens : str or python Function.
Specifiy method to compute sensitivities. To
explictly use pyOptSparse gradient class to do the
derivatives with finite differenes use \'FD\'. \'sens\'
may also be \'CS\' which will cause pyOptSpare to compute
the derivatives using the complex step method. Finally,
\'sens\' may be a python function handle which is expected
to compute the sensitivities directly. For expensive
function evaluations and/or problems with large numbers of
design variables this is the preferred method.
sensStep : float
Set the step size to use for design variables. Defaults to
1e-6 when sens is \'FD\' and 1e-40j when sens is \'CS\'.
sensMode : str
Use \'pgc\' for parallel gradient computations. Only
available with mpi4py and each objective evaluation is
otherwise serial
storeHistory : str
File name of the history file into which the history of
this optimization will be stored
hotStart : str
File name of the history file to "replay" for the
optimziation. The optimization problem used to generate
the history file specified in \'hotStart\' must be
**IDENTICAL** to the currently supplied \'optProb\'. By
identical we mean, **EVERY SINGLE PARAMETER MUST BE
IDENTICAL**. As soon as he requested evaluation point
from ParOpt does not match the history, function and
gradient evaluations revert back to normal evaluations.
storeSens : bool
Flag sepcifying if sensitivities are to be stored in hist.
This is necessay for hot-starting only.
"""
self.callCounter = 0
self.storeSens = storeSens
if len(optProb.constraints) == 0:
# If the problem is unconstrained, add a dummy constraint.
self.unconstrained = True
optProb.dummyConstraint = True
# Save the optimization problem and finalize constraint
# jacobian, in general can only do on root proc
self.optProb = optProb
self.optProb.finalizeDesignVariables()
self.optProb.finalizeConstraints()
self._setInitialCacheValues()
self._setSens(sens, sensStep, sensMode)
blx, bux, xs = self._assembleContinuousVariables()
xs = np.maximum(xs, blx)
xs = np.minimum(xs, bux)
# The number of design variables
n = len(xs)
oneSided = True
if self.unconstrained:
m = 0
else:
indices, blc, buc, fact = self.optProb.getOrdering(
["ne", "le", "ni", "li"], oneSided=oneSided)
m = len(indices)
self.optProb.jacIndices = indices
self.optProb.fact = fact
self.optProb.offset = buc
if self.optProb.comm.rank == 0:
# Set history/hotstart
self._setHistory(storeHistory, hotStart)
class Problem(_ParOpt.Problem):
def __init__(self, ptr, n, m, xs, blx, bux):
super(Problem, self).__init__(MPI.COMM_SELF, n, m)
self.ptr = ptr
self.n = n
self.m = m
self.xs = xs
self.blx = blx
self.bux = bux
self.fobj = 0.0
return
def getVarsAndBounds(self, x, lb, ub):
"""Get the variable values and bounds"""
# Find the average distance between lower and upper bound
bound_sum = 0.0
for i in range(len(x)):
if self.blx[i] <= -1e20 or self.bux[i] >= 1e20:
bound_sum += 1.0
else:
bound_sum += self.bux[i] - self.blx[i]
bound_sum = bound_sum / len(x)
for i in range(len(x)):
x[i] = self.xs[i]
lb[i] = self.blx[i]
ub[i] = self.bux[i]
if self.xs[i] <= self.blx[i]:
x[i] = self.blx[i] + 0.5 * np.min(
(bound_sum, self.bux[i] - self.blx[i]))
elif self.xs[i] >= self.bux[i]:
x[i] = self.bux[i] - 0.5 * np.min(
(bound_sum, self.bux[i] - self.blx[i]))
return
def evalObjCon(self, x):
"""Evaluate the objective and constraint values"""
fobj, fcon, fail = self.ptr._masterFunc(
x[:], ["fobj", "fcon"])
self.fobj = fobj
return fail, fobj, -fcon
def evalObjConGradient(self, x, g, A):
"""Evaluate the objective and constraint gradients"""
gobj, gcon, fail = self.ptr._masterFunc(
x[:], ["gobj", "gcon"])
g[:] = gobj[:]
for i in range(self.m):
A[i][:] = -gcon[i][:]
return fail
optTime = MPI.Wtime()
# Optimize the problem
problem = Problem(self, n, m, xs, blx, bux)
optimizer = _ParOpt.Optimizer(problem, self.set_options)
optimizer.optimize()
x, z, zw, zl, zu = optimizer.getOptimizedPoint()
# Set the total opt time
optTime = MPI.Wtime() - optTime
# Get the obective function value
fobj = problem.fobj
if self.storeHistory:
self.metadata["endTime"] = datetime.datetime.now().strftime(
"%Y-%m-%d %H:%M:%S")
self.metadata["optTime"] = optTime
self.hist.writeData("metadata", self.metadata)
self.hist.close()
