# coding: utf-8
# ------------------------------------------------------------------------------
# Copyright (C) 2020 Maximilian Stahlberg
#
# This file is part of PICOS.
#
# PICOS is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# PICOS is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
# ------------------------------------------------------------------------------
"""Implementation of :class:`Dualization`."""
from itertools import chain
import cvxopt
from ..apidoc import api_end, api_start
from ..constraints import (AffineConstraint, LMIConstraint, RSOCConstraint,
SOCConstraint)
from ..expressions import AffineExpression, Constant
from ..expressions.algebra import rsoc, soc
from ..expressions.variables import (CONTINUOUS_VARTYPES, RealVariable,
SymmetricVariable)
from ..modeling import Problem
from ..modeling.footprint import Footprint, Specification
from ..modeling.solution import PS_INFEASIBLE, PS_UNBOUNDED, Solution
from .reformulation import Reformulation
_API_START = api_start(globals())
# -------------------------------
[docs]class Dualization(Reformulation):
"""Lagrange dual problem reformulation."""
SUPPORTED = Specification.compile(
directions=Specification.ALL,
objectives=[
AffineExpression],
variables=CONTINUOUS_VARTYPES,
constraints=[
AffineConstraint,
SOCConstraint,
RSOCConstraint,
LMIConstraint])
[docs] @classmethod
def supports(cls, footprint):
"""Implement :meth:`~.reformulation.Reformulation.supports`."""
return footprint.options.dualize and footprint << cls.SUPPORTED
[docs] @classmethod
def predict(cls, footprint):
"""Implement :meth:`~.reformulation.Reformulation.predict`."""
vars = []
cons = []
# Objective direction.
dir = "min" if footprint.direction == "max" else "max"
# Objective function.
# HACK: Conversion adds a zero-fixed variable to the objective so that
# it is never constant. This makes prediction succeed in any case.
obj = AffineExpression.make_type(
shape=(1, 1), constant=False, nonneg=False)
# Variables and their bounds.
bound_cons = []
for vartype, k in footprint.variables.items():
cons.append((AffineConstraint.make_type(
dim=vartype.subtype.dim, eq=True), k))
if vartype.subtype.bnd:
bound_cons.append((AffineConstraint.make_type(
dim=vartype.subtype.bnd, eq=False), k))
# Constraints by type.
for contype, k in chain(footprint.constraints.items(), bound_cons):
if contype.clstype is AffineConstraint:
d = contype.subtype.dim
# Dual variable.
vars.append((RealVariable.make_var_type(dim=d, bnd=0), k))
# Conic membership constraint.
if not contype.subtype.eq:
cons.append((
AffineConstraint.make_type(dim=d, eq=False), k))
elif contype.clstype is SOCConstraint:
n = contype.subtype.argdim
d = n + 1
# Dual variable.
vars.append((RealVariable.make_var_type(dim=d, bnd=0), k))
# Conic membership constraint.
cons.append((SOCConstraint.make_type(argdim=n), k))
elif contype.clstype is RSOCConstraint:
n = contype.subtype.argdim
d = n + 2
# Dual variable.
vars.append((RealVariable.make_var_type(dim=d, bnd=0), k))
# Conic membership constraint.
cons.append((RSOCConstraint.make_type(argdim=n), k))
elif contype.clstype is LMIConstraint:
n = contype.subtype.diag
d = n*(n + 1)//2
# Dual variable.
vars.append((SymmetricVariable.make_var_type(dim=d, bnd=0), k))
# Conic membership constraint.
cons.append((LMIConstraint.make_type(diag=n), k))
else:
assert False, "Unexpected constraint type."
# HACK: See above.
vars.append((RealVariable.make_var_type(dim=1, bnd=2), 1))
# Option changes.
nd_opts = footprint.options.updated(
dualize=False,
primals=footprint.options.duals,
duals=footprint.options.primals
).nondefaults
return Footprint.from_types(dir, obj, vars, cons, nd_opts)
[docs] def forward(self):
"""Implement :meth:`~.reformulation.Reformulation.forward`."""
self._pc2dv = {} # Maps primal constraints to dual variables.
self._pv2dc = {} # Maps primal variables to dual constraints.
P = self.input
# Create the dual problem from scratch.
D = self.output = Problem(copyOptions=P.options)
# Reset the 'dualize' option so that solvers can load the dual.
D.options.dualize = False
# Swap 'primals' and 'duals' option.
D.options.primals = P.options.duals
D.options.duals = P.options.primals
# Retrieve the primal objective in standard form.
original_primal_direction, primal_objective = P.objective.normalized
if original_primal_direction == "max":
primal_objective = -primal_objective
# Start building the dual objective function.
dual_objective = primal_objective.cst
# Start building the dual linear equalities for each primal variable.
obj_coefs = primal_objective.lin._coefs
dual_equality_terms = {var: -Constant(obj_coefs[var].T)
if var in obj_coefs else AffineExpression.zero(var.dim)
for var in P.variables.values()}
# Turn variable bounds into affine inequality constraints.
# NOTE: If bound-free variables are required by another reformulation or
# solver in the future, there should be a reformulation for this.
bound_cons = []
for primal_var in P.variables.values():
bound_constraint = primal_var.bound_constraint
if bound_constraint:
bound_cons.append(bound_constraint)
