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/ optimize / _linprog_ip.py
"""Interior-point method for linear programming
The *interior-point* method uses the primal-dual path following algorithm
outlined in [1]_. This algorithm supports sparse constraint matrices and
is typically faster than the simplex methods, especially for large, sparse
problems. Note, however, that the solution returned may be slightly less
accurate than those of the simplex methods and will not, in general,
correspond with a vertex of the polytope defined by the constraints.
.. versionadded:: 1.0.0
References
----------
.. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
"""
# Author: Matt Haberland
from __future__ import print_function, division, absolute_import
import numpy as np
import scipy as sp
import scipy.sparse as sps
import numbers
from warnings import warn
from scipy.linalg import LinAlgError
from .optimize import OptimizeWarning, OptimizeResult, _check_unknown_options
from ._linprog_util import _postsolve
has_umfpack = True
has_cholmod = True
try:
from sksparse.cholmod import cholesky as cholmod
except ImportError:
has_cholmod = False
try:
import scikits.umfpack # test whether to use factorized
except ImportError:
has_umfpack = False
def _get_solver(M, sparse=False, lstsq=False, sym_pos=True,
cholesky=True, permc_spec='MMD_AT_PLUS_A'):
"""
Given solver options, return a handle to the appropriate linear system
solver.
Parameters
----------
M : 2D array
As defined in [4] Equation 8.31
sparse : bool (default = False)
True if the system to be solved is sparse. This is typically set
True when the original ``A_ub`` and ``A_eq`` arrays are sparse.
lstsq : bool (default = False)
True if the system is ill-conditioned and/or (nearly) singular and
thus a more robust least-squares solver is desired. This is sometimes
needed as the solution is approached.
sym_pos : bool (default = True)
True if the system matrix is symmetric positive definite
Sometimes this needs to be set false as the solution is approached,
even when the system should be symmetric positive definite, due to
numerical difficulties.
cholesky : bool (default = True)
True if the system is to be solved by Cholesky, rather than LU,
decomposition. This is typically faster unless the problem is very
small or prone to numerical difficulties.
permc_spec : str (default = 'MMD_AT_PLUS_A')
Sparsity preservation strategy used by SuperLU. Acceptable values are:
- ``NATURAL``: natural ordering.
- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
- ``COLAMD``: approximate minimum degree column ordering.
See SuperLU documentation.
Returns
-------
solve : function
Handle to the appropriate solver function
"""
try:
if sparse:
if lstsq:
def solve(r, sym_pos=False):
return sps.linalg.lsqr(M, r)[0]
elif cholesky:
solve = cholmod(M)
else:
if has_umfpack and sym_pos:
solve = sps.linalg.factorized(M)
else: # factorized doesn't pass permc_spec
solve = sps.linalg.splu(M, permc_spec=permc_spec).solve
else:
if lstsq: # sometimes necessary as solution is approached
def solve(r):
return sp.linalg.lstsq(M, r)[0]
elif cholesky:
L = sp.linalg.cho_factor(M)
def solve(r):
return sp.linalg.cho_solve(L, r)
else:
# this seems to cache the matrix factorization, so solving
# with multiple right hand sides is much faster
def solve(r, sym_pos=sym_pos):
return sp.linalg.solve(M, r, sym_pos=sym_pos)
# There are many things that can go wrong here, and it's hard to say
# what all of them are. It doesn't really matter: if the matrix can't be
# factorized, return None. get_solver will be called again with different
# inputs, and a new routine will try to factorize the matrix.
except KeyboardInterrupt:
raise
except Exception:
return None
return solve
def _get_delta(
A,
b,
c,
x,
y,
z,
tau,
kappa,
gamma,
eta,
sparse=False,
lstsq=False,
sym_pos=True,
cholesky=True,
pc=True,
ip=False,
permc_spec='MMD_AT_PLUS_A'
):
"""
Given standard form problem defined by ``A``, ``b``, and ``c``;
current variable estimates ``x``, ``y``, ``z``, ``tau``, and ``kappa``;
algorithmic parameters ``gamma and ``eta;
and options ``sparse``, ``lstsq``, ``sym_pos``, ``cholesky``, ``pc``
(predictor-corrector), and ``ip`` (initial point improvement),
get the search direction for increments to the variable estimates.
