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1126 lines
45 KiB
Python
1126 lines
45 KiB
Python
"""Interior-point method for linear programming
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The *interior-point* method uses the primal-dual path following algorithm
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outlined in [1]_. This algorithm supports sparse constraint matrices and
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is typically faster than the simplex methods, especially for large, sparse
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problems. Note, however, that the solution returned may be slightly less
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accurate than those of the simplex methods and will not, in general,
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correspond with a vertex of the polytope defined by the constraints.
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.. versionadded:: 1.0.0
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References
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----------
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.. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
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optimizer for linear programming: an implementation of the
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homogeneous algorithm." High performance optimization. Springer US,
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2000. 197-232.
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"""
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# Author: Matt Haberland
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import numpy as np
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import scipy as sp
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import scipy.sparse as sps
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from warnings import warn
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from scipy.linalg import LinAlgError
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from .optimize import OptimizeWarning, OptimizeResult, _check_unknown_options
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from ._linprog_util import _postsolve
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has_umfpack = True
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has_cholmod = True
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try:
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import sksparse
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from sksparse.cholmod import cholesky as cholmod
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from sksparse.cholmod import analyze as cholmod_analyze
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except ImportError:
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has_cholmod = False
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try:
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import scikits.umfpack # test whether to use factorized
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except ImportError:
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has_umfpack = False
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def _get_solver(M, sparse=False, lstsq=False, sym_pos=True,
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cholesky=True, permc_spec='MMD_AT_PLUS_A'):
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"""
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Given solver options, return a handle to the appropriate linear system
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solver.
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Parameters
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----------
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M : 2-D array
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As defined in [4] Equation 8.31
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sparse : bool (default = False)
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True if the system to be solved is sparse. This is typically set
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True when the original ``A_ub`` and ``A_eq`` arrays are sparse.
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lstsq : bool (default = False)
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True if the system is ill-conditioned and/or (nearly) singular and
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thus a more robust least-squares solver is desired. This is sometimes
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needed as the solution is approached.
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sym_pos : bool (default = True)
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True if the system matrix is symmetric positive definite
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Sometimes this needs to be set false as the solution is approached,
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even when the system should be symmetric positive definite, due to
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numerical difficulties.
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cholesky : bool (default = True)
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True if the system is to be solved by Cholesky, rather than LU,
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decomposition. This is typically faster unless the problem is very
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small or prone to numerical difficulties.
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permc_spec : str (default = 'MMD_AT_PLUS_A')
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Sparsity preservation strategy used by SuperLU. Acceptable values are:
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- ``NATURAL``: natural ordering.
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- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
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- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
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- ``COLAMD``: approximate minimum degree column ordering.
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See SuperLU documentation.
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Returns
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-------
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solve : function
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Handle to the appropriate solver function
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"""
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try:
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if sparse:
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if lstsq:
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def solve(r, sym_pos=False):
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return sps.linalg.lsqr(M, r)[0]
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elif cholesky:
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try:
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# Will raise an exception in the first call,
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# or when the matrix changes due to a new problem
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_get_solver.cholmod_factor.cholesky_inplace(M)
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except Exception:
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_get_solver.cholmod_factor = cholmod_analyze(M)
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_get_solver.cholmod_factor.cholesky_inplace(M)
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solve = _get_solver.cholmod_factor
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else:
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if has_umfpack and sym_pos:
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solve = sps.linalg.factorized(M)
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else: # factorized doesn't pass permc_spec
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solve = sps.linalg.splu(M, permc_spec=permc_spec).solve
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else:
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if lstsq: # sometimes necessary as solution is approached
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def solve(r):
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return sp.linalg.lstsq(M, r)[0]
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elif cholesky:
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L = sp.linalg.cho_factor(M)
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def solve(r):
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return sp.linalg.cho_solve(L, r)
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else:
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# this seems to cache the matrix factorization, so solving
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# with multiple right hand sides is much faster
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def solve(r, sym_pos=sym_pos):
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return sp.linalg.solve(M, r, sym_pos=sym_pos)
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# There are many things that can go wrong here, and it's hard to say
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# what all of them are. It doesn't really matter: if the matrix can't be
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# factorized, return None. get_solver will be called again with different
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# inputs, and a new routine will try to factorize the matrix.