# Handle the primal constraints.
# NOTE: Constraint conversion works as follows:
# 1. Transform constraint into conic standard form:
# fx = Ax-b >_K 0.
# 2. Create a dual variable for the constraint.
# 3. Constrain the dual variable to be a member of the dual
# cone K*.
# 4. Extend the dual's linear equalities for each primal variable.
# 5. Extend the dual's objective function.
# TODO: Consider adding a ConicConstraint abstract subclass of
# Constraint with a method conic_form that returns a pair of cone
# member (affine expression) and cone. The conic standard form
# function f(x) and the dual cone can then be obtained by negation
# and through a new Cone.dual method, respectively.
for primal_con in chain(P.constraints.values(), bound_cons):
if isinstance(primal_con, AffineConstraint):
# Transform constraint into conic standard form.
fx = primal_con.ge0.vec
# Create a dual variable for the constraint.
self._pc2dv[primal_con] = dual_var = RealVariable(
"aff_{}".format(primal_con.id), len(fx))
# Constrain the dual variable to be a member of the dual cone.
# NOTE: In the equality case, the dual cone is the real field.
if primal_con.is_inequality():
D.add_constraint(dual_var >= 0)
# Extend the dual's linear equalities for each primal variable.
for var, coef in fx._coefs.items():
dual_equality_terms[var] += coef.T*dual_var
# Extend the dual's objective function.
dual_objective -= fx._const.T*dual_var
elif isinstance(primal_con, SOCConstraint):
# Transform constraint into conic standard form.
fx = (primal_con.ub // primal_con.ne.vec)
# Create a dual variable for the constraint.
self._pc2dv[primal_con] = dual_var = RealVariable(
"soc_{}".format(primal_con.id), len(fx))
# Constrain the dual variable to be a member of the dual cone.
D.add_constraint(dual_var << soc())
# Extend the dual's linear equalities for each primal variable.
for var, coef in fx._coefs.items():
dual_equality_terms[var] += coef.T*dual_var
# Extend the dual's objective function.
dual_objective -= fx._const.T*dual_var
elif isinstance(primal_con, RSOCConstraint):
# Transform constraint into conic standard form.
fx = (primal_con.ub1 // primal_con.ub2 // primal_con.ne.vec)
# Create a dual variable for the constraint.
self._pc2dv[primal_con] = dual_var = RealVariable(
"rso_{}".format(primal_con.id), len(fx))
# Constrain the dual variable to be a member of the dual cone.
D.add_constraint(dual_var << rsoc(4))
# Extend the dual's linear equalities for each primal variable.
for var, coef in fx._coefs.items():
dual_equality_terms[var] += coef.T*dual_var
# Extend the dual's objective function.
dual_objective -= fx._const.T*dual_var
elif isinstance(primal_con, LMIConstraint):
# Transform constraint into conic standard form.
fx = primal_con.psd
# Create a dual variable for the constraint.
self._pc2dv[primal_con] = dual_var = SymmetricVariable(
"lmi_{}".format(primal_con.id), fx.shape)
# Constrain the dual variable to be a member of the dual cone.
D.add_constraint(dual_var >> 0)
# Extend the dual's linear equalities for each primal variable.
for var, coef in fx._coefs.items():
dual_equality_terms[var] += coef.T*dual_var.vec
# Extend the dual's objective function.
dual_objective -= fx._const.T*dual_var.vec
else:
assert False, "Unexpected constraint type."
# Add the finished linear equalities for each variable.
for var, term in dual_equality_terms.items():
self._pv2dc[var] = D.add_constraint(term == 0)
# Adjust dual optimization direction to obtain the duality property.
if original_primal_direction == "max":
dual_direction = "min"
dual_objective = -dual_objective
else:
dual_direction = "max"
# HACK: Add a zero-fixed variable to the objective so that it is never
# constant. This makes prediction succeed in any case.
dual_objective += RealVariable("zero", 1, lower=0, upper=0)
# Set the finished dual objective.
D.objective = dual_direction, dual_objective
[docs] def update(self):
"""Implement :meth:`~.reformulation.Reformulation.update`."""
# TODO: Implement updates to an existing dualization. This is optional.
raise NotImplementedError
[docs] def backward(self, solution):
"""Implement :meth:`~.reformulation.Reformulation.backward`."""
P = self.input
# Require primal values to be properly vectorized.
if not solution.vectorizedPrimals:
raise NotImplementedError(
"Solving with dualize=True is not supported with solvers that "
"produce unvectorized primal solutions.")
# Retrieve primals for the primal problem.
primals = {}
for var in P.variables.values():
con = self._pv2dc[var]
primal = solution.duals[con] if con in solution.duals else None
if primal is not None:
primals[var] = cvxopt.matrix(primal)
else:
primals[var] = None
# Retrieve duals for the primal problem.
duals = {}
for con in P.constraints.values():
var = self._pc2dv[con]
dual = solution.primals[var] if var in solution.primals else None
if dual is not None:
duals[con] = var._vec.devectorize(cvxopt.matrix(dual))
else:
duals[con] = None
# Swap infeasible/unbounded for a problem status.
if solution.problemStatus == PS_UNBOUNDED:
problemStatus = PS_INFEASIBLE
elif solution.problemStatus == PS_INFEASIBLE:
problemStatus = PS_UNBOUNDED
else:
problemStatus = solution.problemStatus
return Solution(
primals=primals,
duals=duals,
problem=solution.problem,
solver=solution.solver,
primalStatus=solution.dualStatus, # Intentional swap.
dualStatus=solution.primalStatus, # Intentional swap.
problemStatus=problemStatus,
searchTime=solution.searchTime,
info=solution.info,
vectorizedPrimals=True)
# --------------------------------------
__all__ = api_end(_API_START, globals())