Parameters
----------
As defined in [4], except:
sparse : bool
True if the system to be solved is sparse. This is typically set
True when the original ``A_ub`` and ``A_eq`` arrays are sparse.
lstsq : bool
True if the system is ill-conditioned and/or (nearly) singular and
thus a more robust least-squares solver is desired. This is sometimes
needed as the solution is approached.
sym_pos : bool
True if the system matrix is symmetric positive definite
Sometimes this needs to be set false as the solution is approached,
even when the system should be symmetric positive definite, due to
numerical difficulties.
cholesky : bool
True if the system is to be solved by Cholesky, rather than LU,
decomposition. This is typically faster unless the problem is very
small or prone to numerical difficulties.
pc : bool
True if the predictor-corrector method of Mehrota is to be used. This
is almost always (if not always) beneficial. Even though it requires
the solution of an additional linear system, the factorization
is typically (implicitly) reused so solution is efficient, and the
number of algorithm iterations is typically reduced.
ip : bool
True if the improved initial point suggestion due to [4] section 4.3
is desired. It's unclear whether this is beneficial.
permc_spec : str (default = 'MMD_AT_PLUS_A')
(Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos =
True``.) A matrix is factorized in each iteration of the algorithm.
This option specifies how to permute the columns of the matrix for
sparsity preservation. Acceptable values are:
- ``NATURAL``: natural ordering.
- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
- ``COLAMD``: approximate minimum degree column ordering.
This option can impact the convergence of the
interior point algorithm; test different values to determine which
performs best for your problem. For more information, refer to
``scipy.sparse.linalg.splu``.
Returns
-------
Search directions as defined in [4]
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
"""
if A.shape[0] == 0:
# If there are no constraints, some solvers fail (understandably)
# rather than returning empty solution. This gets the job done.
sparse, lstsq, sym_pos, cholesky = False, False, True, False
n_x = len(x)
# [4] Equation 8.8
r_P = b * tau - A.dot(x)
r_D = c * tau - A.T.dot(y) - z
r_G = c.dot(x) - b.transpose().dot(y) + kappa
mu = (x.dot(z) + tau * kappa) / (n_x + 1)
# Assemble M from [4] Equation 8.31
Dinv = x / z
if sparse:
M = A.dot(sps.diags(Dinv, 0, format="csc").dot(A.T))
else:
M = A.dot(Dinv.reshape(-1, 1) * A.T)
solve = _get_solver(M, sparse, lstsq, sym_pos, cholesky, permc_spec)
# pc: "predictor-corrector" [4] Section 4.1
# In development this option could be turned off
# but it always seems to improve performance substantially
n_corrections = 1 if pc else 0
i = 0
alpha, d_x, d_z, d_tau, d_kappa = 0, 0, 0, 0, 0
while i <= n_corrections:
# Reference [4] Eq. 8.6
rhatp = eta(gamma) * r_P
rhatd = eta(gamma) * r_D
rhatg = np.array(eta(gamma) * r_G).reshape((1,))
# Reference [4] Eq. 8.7
rhatxs = gamma * mu - x * z
rhattk = np.array(gamma * mu - tau * kappa).reshape((1,))
if i == 1:
if ip: # if the correction is to get "initial point"
# Reference [4] Eq. 8.23
rhatxs = ((1 - alpha) * gamma * mu -
x * z - alpha**2 * d_x * d_z)
rhattk = np.array(
(1 -
alpha) *
gamma *
mu -
tau *
kappa -
alpha**2 *
d_tau *
d_kappa).reshape(
(1,
))
else: # if the correction is for "predictor-corrector"
# Reference [4] Eq. 8.13
rhatxs -= d_x * d_z
rhattk -= d_tau * d_kappa
# sometimes numerical difficulties arise as the solution is approached
# this loop tries to solve the equations using a sequence of functions
# for solve. For dense systems, the order is:
# 1. scipy.linalg.cho_factor/scipy.linalg.cho_solve,
# 2. scipy.linalg.solve w/ sym_pos = True,
# 3. scipy.linalg.solve w/ sym_pos = False, and if all else fails
# 4. scipy.linalg.lstsq
# For sparse systems, the order is:
# 1. sksparse.cholmod.cholesky (if available)
# 2. scipy.sparse.linalg.factorized (if umfpack available)
# 3. scipy.sparse.linalg.splu
# 4. scipy.sparse.linalg.lsqr
solved = False
while(not solved):
try:
# [4] Equation 8.28
p, q = _sym_solve(Dinv, A, c, b, solve)
# [4] Equation 8.29
u, v = _sym_solve(Dinv, A, rhatd -
(1 / x) * rhatxs, rhatp, solve)
if np.any(np.isnan(p)) or np.any(np.isnan(q)):
raise LinAlgError
solved = True
except (LinAlgError, ValueError, TypeError) as e:
