def find_feasible(self):
"""Obtain `x_feasible` satisfying the constraints.
`rhs` must have been specified.
"""
n = self.n
self.log.debug('Obtaining feasible solution...')
self.t_feasible = cputime()
self.rhs[n:] = self.b
self.proj.solve(self.rhs)
self.x_feasible = self.proj.x[:n].copy()
self.t_feasible = cputime() - self.t_feasible
self.check_accurate()
self.log.debug('... done (%-5.2fs)' % self.t_feasible)
return
def perform_factorization(self):
"""Assemble projection matrix and factorize it.
P = [ G A^T ]
[ A 0 ],
where G is the preconditioner, or the identity matrix if no
preconditioner was given.
"""
if self.A is None:
raise ValueError('No linear equality constraints were specified')
# Form projection matrix
P = spmatrix.ll_mat_sym(self.n + self.m, self.nnzA + self.n)
if self.precon is not None:
P[:self.n, :self.n] = self.precon
else:
r = range(self.n)
P.put(1, r, r)
# for i in range(self.n):
# P[i,i] = 1
P[self.n:, :self.n] = self.A
# Add regularization if requested.
if self.dreg > 0.0:
r = range(self.n, self.n + self.m)
P.put(-self.dreg, r, r)
msg = 'Factorizing projection matrix '
msg += '(size %-d, nnz = %-d)...' % (P.shape[0], P.nnz)
self.log.debug(msg)
self.t_fact = cputime()
self.proj = LBLContext(P)
self.t_fact = cputime() - self.t_fact
self.log.debug('... done (%-5.2fs)' % self.t_fact)
self.factorized = True
return
def solve(self, **kwargs):
"""Solve.
:keywords:
:maxiter: maximum number of iterations.
All other keyword arguments are passed directly to the constructor of
the trust-region solver.
"""
nlp = self.nlp
# Gather initial information.
self.f = self.nlp.obj(self.x)
self.f0 = self.f
self.g = self.nlp.grad(self.x)
self.g_old = self.g
self.gNorm = norms.norm2(self.g)
self.g0 = self.gNorm
self.tr.radius = min(max(0.1 * self.gNorm, 1.0), 100)
cgtol = 1.0 if self.inexact else -1.0
stoptol = max(self.abstol, self.reltol * self.g0)
step_status = None
exitUser = False
exitOptimal = self.gNorm <= stoptol
exitIter = self.iter >= self.maxiter
status = ""
# Initialize non-monotonicity parameters.
if not self.monotone:
fMin = fRef = fCan = self.f0
l = 0
sigRef = sigCan = 0
t = cputime()
# Print out header and initial log.
if self.iter % 20 == 0:
self.log.info(self.header)
self.log.info(self.format0, self.iter, self.f, self.gNorm, "", "",
"", self.tr.radius, "")
while not (exitUser or exitOptimal or exitIter):
self.iter += 1
self.alpha = 1.0
if self.save_g:
self.g_old = self.g.copy()
# Iteratively minimize the quadratic model in the trust region
# min m(s) := g's + ½ s'Hs
# Note that m(s) does not include f(x): m(0) = 0.
if self.inexact:
cgtol = max(stoptol, min(0.7 * cgtol, 0.01 * self.gNorm))
qp = QPModel(self.g, self.nlp.hop(self.x, self.nlp.pi0))
self.solver = TrustRegionSolver(qp, self.tr_solver)
self.solver.solve(prec=self.precon,
radius=self.tr.radius,
reltol=cgtol)
step = self.solver.step
snorm = self.solver.step_norm
cgiter = self.solver.niter
# Obtain model value at next candidate
m = self.solver.m
if m is None:
m = qp.obj(step)
self.total_cgiter += cgiter
x_trial = self.x + step
f_trial = nlp.obj(x_trial)
rho = self.tr.ratio(self.f, f_trial, m)
if not self.monotone:
rhoHis = (fRef - f_trial) / (sigRef - m)
rho = max(rho, rhoHis)
step_status = "Rej"
self.step_accepted = False
if rho >= self.tr.eta1:
# Trust-region step is accepted.
self.tr.update_radius(rho, snorm)
self.x = x_trial
self.f = f_trial
self.g = nlp.grad(self.x)
self.gNorm = norms.norm2(self.g)
self.dvars = step
if self.save_g:
self.dgrad = self.g - self.g_old
step_status = "Acc"
self.step_accepted = True
# Update non-monotonicity parameters.
if not self.monotone:
sigRef = sigRef - m
sigCan = sigCan - m
if f_trial < fMin:
fCan = f_trial
fMin = f_trial
sigCan = 0
l = 0
else:
l = l + 1
if f_trial > fCan:
fCan = f_trial
sigCan = 0
if l == self.n_non_monotone:
fRef = fCan
sigRef = sigCan
else:
# Trust-region step is rejected.
if self.ny: # Backtracking linesearch a la Nocedal & Yuan.