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except KeyboardInterrupt:
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raise
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except Exception:
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return None
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return solve
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def _get_delta(A, b, c, x, y, z, tau, kappa, gamma, eta, sparse=False,
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lstsq=False, sym_pos=True, cholesky=True, pc=True, ip=False,
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permc_spec='MMD_AT_PLUS_A'):
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"""
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Given standard form problem defined by ``A``, ``b``, and ``c``;
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current variable estimates ``x``, ``y``, ``z``, ``tau``, and ``kappa``;
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algorithmic parameters ``gamma and ``eta;
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and options ``sparse``, ``lstsq``, ``sym_pos``, ``cholesky``, ``pc``
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(predictor-corrector), and ``ip`` (initial point improvement),
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get the search direction for increments to the variable estimates.
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Parameters
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----------
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As defined in [4], except:
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sparse : bool
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True if the system to be solved is sparse. This is typically set
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True when the original ``A_ub`` and ``A_eq`` arrays are sparse.
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lstsq : bool
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True if the system is ill-conditioned and/or (nearly) singular and
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thus a more robust least-squares solver is desired. This is sometimes
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needed as the solution is approached.
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sym_pos : bool
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True if the system matrix is symmetric positive definite
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Sometimes this needs to be set false as the solution is approached,
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even when the system should be symmetric positive definite, due to
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numerical difficulties.
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cholesky : bool
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True if the system is to be solved by Cholesky, rather than LU,
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decomposition. This is typically faster unless the problem is very
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small or prone to numerical difficulties.
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pc : bool
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True if the predictor-corrector method of Mehrota is to be used. This
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is almost always (if not always) beneficial. Even though it requires
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the solution of an additional linear system, the factorization
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is typically (implicitly) reused so solution is efficient, and the
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number of algorithm iterations is typically reduced.
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ip : bool
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True if the improved initial point suggestion due to [4] section 4.3
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is desired. It's unclear whether this is beneficial.
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permc_spec : str (default = 'MMD_AT_PLUS_A')
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(Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos =
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True``.) A matrix is factorized in each iteration of the algorithm.
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This option specifies how to permute the columns of the matrix for
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sparsity preservation. Acceptable values are:
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- ``NATURAL``: natural ordering.
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- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
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- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
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- ``COLAMD``: approximate minimum degree column ordering.
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This option can impact the convergence of the
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interior point algorithm; test different values to determine which
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performs best for your problem. For more information, refer to
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``scipy.sparse.linalg.splu``.
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Returns
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-------
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Search directions as defined in [4]
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References
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----------
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.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point
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optimizer for linear programming: an implementation of the
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homogeneous algorithm." High performance optimization. Springer US,
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2000. 197-232.
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"""
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if A.shape[0] == 0:
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# If there are no constraints, some solvers fail (understandably)
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# rather than returning empty solution. This gets the job done.
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sparse, lstsq, sym_pos, cholesky = False, False, True, False
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n_x = len(x)
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# [4] Equation 8.8
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r_P = b * tau - A.dot(x)
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r_D = c * tau - A.T.dot(y) - z
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r_G = c.dot(x) - b.transpose().dot(y) + kappa
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mu = (x.dot(z) + tau * kappa) / (n_x + 1)
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# Assemble M from [4] Equation 8.31
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Dinv = x / z
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if sparse:
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M = A.dot(sps.diags(Dinv, 0, format="csc").dot(A.T))
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else:
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M = A.dot(Dinv.reshape(-1, 1) * A.T)
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solve = _get_solver(M, sparse, lstsq, sym_pos, cholesky, permc_spec)
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# pc: "predictor-corrector" [4] Section 4.1
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# In development this option could be turned off
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# but it always seems to improve performance substantially
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n_corrections = 1 if pc else 0
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i = 0
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alpha, d_x, d_z, d_tau, d_kappa = 0, 0, 0, 0, 0
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while i <= n_corrections:
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# Reference [4] Eq. 8.6
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rhatp = eta(gamma) * r_P
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rhatd = eta(gamma) * r_D
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rhatg = eta(gamma) * r_G
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# Reference [4] Eq. 8.7