# Usually this doesn't happen. If it does, it happens when
# there are redundant constraints or when approaching the
# solution. If so, change solver.
if cholesky:
cholesky = False
warn(
"Solving system with option 'cholesky':True "
"failed. It is normal for this to happen "
"occasionally, especially as the solution is "
"approached. However, if you see this frequently, "
"consider setting option 'cholesky' to False.",
OptimizeWarning, stacklevel=5)
elif sym_pos:
sym_pos = False
warn(
"Solving system with option 'sym_pos':True "
"failed. It is normal for this to happen "
"occasionally, especially as the solution is "
"approached. However, if you see this frequently, "
"consider setting option 'sym_pos' to False.",
OptimizeWarning, stacklevel=5)
elif not lstsq:
lstsq = True
warn(
"Solving system with option 'sym_pos':False "
"failed. This may happen occasionally, "
"especially as the solution is "
"approached. However, if you see this frequently, "
"your problem may be numerically challenging. "
"If you cannot improve the formulation, consider "
"setting 'lstsq' to True. Consider also setting "
"`presolve` to True, if it is not already.",
OptimizeWarning, stacklevel=5)
else:
raise e
solve = _get_solver(M, sparse, lstsq, sym_pos,
cholesky, permc_spec)
# [4] Results after 8.29
d_tau = ((rhatg + 1 / tau * rhattk - (-c.dot(u) + b.dot(v))) /
(1 / tau * kappa + (-c.dot(p) + b.dot(q))))
d_x = u + p * d_tau
d_y = v + q * d_tau
# [4] Relations between after 8.25 and 8.26
d_z = (1 / x) * (rhatxs - z * d_x)
d_kappa = 1 / tau * (rhattk - kappa * d_tau)
# [4] 8.12 and "Let alpha be the maximal possible step..." before 8.23
alpha = _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, 1)
if ip: # initial point - see [4] 4.4
gamma = 10
else: # predictor-corrector, [4] definition after 8.12
beta1 = 0.1 # [4] pg. 220 (Table 8.1)
gamma = (1 - alpha)**2 * min(beta1, (1 - alpha))
i += 1
return d_x, d_y, d_z, d_tau, d_kappa
def _sym_solve(Dinv, A, r1, r2, solve):
"""
An implementation of [4] equation 8.31 and 8.32
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
"""
# [4] 8.31
r = r2 + A.dot(Dinv * r1)
v = solve(r)
# [4] 8.32
u = Dinv * (A.T.dot(v) - r1)
return u, v
def _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, alpha0):
"""
An implementation of [4] equation 8.21
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
"""
# [4] 4.3 Equation 8.21, ignoring 8.20 requirement
# same step is taken in primal and dual spaces
# alpha0 is basically beta3 from [4] Table 8.1, but instead of beta3
# the value 1 is used in Mehrota corrector and initial point correction
i_x = d_x < 0
i_z = d_z < 0
alpha_x = alpha0 * np.min(x[i_x] / -d_x[i_x]) if np.any(i_x) else 1
alpha_tau = alpha0 * tau / -d_tau if d_tau < 0 else 1
alpha_z = alpha0 * np.min(z[i_z] / -d_z[i_z]) if np.any(i_z) else 1
alpha_kappa = alpha0 * kappa / -d_kappa if d_kappa < 0 else 1
alpha = np.min([1, alpha_x, alpha_tau, alpha_z, alpha_kappa])
return alpha
def _get_message(status):
"""
Given problem status code, return a more detailed message.
Parameters
----------
status : int
An integer representing the exit status of the optimization::
0 : Optimization terminated successfully
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
Returns
-------
message : str
A string descriptor of the exit status of the optimization.
"""
messages = (
["Optimization terminated successfully.",
"The iteration limit was reached before the algorithm converged.",
"The algorithm terminated successfully and determined that the "
"problem is infeasible.",
"The algorithm terminated successfully and determined that the "
"problem is unbounded.",
"Numerical difficulties were encountered before the problem "
"converged. Please check your problem formulation for errors, "
"independence of linear equality constraints, and reasonable "
"scaling and matrix condition numbers. If you continue to "
"encounter this error, please submit a bug report."