slope = np.dot(self.g, step)
bk = 0
armijo = self.f + 1.0e-4 * self.alpha * slope
while bk < self.nbk and f_trial >= armijo:
bk = bk + 1
self.alpha /= 1.2
x_trial = self.x + self.alpha * step
f_trial = nlp.obj(x_trial)
self.x = x_trial
self.f = f_trial
self.g = nlp.grad(self.x)
self.gNorm = norms.norm2(self.g)
self.tr.radius = self.alpha * snorm
snorm /= self.alpha
step_status = "N-Y"
self.step_accepted = True
self.dvars = self.alpha * step
if self.save_g:
self.dgrad = self.g - self.g_old
else:
self.tr.update_radius(rho, snorm)
self.step_status = step_status
self.radii.append(self.tr.radius)
status = ""
try:
self.post_iteration()
except UserExitRequest:
status = "usr"
# Print out header, say, every 20 iterations
if self.iter % 20 == 0:
self.log.info(self.header)
pstatus = step_status if step_status != "Acc" else ""
self.log.info(self.format % (self.iter, self.f, self.gNorm, cgiter,
rho, snorm, self.tr.radius, pstatus))
exitOptimal = self.gNorm <= stoptol
exitIter = self.iter > self.maxiter
exitUser = status == "usr"
self.tsolve = cputime() - t # Solve time
# Set final solver status.
if status == "usr":
pass
elif self.gNorm <= stoptol:
status = "opt"
else: # self.iter > self.maxiter:
status = "itr"
self.status = status
def solve(self, **kwargs):
"""Solve method.
All keyword arguments are passed directly to the constructor of the
bound constraint solver.
"""
al_model = self.model
slack_model = self.model.model
on = slack_model.original_n
# Move starting point into the feasible box
self.x = project(self.x, al_model.Lvar, al_model.Uvar)
# "Smart" initialization of slack variables using the magical step
# function that is already available
(self.x, m_step_init) = self.model.magical_step(self.x)
dL = al_model.dual_feasibility(self.x)
self.f = self.f0 = self.model.model.model.obj(self.x[:on])
PdL = self.project_gradient(self.x, dL)
Pmax = np.max(np.abs(PdL))
self.pg0 = self.pgnorm = Pmax
# Specific handling for the case where the original NLP is
# unconstrained
if slack_model.m == 0:
max_cons = 0.
else:
max_cons = np.max(np.abs(slack_model.cons(self.x)))
cons_norm_ref = max_cons
self.cons0 = max_cons
self.omega = self.omega_init
self.eta = self.eta_init
self.omega_opt = self.omega_rel * self.pg0 + self.omega_abs
self.eta_opt = self.eta_rel * max_cons + self.eta_abs
self.iter = 0
self.inner_fail_count = 0
self.niter_total = 0
infeas_iter = 0
exitIter = False
exitTime = False
# Convergence check
exitOptimal = (Pmax <= self.omega_opt and max_cons <= self.eta_opt)
if exitOptimal:
self.status = "opt"
tick = cputime()
# Print out header and initial log.
if self.iter % 20 == 0:
self.log.info(self.header)
self.log.info(self.format0, self.iter, self.f, self.pg0,
self.cons0, al_model.penalty, "", "", self.omega,
self.eta)
while not (exitOptimal or exitIter or exitTime):
self.iter += 1
# Perform bound-constrained minimization
# TODO: set appropriate stopping conditions
bc_solver = self.setup_bc_solver()
bc_solver.solve()
self.x = bc_solver.x.copy() # may not be useful.
self.niter_total += bc_solver.iter + 1
dL = al_model.dual_feasibility(self.x)
PdL = self.project_gradient(self.x, dL)
Pmax = np.max(np.abs(PdL))
convals = slack_model.cons(self.x)
# Specific handling for the case where the original NLP is
# unconstrained
if slack_model.m == 0:
max_cons = 0.
else:
max_cons = np.max(np.abs(convals))
self.f = self.model.model.model.obj(self.x[:on])