# Create the optimization solution. Note that the signs on the multipliers
# are switch since ParOpt uses a formulation with c(x) >= 0, while pyOpt
# uses g(x) = -c(x) <= 0. Therefore the multipliers are reversed.
sol_inform = {}
# If number of constraints is zero, ParOpt returns z as None.
# Thus if there is no constraints, should pass an empty list
# to multipliers instead of z.
if z is not None:
sol = self._createSolution(optTime,
sol_inform,
fobj,
x[:],
multipliers=-z)
else:
sol = self._createSolution(optTime,
sol_inform,
fobj,
x[:],
multipliers=[])
# Indicate solution finished
self.optProb.comm.bcast(-1, root=0)
else: # We are not on the root process so go into waiting loop:
self._waitLoop()
sol = None
# Communication solution and return
sol = self._communicateSolution(sol)
return sol
def plot_it_all(problem, use_tr=False):
"""
Plot a carpet plot with the search histories for steepest descent,
conjugate gradient and BFGS from the same starting point.
"""
# Check the problem gradients
problem.checkGradients(1e-6)
# Create the data for the carpet plot
n = 150
xlow = -4.0
xhigh = 4.0
x1 = np.linspace(xlow, xhigh, n)
ylow = -3.0
yhigh = 3.0
x2 = np.linspace(ylow, yhigh, n)
r = np.zeros((n, n))
for j in range(n):
for i in range(n):
fail, fobj, con = problem.evalObjCon([x1[i], x2[j]])
r[j, i] = fobj
# Assign the contour levels
levels = np.min(r) + np.linspace(0, 1.0, 75)**2 * (np.max(r) - np.min(r))
# Create the plot
fig = plt.figure(facecolor='w')
plt.contour(x1, x2, r, levels)
plt.plot([0.5 - yhigh, 0.5 - ylow], [yhigh, ylow], '-k')
colours = [
'-bo', '-ko', '-co', '-mo', '-yo', '-bx', '-kx', '-cx', '-mx', '-yx'
]
for k in range(len(colours)):
# Optimize the problem
problem.x_hist = []
# Set the trust region parameters
filename = 'paropt.out'
options = {
'algorithm': 'ip',
'abs_res_tol': 1e-8,
'starting_point_strategy': 'affine_step',
'barrier_strategy': 'monotone',
'start_affine_multiplier_min': 0.01,
'penalty_gamma': 1000.0,
'qn_subspace_size': 10,
'qn_type': 'bfgs'
}
if use_tr:
options = {
'algorithm': 'tr',
'tr_init_size': 0.05,
'tr_min_size': 1e-6,
'tr_max_size': 10.0,
'tr_eta': 0.25,
'penalty_gamma': 10.0,
'qn_subspace_size': 10,
'qn_type': 'bfgs',
'abs_res_tol': 1e-8,
'output_file': filename,
'tr_output_file': os.path.splitext(filename)[0] + '.tr',
'starting_point_strategy': 'affine_step',
'barrier_strategy': 'monotone',
'start_affine_multiplier_min': 0.01
}
opt = ParOpt.Optimizer(problem, options)
# Set a new starting point
opt.optimize()
x, z, zw, zl, zu = opt.getOptimizedPoint()
# Copy out the steepest descent points
popt = np.zeros((2, len(problem.x_hist)))
for i in range(len(problem.x_hist)):
popt[0, i] = problem.x_hist[i][0]
popt[1, i] = problem.x_hist[i][1]
plt.plot(popt[0, :],
popt[1, :],
colours[k],
label='ParOpt %d' % (popt.shape[1]))
plt.plot(popt[0, -1], popt[1, -1], '-ro')
# Print the data to the screen
g = np.zeros(2)
A = np.zeros((1, 2))
problem.evalObjConGradient(x, g, A)
print('The design variables: ', x[:])
print('The multipliers: ', z[:])
print('The objective gradient: ', g[:])
print('The constraint gradient: ', A[:])
ax = fig.axes[0]
ax.set_aspect('equal', 'box')
plt.legend()
'use_backtracking_alpha': True,
'output_level': 1,
'max_major_iters': 1000
}
# use trust region algorithm
if use_tr:
options = {
'algorithm': 'tr',
'qn_type': 'bfgs',
'abs_res_tol': 1e-8,
'output_level': 0,
'use_backtracking_alpha': True,
'max_major_iters': 100,
'tr_init_size': 0.1,
'tr_min_size': 1e-6,
'tr_max_size': 1.0,
'tr_eta': 0.25,
'penalty_gamma': 1.0,