])
return messages[status]
def _do_step(x, y, z, tau, kappa, d_x, d_y, d_z, d_tau, d_kappa, alpha):
"""
An implementation of [4] Equation 8.9
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
"""
x = x + alpha * d_x
tau = tau + alpha * d_tau
z = z + alpha * d_z
kappa = kappa + alpha * d_kappa
y = y + alpha * d_y
return x, y, z, tau, kappa
def _get_blind_start(shape):
"""
Return the starting point from [4] 4.4
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
"""
m, n = shape
x0 = np.ones(n)
y0 = np.zeros(m)
z0 = np.ones(n)
tau0 = 1
kappa0 = 1
return x0, y0, z0, tau0, kappa0
def _indicators(A, b, c, c0, x, y, z, tau, kappa):
"""
Implementation of several equations from [4] used as indicators of
the status of optimization.
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
"""
# residuals for termination are relative to initial values
x0, y0, z0, tau0, kappa0 = _get_blind_start(A.shape)
# See [4], Section 4 - The Homogeneous Algorithm, Equation 8.8
def r_p(x, tau):
return b * tau - A.dot(x)
def r_d(y, z, tau):
return c * tau - A.T.dot(y) - z
def r_g(x, y, kappa):
return kappa + c.dot(x) - b.dot(y)
# np.dot unpacks if they are arrays of size one
def mu(x, tau, z, kappa):
return (x.dot(z) + np.dot(tau, kappa)) / (len(x) + 1)
obj = c.dot(x / tau) + c0
def norm(a):
return np.linalg.norm(a)
# See [4], Section 4.5 - The Stopping Criteria
r_p0 = r_p(x0, tau0)
r_d0 = r_d(y0, z0, tau0)
r_g0 = r_g(x0, y0, kappa0)
mu_0 = mu(x0, tau0, z0, kappa0)
rho_A = norm(c.T.dot(x) - b.T.dot(y)) / (tau + norm(b.T.dot(y)))
rho_p = norm(r_p(x, tau)) / max(1, norm(r_p0))
rho_d = norm(r_d(y, z, tau)) / max(1, norm(r_d0))
rho_g = norm(r_g(x, y, kappa)) / max(1, norm(r_g0))
rho_mu = mu(x, tau, z, kappa) / mu_0
return rho_p, rho_d, rho_A, rho_g, rho_mu, obj
def _display_iter(rho_p, rho_d, rho_g, alpha, rho_mu, obj, header=False):
"""
Print indicators of optimization status to the console.
Parameters
----------
rho_p : float
The (normalized) primal feasibility, see [4] 4.5
rho_d : float
The (normalized) dual feasibility, see [4] 4.5
rho_g : float
The (normalized) duality gap, see [4] 4.5
alpha : float
The step size, see [4] 4.3
rho_mu : float
The (normalized) path parameter, see [4] 4.5
obj : float
The objective function value of the current iterate
header : bool
True if a header is to be printed
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
"""
if header:
print("Primal Feasibility ",
"Dual Feasibility ",
"Duality Gap ",
"Step ",
"Path Parameter ",
"Objective ")
# no clue why this works
fmt = '{0:<20.13}{1:<20.13}{2:<20.13}{3:<17.13}{4:<20.13}{5:<20.13}'
print(fmt.format(
float(rho_p),
float(rho_d),
float(rho_g),
alpha if isinstance(alpha, str) else float(alpha),
float(rho_mu),
float(obj)))
def _ip_hsd(A, b, c, c0, alpha0, beta, maxiter, disp, tol, sparse, lstsq,
sym_pos, cholesky, pc, ip, permc_spec, callback, _T_o):
r"""
Solve a linear programming problem in standard form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
using the interior point method of [4].
Parameters
----------
A : 2D array
2D array such that ``A @ x``, gives the values of the equality
constraints at ``x``.
b : 1D array
1D array of values representing the RHS of each equality constraint
(row) in ``A`` (for standard form problem).
c : 1D array
Coefficients of the linear objective function to be minimized (for
standard form problem).
c0 : float
Constant term in objective function due to fixed (and eliminated)
variables. (Purely for display.)
alpha0 : float
The maximal step size for Mehrota's predictor-corrector search
direction; see :math:`\beta_3`of [4] Table 8.1
beta : float
The desired reduction of the path parameter :math:`\mu` (see [6]_)
maxiter : int
The maximum number of iterations of the algorithm.