self.pgnorm = Pmax
# Print out header, say, every 20 iterations.
if self.iter % 20 == 0:
self.log.info(self.header)
self.log.info(self.format % (self.iter, self.f,
self.pgnorm, max_cons,
al_model.penalty,
bc_solver.iter, bc_solver.status,
self.omega, self.eta))
# Update penalty parameter or multipliers based on result
if max_cons <= np.maximum(self.eta, self.eta_opt):
# Update convergence check
if max_cons <= self.eta_opt and Pmax <= self.omega_opt:
exitOptimal = True
break
self.update_multipliers(convals, bc_solver.status)
# Update reference constraint norm on successful reduction
cons_norm_ref = max_cons
infeas_iter = 0
# If optimality of the inner loop is not achieved within 10
# major iterations, exit immediately
if self.inner_fail_count == 10:
if max_cons <= self.eta_opt:
self.status = "feas"
self.log.debug("cannot improve current point, exiting")
else:
self.status = "fail"
self.log.debug("cannot improve current point, exiting")
break
self.log.debug("updating multipliers estimates")
else:
self.update_penalty_parameter()
self.log.debug("keeping current multipliers estimates")
if max_cons > 0.99 * cons_norm_ref and self.iter != 1:
infeas_iter += 1
else:
cons_norm_ref = max_cons
infeas_iter = 0
if infeas_iter == 10:
self.status = "stal"
self.log.debug("problem appears infeasible, exiting")
break
# Safeguard: tightest tolerance should be near optimality to
# prevent excessive inner loop iterations at the end of the
# algorithm
if self.omega < self.omega_opt:
self.omega = self.omega_opt
if self.eta < self.eta_opt:
self.eta = self.eta_opt
try:
self.post_iteration()
except UserExitRequest:
self.status = "usr"
exitIter = self.niter_total > self.maxiter or self.iter > self.maxupdate
exitTime = (cputime() - tick) > self.maxtime
self.tsolve = cputime() - tick # Solve time
# Solution output, etc.
if exitOptimal:
self.status = "opt"
self.log.debug("optimal solution found")
elif not exitOptimal and exitTime:
self.status = "time"
self.log.debug("maximum run time exceeded")
elif not exitOptimal and exitIter:
self.status = "iter"
self.log.debug("maximum number of iterations reached")
self.log.info("f = %12.8g" % self.f)
if slack_model.m != 0:
self.log.info("pi_max = %12.8g" % np.max(al_model.pi))
self.log.info("max infeas. = %12.8g" % max_cons)
def solve(self):
"""Solve."""
n = self.n
x_norm2 = 0.0 # Squared norm of current iterate x, not counting x_feas
# Obtain initial projected residual
self.t_solve = cputime()
if self.qp.A is not None:
if self.factorize and not self.factorized:
self.perform_factorization()
if self.b is not None:
self.rhs[:n] = self.qp.grad(self.x_feasible)
self.rhs[n:] = 0.0
else:
self.rhs[:n] = self.qp.c
self.proj.solve(self.rhs)
r = g = self.proj.x[:n]
self.v = self.proj.x[n:]
# self.CheckAccurate()
else:
g = self.qp.c
r = g.copy()
# Initialize search direction
p = -g
pHp = None
self.resid_norm0 = np.dot(r, g)
rg = self.resid_norm0
threshold = max(self.abstol, self.reltol * sqrt(self.resid_norm0))
iter = 0
on_boundary = False
self.log.info(self.header)
self.log.info('-' * len(self.header))
self.log.info(self.fmt1, iter, rg)
while sqrt(rg) > threshold and iter < self.maxiter and not on_boundary:
Hp = self.qp.H * p
pHp = np.dot(p, Hp)
# Display current iteration info
self.log.info(self.fmt, iter, rg, pHp)
if self.radius is not None:
# Compute steplength to the boundary
sigma = to_boundary(self.x, p, self.radius, xx=x_norm2)
if (self.btol is not None) and (self.cur_iter is not None):
sigma = min(sigma, self.ftb(self.x, p))
elif pHp <= 0.0:
self.log.error('Problem is not second-order sufficient')
status = 'problem not SOS'
self.inf_descent = True
self.inf_descent_dir = p
continue
alpha = rg / pHp if pHp != 0.0 else np.inf
if self.radius is not None and (pHp <= 0.0 or alpha > sigma):
# p is a direction of singularity or negative curvature or
# next iterate will lie past the boundary of the trust region
# Move to boundary of trust-region
self.x += sigma * p
x_norm2 = self.radius * self.radius
status = u'on boundary (σ = %g)' % sigma
self.inf_descent = (pHp <= 0.0)
on_boundary = True
continue
# Make sure nonnegativity bounds remain enforced, if requested
if (self.btol is not None) and (self.cur_iter is not None):
step_bnd = self.ftb(self.x, p)
if step_bnd < alpha:
self.x += step_bnd * p
status = 'on boundary'
on_boundary = True
continue
# Move on
self.x += alpha * p
r += alpha * Hp
if self.qp.A is not None:
# Project current residual
self.rhs[:n] = r
self.proj.solve(self.rhs)
# Perform actual iterative refinement, if necessary
# self.proj.refine(self.rhs, nitref=self.max_itref,
# tol=self.itref_tol)
# Obtain new projected gradient
g = self.proj.x[:n]
if self.precon is not None:
# Prepare for iterative semi-refinement
self.qp.jtprod(self.proj.x[n:], self.v)
else:
g = r
rg_next = np.dot(r, g)
beta = rg_next / rg
p = -g + beta * p
if self.precon is not None:
# Perform iterative semi-refinement
r = r - self.v
else:
r = g
rg = rg_next
if self.radius is not None:
x_norm2 = np.dot(self.x, self.x)
iter += 1
# Output info about the last iteration
if iter > 0:
self.log.info(self.fmt, iter, rg, pHp)
# Obtain final solution x
self.x_norm2 = x_norm2
self.step_norm = sqrt(x_norm2)
if self.x_feasible is not None:
self.x += self.x_feasible
if self.qp.A is not None:
# Find (weighted) least-squares Lagrange multipliers
self.rhs[:n] = -self.qp.grad(self.x)
self.rhs[n:] = 0.0
self.proj.solve(self.rhs)
self.v = self.proj.x[n:].copy()
self.t_solve = cputime() - self.t_solve
self.step = self.x # Alias for consistency with TruncatedCG.