'tr_adaptive_gamma_update': True,
'tr_penalty_gamma_max': 1e5,
'tr_penalty_gamma_min': 1e-5,
'tr_max_iterations': 500,
'use_line_search': False
}
polygon = Polygon(args.n)
polygon.checkGradients()
opt = ParOpt.Optimizer(polygon, options)
opt.optimize()
def solve_problem(n,
ndv,
N,
rho,
filename=None,
use_quadratic_approx=True,
verify=False):
problem = SpectralAggregate(n, ndv, rho=rho)
if verify:
x0 = np.random.uniform(size=ndv)
problem.verify_derivatives(x0)
options = {
'algorithm': 'tr',
'tr_init_size': 0.05,
'tr_min_size': 1e-6,
'tr_max_size': 10.0,
'tr_eta': 0.1,
'tr_output_file': os.path.splitext(filename)[0] + '.tr',
'tr_penalty_gamma_max': 1e6,
'tr_adaptive_gamma_update': True,
'tr_infeas_tol': 1e-6,
'tr_l1_tol': 5e-4,
'tr_linfty_tol': 5e-4,
'tr_max_iterations': 200,
'output_level': 2,
'penalty_gamma': 10.0,
'qn_subspace_size': 10,
'qn_type': 'bfgs',
'abs_res_tol': 1e-8,
'output_file': filename,
'starting_point_strategy': 'affine_step',
'barrier_strategy': 'monotone',
'start_affine_multiplier_min': 0.01
}
if use_quadratic_approx:
options['qn_subspace_size'] = 0
options['qn_type'] = 'none'
opt = ParOpt.Optimizer(problem, options)
if use_quadratic_approx:
# Create the BFGS approximation
qn = ParOpt.LBFGS(problem, subspace=10)
# Create the quadratic eigenvalue approximation object
approx = ParOptEig.CompactEigenApprox(problem, N)
# Set up the corresponding quadratic approximation, specifying the index
# of the eigenvalue constraint
eig_qn = ParOptEig.EigenQuasiNewton(qn, approx, index=0)
# Set up the eigenvalue optimization subproblem
subproblem = ParOptEig.EigenSubproblem(problem, eig_qn)
subproblem.setUpdateEigenModel(problem.updateModel)
opt.setTrustRegionSubproblem(subproblem)
opt.optimize()
# Get the optimized point from the trust-region subproblem
x, z, zw, zl, zu = opt.getOptimizedPoint()
print('max(x) = %15.6e' % (np.max(x[:])))
print('avg(x) = %15.6e' % (np.average(x[:])))
print('sum(x) = %15.6e' % (np.sum(x[:])))
if verify:
for h in [1e-6, 1e-7, 1e-8, 1e-9, 1e-10]:
problem.checkGradients(h)
problem.verify_derivatives(x[:], h)
return x
def solve_problem(eigs, filename=None, data_type='orthogonal', use_tr=False):
# Create a random orthogonal Q vector
if data_type == 'orthogonal':
B = np.random.uniform(size=(n, n))
Q, s, v = np.linalg.svd(B)
# Create a random Affine matrix
Affine = create_random_spd(eigs)
else:
Q = np.random.uniform(size=(n, n))
Affine = np.diag(1e-3 * np.ones(n))
# Create the random right-hand-side
b = np.random.uniform(size=n)
# Create the constraint data
Acon = np.random.uniform(size=n)
bcon = 0.25 * np.sum(Acon)
# Create the convex problem
problem = ConvexProblem(Q, Affine, b, Acon, bcon)
options = {
'algorithm': 'ip',
'abs_res_tol': 1e-8,
'starting_point_strategy': 'affine_step',
'barrier_strategy': 'monotone',
'start_affine_multiplier_min': 0.01,
'penalty_gamma': 1000.0,
'qn_subspace_size': 10,
'qn_type': 'bfgs',
'output_file': filename
}
if use_tr:
options = {
'algorithm': 'tr',
'tr_init_size': 0.05,
'tr_min_size': 1e-6,
'tr_max_size': 10.0,
'tr_eta': 0.25,
'tr_adaptive_gamma_update': True,
'tr_max_iterations': 200,
'penalty_gamma': 10.0,
'qn_subspace_size': 10,
'qn_type': 'bfgs',
'abs_res_tol': 1e-8,
'output_file': filename,
'tr_output_file': os.path.splitext(filename)[0] + '.tr',
'starting_point_strategy': 'affine_step',
'barrier_strategy': 'monotone',
'start_affine_multiplier_min': 0.01
}
opt = ParOpt.Optimizer(problem, options)
# Set a new starting point
opt.optimize()
x, z, zw, zl, zu = opt.getOptimizedPoint()
return x