disp : bool
Set to ``True`` if indicators of optimization status are to be printed
to the console each iteration.
tol : float
Termination tolerance; see [4]_ Section 4.5.
sparse : bool
Set to ``True`` if the problem is to be treated as sparse. However,
the inputs ``A_eq`` and ``A_ub`` should nonetheless be provided as
(dense) arrays rather than sparse matrices.
lstsq : bool
Set to ``True`` if the problem is expected to be very poorly
conditioned. This should always be left as ``False`` unless severe
numerical difficulties are frequently encountered, and a better option
would be to improve the formulation of the problem.
sym_pos : bool
Leave ``True`` if the problem is expected to yield a well conditioned
symmetric positive definite normal equation matrix (almost always).
cholesky : bool
Set to ``True`` if the normal equations are to be solved by explicit
Cholesky decomposition followed by explicit forward/backward
substitution. This is typically faster for moderate, dense problems
that are numerically well-behaved.
pc : bool
Leave ``True`` if the predictor-corrector method of Mehrota is to be
used. This is almost always (if not always) beneficial.
ip : bool
Set to ``True`` if the improved initial point suggestion due to [4]_
Section 4.3 is desired. It's unclear whether this is beneficial.
permc_spec : str (default = 'MMD_AT_PLUS_A')
(Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos =
True``.) A matrix is factorized in each iteration of the algorithm.
This option specifies how to permute the columns of the matrix for
sparsity preservation. Acceptable values are:
- ``NATURAL``: natural ordering.
- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
- ``COLAMD``: approximate minimum degree column ordering.
This option can impact the convergence of the
interior point algorithm; test different values to determine which
performs best for your problem. For more information, refer to
``scipy.sparse.linalg.splu``.
callback : callable, optional
If a callback function is provided, it will be called within each
iteration of the algorithm. The callback function must accept a single
`scipy.optimize.OptimizeResult` consisting of the following fields:
x : 1D array
Current solution vector
fun : float
Current value of the objective function
success : bool
True only when an algorithm has completed successfully,
so this is always False as the callback function is called
only while the algorithm is still iterating.
slack : 1D array
The values of the slack variables. Each slack variable
corresponds to an inequality constraint. If the slack is zero,
the corresponding constraint is active.
con : 1D array
The (nominally zero) residuals of the equality constraints,
that is, ``b - A_eq @ x``
phase : int
The phase of the algorithm being executed. This is always
1 for the interior-point method because it has only one phase.
status : int
For revised simplex, this is always 0 because if a different
status is detected, the algorithm terminates.
nit : int
The number of iterations performed.
message : str
A string descriptor of the exit status of the optimization.
Returns
-------
x_hat : float
Solution vector (for standard form problem).
status : int
An integer representing the exit status of the optimization::
0 : Optimization terminated successfully
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
message : str
A string descriptor of the exit status of the optimization.
iteration : int
The number of iterations taken to solve the problem
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
.. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear
Programming based on Newton's Method." Unpublished Course Notes,
March 2004. Available 2/25/2017 at:
https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf
"""
iteration = 0
# default initial point
x, y, z, tau, kappa = _get_blind_start(A.shape)
# first iteration is special improvement of initial point
ip = ip if pc else False
# [4] 4.5
rho_p, rho_d, rho_A, rho_g, rho_mu, obj = _indicators(
A, b, c, c0, x, y, z, tau, kappa)
go = rho_p > tol or rho_d > tol or rho_A > tol # we might get lucky : )
if disp:
_display_iter(rho_p, rho_d, rho_g, "-", rho_mu, obj, header=True)
if callback is not None:
x_o, fun, slack, con, _, _ = _postsolve(x/tau, *_T_o,
tol=tol)
res = OptimizeResult({'x': x_o, 'fun': fun, 'slack': slack,
'con': con, 'nit': iteration, 'phase': 1,
'complete': False, 'status': 0,
'message': "", 'success': False})
callback(res)
status = 0
message = "Optimization terminated successfully."