self.on_boundary = on_boundary
self.converged = (iter < self.maxiter)
if iter < self.maxiter and not on_boundary:
status = 'residual small'
elif iter >= self.maxiter:
status = 'max iter'
self.iter = iter
self.n_matvec = iter
self.resid_norm = sqrt(rg)
self.status = status
return
def solve(self):
"""Solve method.
All keyword arguments are passed directly to the constructor of the
trust-region solver.
"""
self.log.debug("entering solve")
model = self.model
ls_fmt = "%7.1e %8.1e"
# Project the initial point into [l,u].
self.x = project(self.x, model.Lvar, model.Uvar)
# Gather initial information.
self.f = model.obj(self.x)
self.f0 = self.f
self.g = model.grad(self.x) # Current gradient
self.g_old = self.g.copy()
self.x_old = self.x.copy()
pgnorm = projected_gradient_norm2(self.x, self.g, model.Lvar,
model.Uvar)
self.pg0 = pgnorm
cgtol = self.cgtol
cg_iter = 0
cgitermax = model.n
# Initialize the trust region radius
self.tr.radius = min(max(0.1 * self.pg0, 1.0), 100)
# Test for convergence or termination
stoptol = max(self.gabstol, self.greltol * self.pg0)
# stoptol = self.greltol * pgnorm
exitUser = False
exitOptimal = pgnorm <= stoptol
exitIter = self.iter >= self.maxiter
exitFunCall = model.obj.ncalls >= self.maxfuncall
status = ""
tick = cputime()
# Print out header and initial log.
if self.iter % 20 == 0:
self.log.info(self.header)
self.log.info(self.format0, self.iter, self.f, pgnorm, "", "", "",
self.tr.radius, "")
while not (exitUser or exitOptimal or exitIter or exitFunCall):
self.iter += 1
self.step_accepted = False
if self.save_g:
self.g_old = self.g.copy()
self.x_old = self.x.copy()
# Wrap Hessian into an operator.
H = model.hop(self.x.copy())
# Compute the Cauchy step and store in s.
(s, self.alphac) = self.cauchy(self.x, self.g, H, model.Lvar,
model.Uvar, self.tr.radius,
self.alphac)
# Compute the projected Newton step.
(x, s, cg_iter,
_) = self.projected_newton_step(self.x, self.g, H, self.tr.radius,
model.Lvar, model.Uvar, s, cgtol,
cgitermax)
snorm = norms.norm2(s)
self.total_cgiter += cg_iter
# Compute the predicted reduction.
m = np.dot(s, self.g) + .5 * np.dot(s, H * s)
# Evaluate actual objective.
x_trial = project(self.x + s, model.Lvar, model.Uvar)
f_trial = model.obj(x_trial)
# Incorporate a magical step to further improve the trial
# (if possible) and modify the predicted reduction to
# take the extra improvement into account
if "magical_step" in dir(model):
(x_trial, s_magic) = model.magical_step(x_trial)
if s_magic is not None:
s += s_magic
m -= f_trial
f_trial = model.obj(x_trial)