if sparse:
A = sps.csc_matrix(A)
A.T = A.transpose() # A.T is defined for sparse matrices but is slow
# Redefine it to avoid calculating again
# This is fine as long as A doesn't change
while go:
iteration += 1
if ip: # initial point
# [4] Section 4.4
gamma = 1
def eta(g):
return 1
else:
# gamma = 0 in predictor step according to [4] 4.1
# if predictor/corrector is off, use mean of complementarity [6]
# 5.1 / [4] Below Figure 10-4
gamma = 0 if pc else beta * np.mean(z * x)
# [4] Section 4.1
def eta(g=gamma):
return 1 - g
try:
# Solve [4] 8.6 and 8.7/8.13/8.23
d_x, d_y, d_z, d_tau, d_kappa = _get_delta(
A, b, c, x, y, z, tau, kappa, gamma, eta,
sparse, lstsq, sym_pos, cholesky, pc, ip, permc_spec)
if ip: # initial point
# [4] 4.4
# Formula after 8.23 takes a full step regardless if this will
# take it negative
alpha = 1.0
x, y, z, tau, kappa = _do_step(
x, y, z, tau, kappa, d_x, d_y,
d_z, d_tau, d_kappa, alpha)
x[x < 1] = 1
z[z < 1] = 1
tau = max(1, tau)
kappa = max(1, kappa)
ip = False # done with initial point
else:
# [4] Section 4.3
alpha = _get_step(x, d_x, z, d_z, tau,
d_tau, kappa, d_kappa, alpha0)
# [4] Equation 8.9
x, y, z, tau, kappa = _do_step(
x, y, z, tau, kappa, d_x, d_y, d_z, d_tau, d_kappa, alpha)
except (LinAlgError, FloatingPointError,
ValueError, ZeroDivisionError):
# this can happen when sparse solver is used and presolve
# is turned off. Also observed ValueError in AppVeyor Python 3.6
# Win32 build (PR #8676). I've never seen it otherwise.
status = 4
message = _get_message(status)
break
# [4] 4.5
rho_p, rho_d, rho_A, rho_g, rho_mu, obj = _indicators(
A, b, c, c0, x, y, z, tau, kappa)
go = rho_p > tol or rho_d > tol or rho_A > tol
if disp:
_display_iter(rho_p, rho_d, rho_g, alpha, rho_mu, obj)
if callback is not None:
x_o, fun, slack, con, _, _ = _postsolve(x/tau, *_T_o,
tol=tol)
res = OptimizeResult({'x': x_o, 'fun': fun, 'slack': slack,
'con': con, 'nit': iteration, 'phase': 1,
'complete': False, 'status': 0,
'message': "", 'success': False})
callback(res)
# [4] 4.5
inf1 = (rho_p < tol and rho_d < tol and rho_g < tol and tau < tol *
max(1, kappa))
inf2 = rho_mu < tol and tau < tol * min(1, kappa)
if inf1 or inf2:
# [4] Lemma 8.4 / Theorem 8.3
if b.transpose().dot(y) > tol:
status = 2
else: # elif c.T.dot(x) < tol: ? Probably not necessary.
status = 3
message = _get_message(status)
break
elif iteration >= maxiter:
status = 1
message = _get_message(status)
break
x_hat = x / tau
# [4] Statement after Theorem 8.2
return x_hat, status, message, iteration
def _linprog_ip(
c,
c0=0,
A=None,
b=None,
callback=None,
_T_o=[],
alpha0=.99995,
beta=0.1,
maxiter=1000,
disp=False,
tol=1e-8,
sparse=False,
lstsq=False,
sym_pos=True,
cholesky=None,
pc=True,
ip=False,
permc_spec='MMD_AT_PLUS_A',
**unknown_options):
r"""
Minimize a linear objective function subject to linear
equality and non-negativity constraints using the interior point method
of [4]_. Linear programming is intended to solve problems
of the following form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
Parameters
----------
c : 1D array
Coefficients of the linear objective function to be minimized.
c0 : float
Constant term in objective function due to fixed (and eliminated)
variables. (Purely for display.)
A : 2D array
2D array such that ``A @ x``, gives the values of the equality
constraints at ``x``.
b : 1D array
1D array of values representing the right hand side of each equality
constraint (row) in ``A``.