m += f_trial
# Evaluate the step and determine if the step is successful.
# Compute the actual reduction.
rho = self.tr.ratio(self.f, f_trial, m)
ared = self.f - f_trial
# On the first iteration, adjust the initial step bound.
snorm = norms.norm2(s)
if self.iter == 1:
self.tr.radius = min(self.tr.radius, snorm)
# Update the trust region bound
slope = np.dot(self.g, s)
if f_trial - self.f - slope <= 0:
alpha = self.tr.gamma3
else:
alpha = max(self.tr.gamma1,
-0.5 * (slope / (f_trial - self.f - slope)))
# Update the trust region bound according to the ratio
# of actual to predicted reduction
self.tr.update_radius(rho, snorm, alpha)
# Update the iterate.
if rho > self.tr.eta0:
# Successful iterate
# Trust-region step is accepted.
self.x = x_trial
self.f = f_trial
self.g = model.grad(self.x)
step_status = "Acc"
self.step_accepted = True
self.dvars = s
if self.save_g:
self.dgrad = self.g - self.g_old
elif self.ny:
try:
# Trust-region step is rejected; backtrack.
line_model = C1LineModel(model, self.x, s)
ls = ArmijoLineSearch(line_model, bkmax=5, decr=1.75)
for step in ls:
self.log.debug(ls_fmt, step, ls.trial_value)
ared = self.f - ls.trial_value
self.x = ls.iterate
self.f = ls.trial_value
self.g = model.grad(self.x)
snorm *= ls.step
self.tr.radius = snorm
step_status = "N-Y"
self.dvars = ls.step * s
self.step_accepted = True
if self.save_g:
self.dgrad = self.g - self.g_old
except (LineSearchFailure, ValueError):
step_status = "Rej"
else:
# Fall back on trust-region rule.
step_status = "Rej"
self.step_status = step_status
status = ""
try:
self.post_iteration()
except UserExitRequest:
status = "usr"
# Print out header, say, every 20 iterations
if self.iter % 20 == 0:
self.log.info(self.header)
pstatus = step_status if step_status != "Acc" else ""
# Test for convergence.
pgnorm = projected_gradient_norm2(self.x, self.g, model.Lvar,
model.Uvar)
if pstatus == "" or pstatus == "N-Y":
if pgnorm <= stoptol:
exitOptimal = True
status = "gtol"
elif abs(ared) <= self.abstol and -m <= self.abstol:
exitOptimal = True
status = "fatol"
elif abs(ared) <= self.reltol * abs(self.f) and \
(-m <= self.reltol * abs(self.f)):
exitOptimal = True
status = "frtol"
else:
self.iter -= 1 # to match TRON iteration number
exitIter = self.iter > self.maxiter
exitFunCall = model.obj.ncalls >= self.maxfuncall
exitUser = status == "usr"
self.log.info(self.format, self.iter, self.f, pgnorm, cg_iter, rho,
snorm, self.tr.radius, pstatus)
self.tsolve = cputime() - tick # Solve time
self.pgnorm = pgnorm
# Set final solver status.
if status == "usr":
pass
elif self.iter > self.maxiter:
status = "itr"
elif status == "": # corner case; initial guess was optimal
status = "gtol"
self.status = status
self.log.info("final status: %s", self.status)
def solve(self):
"""Solve model with the L-BFGS method."""
model = self.model
x = self.x
self.logger.info(self.hdr)
tstart = cputime()
self.f0 = self.f = f = model.obj(x)
self.g = g = model.grad(x)
self.g_norm0 = g_norm = norms.norm2(g)
stoptol = max(self.abstol, self.reltol * self.g_norm0)
exitUser = False
exitLS = False
exitOptimal = g_norm <= stoptol
exitIter = self.iter >= self.maxiter
status = ""
while not (exitUser or exitOptimal or exitIter or exitLS):
# Obtain search direction
H = model.hop(x)
d = -(H * g)
# Prepare for modified linesearch
step0 = max(1.0e-3, 1.0 / g_norm) if self.iter == 0 else 1.0
line_model = C1LineModel(self.model, x, d)
ls = self.setup_linesearch(line_model, step0)
try:
for step in ls:
self.logger.debug(self.ls_fmt, step, ls.trial_value)
except LineSearchFailure:
exitLS = True
continue
self.logger.info(self.fmt, self.iter, f, g_norm, ls.slope, ls.step)
# Prepare new pair {s,y} to be inserted into L-BFGS operator.
self.s = ls.step * d
x = ls.iterate
g_next = line_model.gradval
self.y = g_next - g
status = ""
try:
self.post_iteration()
except UserExitRequest:
status = "usr"
# Prepare for next round.
g = g_next
g_norm = norms.norm2(g)
f = ls.trial_value
self.iter += 1
exitOptimal = g_norm <= stoptol
exitIter = self.iter >= self.maxiter
exitUser = status == "usr"
self.tsolve = cputime() - tstart
self.logger.info(self.fmt_short, self.iter, f, g_norm)
self.x = x
self.f = f
self.g = g
self.g_norm = g_norm
# Set final solver status.
if status == "usr":
pass
elif self.g_norm <= stoptol:
status = "opt"
elif exitLS:
status = "lsf"
else: # self.iter > self.maxiter:
status = "itr"
self.status = status
def solve(self, **kwargs):
"""Solve current problem with trust-funnel framework.
:keywords:
:ny: Enable Nocedal-Yuan backtracking linesearch.
:returns:
This method sets the following members of the instance:
:f: Final objective value
:optimal: Flag indicating whether normal stopping conditions were
attained
:p_resid: Final primal residual
:d_resid: Final dual residual
:niter: Total number of iterations
:tsolve: Solve time.
Warning: backtracking after rejected trust-region steps fails with a
"Not a descent direction" flag.
"""
ny = kwargs.get('ny', False)
reg = kwargs.get('reg', 0.0)
tsolve = cputime()
# Set some shortcuts.
model = self.model
n = model.n
m = model.m
x = self.x
f = self.f
c = model.cons_pos(x)
y = model.pi0.copy()
# Initialize some constants.
kappa_n = 1.0e+2 # Factor of p_norm in normal step TR radius.
kappa_b = 0.99 # Fraction of TR to compute tangential step.
kappa_delta = 0.1 # Progress factor to compute tangential step.
# Trust-region parameters.
eta_1 = 1.0e-5
eta_2 = 0.95
eta_3 = 0.5
gamma_1 = 0.25
gamma_3 = 2.5
kappa_tx1 = 0.9 # Factor of theta_max in max acceptable infeasibility.
kappa_tx2 = 0.5 # Convex combination factor of theta and theta_trial.
# Compute constraint violation.
theta = 0.5 * np.dot(c, c)
# Set initial funnel radius.
kappa_ca = 1.0e+3 # Max initial funnel radius.
kappa_cr = 2.0 # Infeasibility tolerance factor.
theta_max = max(kappa_ca, kappa_cr * theta)
# Evaluate first-order derivatives.
g = model.grad(x)
J = model.jac(x)
Jop = model.jop(x)
# Initial radius for f- and c-iterations.
radius_f = max(self.radius_f, .1 * np.linalg.norm(g))
radius_c = max(self.radius_c, .1 * sqrt(2 * theta))
# Reset initial multipliers to least-squares estimates by
# approximately solving:
# [ I Jᵀ ] [ w ] [ -g ]
# [ J 0 ] [ y ] = [ 0 ].
# This is equivalent to solving
# minimize |g + J'y|.
if m > 0:
y, _, _ = self.lsq(Jop.T, -g, reg=reg)
p_norm = c_norm = 0
if m > 0:
p_norm = np.linalg.norm(c)
c_norm = np.linalg.norm(c, np.inf)
grad_lag = g + Jop.T * y
else:
grad_lag = g.copy()
d_norm = np.linalg.norm(grad_lag) / (1 + np.linalg.norm(y))
# Display current info if requested.
self.log.info(self.hdr)
self.log.info(self.linefmt1, 0, ' ', ' ', ' ', ' ', f, p_norm, d_norm,
radius_f, radius_c, theta_max, 0)
# Compute primal stopping tolerance.
stop_p = max(self.atol, self.stop_p * p_norm)
self.log.debug('p_norm=%7.1e, c_norm=%7.1e, d_norm=%7.1e', p_norm,
c_norm, d_norm)
optimal = (p_norm <= stop_p) and (d_norm <= self.stop_d)
self.log.debug('optimal: %s', repr(optimal))
# Start of main iteration.
while not optimal and (self.iter < self.maxiter):
self.iter += 1
radius = min(radius_f, radius_c)
cgiter = 0
# 1. Compute normal step as an (approximate) solution to
# minimize |c + J n| subject to |n| <= min(radius_c, kN |c|).
if self.iter > 1 and \
p_norm <= stop_p and \
d_norm >= 1.0e+4 * self.stop_d:
self.log.debug('Setting step_n=0 b/c must work on optimality')
step_n = np.zeros(n)
step_n_norm = 0.0
status_n = '0'
m_xpn = 0 # Model value at x+n.
else:
step_n_max = min(radius_c, kappa_n * p_norm)
step_n, step_n_norm, lsq_status = self.lsq(Jop,
-c,
radius=step_n_max,
reg=reg)
if lsq_status == 'residual small':
status_n = 'r'
elif lsq_status == 'trust-region boundary active':
status_n = 'b'
else:
status_n = '?'
# Evaluate the model of the obective after the normal step.
_Hv = model.hprod(x, -y, step_n) # H*step_n
m_xpn = np.dot(g, step_n) + 0.5 * np.dot(step_n, _Hv)
self.log.debug('Normal step norm = %8.2e', step_n_norm)
self.log.debug('Model value: %9.2e', m_xpn)
# 2. Compute tangential step if normal step is not too long.
if step_n_norm <= kappa_b * radius:
# 2.1. Compute Lagrange multiplier estimates and dual residuals
# by minimizing |(g + H n) + J'y|
if step_n_norm == 0.0:
g_n = g # Just a pointer ; g will not be modified below.