Options
-------
maxiter : int (default = 1000)
The maximum number of iterations of the algorithm.
disp : bool (default = False)
Set to ``True`` if indicators of optimization status are to be printed
to the console each iteration.
tol : float (default = 1e-8)
Termination tolerance to be used for all termination criteria;
see [4]_ Section 4.5.
alpha0 : float (default = 0.99995)
The maximal step size for Mehrota's predictor-corrector search
direction; see :math:`\beta_{3}` of [4]_ Table 8.1.
beta : float (default = 0.1)
The desired reduction of the path parameter :math:`\mu` (see [6]_)
when Mehrota's predictor-corrector is not in use (uncommon).
sparse : bool (default = False)
Set to ``True`` if the problem is to be treated as sparse after
presolve. If either ``A_eq`` or ``A_ub`` is a sparse matrix,
this option will automatically be set ``True``, and the problem
will be treated as sparse even during presolve. If your constraint
matrices contain mostly zeros and the problem is not very small (less
than about 100 constraints or variables), consider setting ``True``
or providing ``A_eq`` and ``A_ub`` as sparse matrices.
lstsq : bool (default = False)
Set to ``True`` if the problem is expected to be very poorly
conditioned. This should always be left ``False`` unless severe
numerical difficulties are encountered. Leave this at the default
unless you receive a warning message suggesting otherwise.
sym_pos : bool (default = True)
Leave ``True`` if the problem is expected to yield a well conditioned
symmetric positive definite normal equation matrix
(almost always). Leave this at the default unless you receive
a warning message suggesting otherwise.
cholesky : bool (default = True)
Set to ``True`` if the normal equations are to be solved by explicit
Cholesky decomposition followed by explicit forward/backward
substitution. This is typically faster for problems
that are numerically well-behaved.
pc : bool (default = True)
Leave ``True`` if the predictor-corrector method of Mehrota is to be
used. This is almost always (if not always) beneficial.
ip : bool (default = False)
Set to ``True`` if the improved initial point suggestion due to [4]_
Section 4.3 is desired. Whether this is beneficial or not
depends on the problem.
permc_spec : str (default = 'MMD_AT_PLUS_A')
(Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos =
True``, and no SuiteSparse.)
A matrix is factorized in each iteration of the algorithm.
This option specifies how to permute the columns of the matrix for
sparsity preservation. Acceptable values are:
- ``NATURAL``: natural ordering.
- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
- ``COLAMD``: approximate minimum degree column ordering.
This option can impact the convergence of the
interior point algorithm; test different values to determine which
performs best for your problem. For more information, refer to
``scipy.sparse.linalg.splu``.
Returns
-------
x : 1D array
Solution vector.
status : int
An integer representing the exit status of the optimization::
0 : Optimization terminated successfully
1 : Iteration limit reached
2 : Problem appears to be infeasible
3 : Problem appears to be unbounded
4 : Serious numerical difficulties encountered
message : str
A string descriptor of the exit status of the optimization.
iteration : int
The number of iterations taken to solve the problem.
Notes
-----
This method implements the algorithm outlined in [4]_ with ideas from [8]_
and a structure inspired by the simpler methods of [6]_.
The primal-dual path following method begins with initial 'guesses' of
the primal and dual variables of the standard form problem and iteratively
attempts to solve the (nonlinear) Karush-Kuhn-Tucker conditions for the
problem with a gradually reduced logarithmic barrier term added to the
objective. This particular implementation uses a homogeneous self-dual
formulation, which provides certificates of infeasibility or unboundedness
where applicable.
The default initial point for the primal and dual variables is that
defined in [4]_ Section 4.4 Equation 8.22. Optionally (by setting initial
point option ``ip=True``), an alternate (potentially improved) starting
point can be calculated according to the additional recommendations of
[4]_ Section 4.4.
A search direction is calculated using the predictor-corrector method
(single correction) proposed by Mehrota and detailed in [4]_ Section 4.1.
(A potential improvement would be to implement the method of multiple
corrections described in [4]_ Section 4.2.) In practice, this is
accomplished by solving the normal equations, [4]_ Section 5.1 Equations
8.31 and 8.32, derived from the Newton equations [4]_ Section 5 Equations
8.25 (compare to [4]_ Section 4 Equations 8.6-8.8). The advantage of
solving the normal equations rather than 8.25 directly is that the
matrices involved are symmetric positive definite, so Cholesky
decomposition can be used rather than the more expensive LU factorization.
With default options, the solver used to perform the factorization depends
on third-party software availability and the conditioning of the problem.