else:
g_n = g + _Hv
y_new, _, _ = self.lsq(Jop.T, -g_n, reg=reg)
r = g_n + Jop.T * y_new
# Here Nick does iterative refinement to improve r and y_new...
# Compute dual optimality measure.
resid_norm = np.linalg.norm(r)
pi = 0.0
if resid_norm > 0:
pi = abs(np.dot(g_n, r)) / resid_norm
# 2.2. If the dual residuals are large, compute a suitable
# tangential step as a solution to:
# minimize g't + 1/2 t' H t
# subject to Jt = 0, |n+t| <= radius.
if pi > self.forcing(3, theta):
self.log.debug('Computing step_t...')
radius_within = radius - step_n_norm
Hop = model.hop(x, -y_new)
qp = QPModel(g_n, Hop, A=J.matrix)
# PPCG = ProjectedCG(g_n, Hop,
# A=J.matrix if m > 0 else None,
# radius=radius_within, dreg=reg)
PPCG = ProjectedCG(qp, radius=radius_within, dreg=reg)
PPCG.solve()
step_t = PPCG.step
step_t_norm = PPCG.step_norm
cgiter = PPCG.iter
self.log.debug(u'‖t‖ = %8.2e', step_t_norm)
if PPCG.status == 'residual small':
status_t = 'r'
elif PPCG.on_boundary and not PPCG.inf_descent:
status_t = 'b'
elif PPCG.inf_descent:
status_t = '-'
elif PPCG.status == 'max iter':
status_t = '>'
else:
status_t = '?'
# Compute total step and model decrease.
step = step_n + step_t
step_norm = np.linalg.norm(step)
# _Hv = model.hprod(x, -y, step) # y or y_new?
# m_xps = np.dot(g, step) + 0.5 * np.dot(step, _Hv)
m_xps = qp.obj(step)
else:
self.log.debug('Setting step_t=0 b/c pi sufficiently' +
'small')
step_t_norm = 0
status_t = '0'
step = step_n
step_norm = step_n_norm
m_xps = m_xpn
y = y_new
else:
# No need to compute a tangential step.
self.log.debug('Setting step_t=0 b/c normal step too large')
status_t = '0'
y = np.zeros(m)
step_t_norm = 0.0
step = step_n
step_norm = step_n_norm
m_xps = m_xpn
self.log.debug('Model decrease = %9.2e', m_xps)
# Compute trial point and evaluate local data.
x_trial = x + step
f_trial = model.obj(x_trial)
c_trial = model.cons_pos(x_trial)
theta_trial = 0.5 * np.dot(c_trial, c_trial)
delta_f = -m_xps # Overall improvement in the model.
delta_ft = m_xpn - m_xps # Improvement due to tangential step.
Jspc = c + Jop * step
# Compute improvement in linearized feasibility.
delta_feas = theta - 0.5 * np.dot(Jspc, Jspc)
# Decide whether to consider the current iteration
# an f- or a c-iteration.
if step_t_norm > 0 and (delta_f >= self.forcing(2, theta)) and \
delta_f >= kappa_delta * delta_ft and \
theta_trial <= theta_max:
# Step 3. Consider that this is an f-iteration.
it_type = 'f'
# Decide whether trial point is accepted.
ratio = (f - f_trial) / delta_f
self.log.debug('f-iter ratio = %9.2e', ratio)
if ratio >= eta_1: # Successful step.
suc = 's'
x = x_trial
f = f_trial
c = c_trial
self.step = step.copy()
theta = theta_trial
self.log.debug('stepnorm = %8.2e %8.2e', step_norm,
radius_f)
# Decide whether to update f-trust-region radius.
if ratio >= eta_2:
suc = 'v'
radius_f = min(max(radius_f, gamma_3 * step_norm),
1.0e+10)
# Decide whether to update c-trust-region radius.
if theta_trial < eta_3 * theta_max:
ns = step_n_norm if step_n_norm > 0 else radius_c
radius_c = min(max(radius_c, gamma_3 * ns), 1.0e+10)
self.log.debug('New radius_f = %8.2e', radius_f)
self.log.debug('New radius_c = %8.2e', radius_c)
else: # Unsuccessful step (ratio < eta_1).
attempt_soc = True
suc = 'u'
if attempt_soc:
self.log.debug(' Attempting second-order correction')
# Attempt a second-order correction by solving
# minimize |c_trial + J n| subject to |n| <= radius_c.
step_soc, _, status_soc = \
self.lsq(Jop, -c_trial, radius=radius_c, reg=reg)
self.log.debug("lsq status: %20s", status_soc)
if status_soc != 'trust-region boundary active':
# Consider SOC step as candidate step.
x_soc = x_trial + step_soc
f_soc = model.obj(x_soc)
c_soc = model.cons_pos(x_soc)
theta_soc = 0.5 * np.dot(c_soc, c_soc)
ratio = (f - f_soc) / delta_f
# Decide whether to accept SOC step.
if ratio >= eta_1 and theta_soc <= theta_max:
suc = '2'
x = x_soc
f = f_soc
c = c_soc
theta = theta_soc
self.step = step + step_soc
else:
# Backtracking linesearch a la Nocedal & Yuan.
# Abandon SOC step. Backtrack from x+step.
if ny:
(x, f,
alpha) = self.nyf(x, f, f_trial, g, step)
# g = model.grad(x)
c = model.cons_pos(x)
theta = 0.5 * np.dot(c, c)
self.step = step + alpha * step_soc
radius_f = min(alpha, .8) * step_norm
suc = 'y'
else:
radius_f = gamma_1 * radius_f
else: # SOC step lies on boundary of trust region.
radius_f = gamma_1 * radius_f
else: # SOC step not attempted.
# Backtracking linesearch a la Nocedal & Yuan.
if ny:
(x, f, alpha) = self.nyf(x, f, f_trial, g, step)
c = model.cons_pos(x)
theta = 0.5 * np.dot(c, c)
self.step = alpha * step
radius_f = min(alpha, .8) * step_norm
suc = 'y'
else:
radius_f = gamma_1 * radius_f
else:
# Step 4. Consider that this is a c-iteration.
it_type = 'c'
# Display information.
self.log.debug('c-iteration because ')
if step_t_norm == 0.0:
self.log.debug('|t|=0')
if delta_f < self.forcing(2, theta):
self.log.debug('delta_f=%8.2e < forcing=%8.2e', delta_f,
self.forcing(2, theta))
if delta_f < kappa_delta * delta_ft:
self.log.debug('delta_f=%8.2e < frac * delta_ft=%8.2e',
delta_f, delta_ft)
if theta_trial > theta_max:
self.log.debug('theta_trial=%8.2e > theta_max=%8.2e',
theta_trial, theta_max)
# Step 4.1. Check trial point for acceptability.
if delta_feas < 0:
self.log.debug(' !!! Warning: delta_feas is negative !!!')
ratio = (theta - theta_trial + 1.e-16) / (delta_feas + 1.e-16)
self.log.debug('c-iter ratio = %9.2e', ratio)
if ratio >= eta_1: # Successful step.
x = x_trial
f = f_trial
c = c_trial
self.step = step.copy()
suc = 's'
# Step 4.2. Update radius_c.
if ratio >= eta_2: # Very successful step.
ns = step_n_norm if step_n_norm > 0 else radius_c
radius_c = min(max(radius_c, gamma_3 * ns), 1.0e+10)
suc = 'v'
# Step 4.3. Update maximum acceptable infeasibility.
theta_max = max(
kappa_tx1 * theta_max,
kappa_tx2 * theta + (1 - kappa_tx2) * theta_trial)
theta = theta_trial
else: # Unsuccessful step.
# Backtracking linesearch a la Nocedal & Yuan.
ns = step_n_norm if step_n_norm > 0 else radius_c
if ny:
(x, c, theta,
alpha) = self.nyc(x, theta, theta_trial, c, Jop.T * c,
step)
f = model.obj(x)
self.step = alpha * step
radius_c = min(alpha, .8) * ns
suc = 'y'
else:
radius_c = gamma_1 * ns
suc = 'u'
self.log.debug('New radius_c = %8.2e', radius_c)
self.log.debug('New theta_max = %8.2e', theta_max)
# Step 5. Book keeping.
if ratio >= eta_1 or ny:
if self.save_g:
self.g_old = g.copy()
g = model.grad(x)
self.dgrad = g - self.g_old
else:
g = model.grad(x)
J = model.jac(x)
Jop = model.jop(x)
try:
self.post_iteration()
except UserExitRequest:
self.status = "usr"
p_norm = c_norm = 0
if m > 0:
p_norm = np.linalg.norm(c)
c_norm = np.linalg.norm(c, np.inf)
grad_lag = g + Jop.T * y
else:
grad_lag = g.copy()
d_norm = np.linalg.norm(grad_lag) / (1 + np.linalg.norm(y))
if self.iter % 20 == 0:
self.log.info(self.hdr)
self.log.info(self.linefmt, self.iter, it_type, suc, status_n,
status_t, f, p_norm, d_norm, radius_f, radius_c,
theta_max, cgiter)
optimal = (p_norm <= stop_p) and (d_norm <= self.stop_d)
# End while.
self.tsolve = cputime() - tsolve
# Set final solver status.
if self.status == "usr":
pass
elif optimal:
self.status = "opt" # Successful solve.
self.log.info('Found an optimal solution! Yeah!')
else: # self.iter > self.maxiter:
self.status = "itr"
self.x = x
self.f = f
self.optimal = optimal
self.p_resid = p_norm
self.d_resid = d_norm
return