For dense problems, solvers are tried in the following order:
1. ``scipy.linalg.cho_factor``
2. ``scipy.linalg.solve`` with option ``sym_pos=True``
3. ``scipy.linalg.solve`` with option ``sym_pos=False``
4. ``scipy.linalg.lstsq``
For sparse problems:
1. ``sksparse.cholmod.cholesky`` (if scikit-sparse and SuiteSparse are installed)
2. ``scipy.sparse.linalg.factorized`` (if scikit-umfpack and SuiteSparse are installed)
3. ``scipy.sparse.linalg.splu`` (which uses SuperLU distributed with SciPy)
4. ``scipy.sparse.linalg.lsqr``
If the solver fails for any reason, successively more robust (but slower)
solvers are attempted in the order indicated. Attempting, failing, and
re-starting factorization can be time consuming, so if the problem is
numerically challenging, options can be set to bypass solvers that are
failing. Setting ``cholesky=False`` skips to solver 2,
``sym_pos=False`` skips to solver 3, and ``lstsq=True`` skips
to solver 4 for both sparse and dense problems.
Potential improvements for combatting issues associated with dense
columns in otherwise sparse problems are outlined in [4]_ Section 5.3 and
[10]_ Section 4.1-4.2; the latter also discusses the alleviation of
accuracy issues associated with the substitution approach to free
variables.
After calculating the search direction, the maximum possible step size
that does not activate the non-negativity constraints is calculated, and
the smaller of this step size and unity is applied (as in [4]_ Section
4.1.) [4]_ Section 4.3 suggests improvements for choosing the step size.
The new point is tested according to the termination conditions of [4]_
Section 4.5. The same tolerance, which can be set using the ``tol`` option,
is used for all checks. (A potential improvement would be to expose
the different tolerances to be set independently.) If optimality,
unboundedness, or infeasibility is detected, the solve procedure
terminates; otherwise it repeats.
The expected problem formulation differs between the top level ``linprog``
module and the method specific solvers. The method specific solvers expect a
problem in standard form:
Minimize::
c @ x
Subject to::
A @ x == b
x >= 0
Whereas the top level ``linprog`` module expects a problem of form:
Minimize::
c @ x
Subject to::
A_ub @ x <= b_ub
A_eq @ x == b_eq
lb <= x <= ub
where ``lb = 0`` and ``ub = None`` unless set in ``bounds``.
The original problem contains equality, upper-bound and variable constraints
whereas the method specific solver requires equality constraints and
variable non-negativity.
``linprog`` module converts the original problem to standard form by
converting the simple bounds to upper bound constraints, introducing
non-negative slack variables for inequality constraints, and expressing
unbounded variables as the difference between two non-negative variables.
References
----------
.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
optimizer for linear programming: an implementation of the
homogeneous algorithm." High performance optimization. Springer US,
2000. 197-232.
.. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear
Programming based on Newton's Method." Unpublished Course Notes,
March 2004. Available 2/25/2017 at
https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf
.. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear
programming." Mathematical Programming 71.2 (1995): 221-245.
.. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear
programming." Athena Scientific 1 (1997): 997.
.. [10] Andersen, Erling D., et al. Implementation of interior point methods
for large scale linear programming. HEC/Universite de Geneve, 1996.
"""
_check_unknown_options(unknown_options)
# These should be warnings, not errors
if (cholesky or cholesky is None) and sparse and not has_cholmod:
if cholesky:
warn("Sparse cholesky is only available with scikit-sparse. "
"Setting `cholesky = False`",
OptimizeWarning, stacklevel=3)
cholesky = False
if sparse and lstsq:
warn("Option combination 'sparse':True and 'lstsq':True "
"is not recommended.",
OptimizeWarning, stacklevel=3)
if lstsq and cholesky:
warn("Invalid option combination 'lstsq':True "
"and 'cholesky':True; option 'cholesky' has no effect when "
"'lstsq' is set True.",
OptimizeWarning, stacklevel=3)
valid_permc_spec = ('NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A', 'COLAMD')
if permc_spec.upper() not in valid_permc_spec:
warn("Invalid permc_spec option: '" + str(permc_spec) + "'. "
"Acceptable values are 'NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A', "
"and 'COLAMD'. Reverting to default.",
OptimizeWarning, stacklevel=3)
permc_spec = 'MMD_AT_PLUS_A'
# This can be an error
if not sym_pos and cholesky:
raise ValueError(
"Invalid option combination 'sym_pos':False "
"and 'cholesky':True: Cholesky decomposition is only possible "
"for symmetric positive definite matrices.")
cholesky = cholesky or (cholesky is None and sym_pos and not lstsq)
x, status, message, iteration = _ip_hsd(A, b, c, c0, alpha0, beta,
maxiter, disp, tol, sparse,
lstsq, sym_pos, cholesky,
pc, ip, permc_spec, callback,
_T_o)
return x, status, message, iteration