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rhatxs = gamma * mu - x * z
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rhattk = gamma * mu - tau * kappa
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if i == 1:
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if ip: # if the correction is to get "initial point"
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# Reference [4] Eq. 8.23
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rhatxs = ((1 - alpha) * gamma * mu -
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x * z - alpha**2 * d_x * d_z)
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rhattk = ((1 - alpha) * gamma * mu -
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tau * kappa -
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alpha**2 * d_tau * d_kappa)
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else: # if the correction is for "predictor-corrector"
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# Reference [4] Eq. 8.13
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rhatxs -= d_x * d_z
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rhattk -= d_tau * d_kappa
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# sometimes numerical difficulties arise as the solution is approached
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# this loop tries to solve the equations using a sequence of functions
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# for solve. For dense systems, the order is:
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# 1. scipy.linalg.cho_factor/scipy.linalg.cho_solve,
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# 2. scipy.linalg.solve w/ sym_pos = True,
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# 3. scipy.linalg.solve w/ sym_pos = False, and if all else fails
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# 4. scipy.linalg.lstsq
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# For sparse systems, the order is:
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# 1. sksparse.cholmod.cholesky (if available)
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# 2. scipy.sparse.linalg.factorized (if umfpack available)
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# 3. scipy.sparse.linalg.splu
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# 4. scipy.sparse.linalg.lsqr
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solved = False
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while(not solved):
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try:
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# [4] Equation 8.28
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p, q = _sym_solve(Dinv, A, c, b, solve)
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# [4] Equation 8.29
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u, v = _sym_solve(Dinv, A, rhatd -
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(1 / x) * rhatxs, rhatp, solve)
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if np.any(np.isnan(p)) or np.any(np.isnan(q)):
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raise LinAlgError
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solved = True
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except (LinAlgError, ValueError, TypeError) as e:
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# Usually this doesn't happen. If it does, it happens when
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# there are redundant constraints or when approaching the
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# solution. If so, change solver.
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if cholesky:
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cholesky = False
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warn(
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"Solving system with option 'cholesky':True "
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"failed. It is normal for this to happen "
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"occasionally, especially as the solution is "
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"approached. However, if you see this frequently, "
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"consider setting option 'cholesky' to False.",
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OptimizeWarning, stacklevel=5)
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elif sym_pos:
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sym_pos = False
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warn(
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"Solving system with option 'sym_pos':True "
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"failed. It is normal for this to happen "
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"occasionally, especially as the solution is "
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"approached. However, if you see this frequently, "
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"consider setting option 'sym_pos' to False.",
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OptimizeWarning, stacklevel=5)
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elif not lstsq:
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lstsq = True
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warn(
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"Solving system with option 'sym_pos':False "
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"failed. This may happen occasionally, "
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"especially as the solution is "
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"approached. However, if you see this frequently, "
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"your problem may be numerically challenging. "
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"If you cannot improve the formulation, consider "
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"setting 'lstsq' to True. Consider also setting "
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"`presolve` to True, if it is not already.",
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OptimizeWarning, stacklevel=5)
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else:
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raise e
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solve = _get_solver(M, sparse, lstsq, sym_pos,
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cholesky, permc_spec)
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# [4] Results after 8.29
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d_tau = ((rhatg + 1 / tau * rhattk - (-c.dot(u) + b.dot(v))) /
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(1 / tau * kappa + (-c.dot(p) + b.dot(q))))
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d_x = u + p * d_tau
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d_y = v + q * d_tau
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# [4] Relations between after 8.25 and 8.26
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d_z = (1 / x) * (rhatxs - z * d_x)
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d_kappa = 1 / tau * (rhattk - kappa * d_tau)
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# [4] 8.12 and "Let alpha be the maximal possible step..." before 8.23
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alpha = _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, 1)
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if ip: # initial point - see [4] 4.4
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gamma = 10
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else: # predictor-corrector, [4] definition after 8.12
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beta1 = 0.1 # [4] pg. 220 (Table 8.1)
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gamma = (1 - alpha)**2 * min(beta1, (1 - alpha))
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i += 1
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return d_x, d_y, d_z, d_tau, d_kappa
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def _sym_solve(Dinv, A, r1, r2, solve):
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"""
|
|
An implementation of [4] equation 8.31 and 8.32
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References
|
|
----------
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|
.. [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.
|
|
|
|
"""
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# [4] 8.31
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r = r2 + A.dot(Dinv * r1)
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v = solve(r)
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# [4] 8.32
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u = Dinv * (A.T.dot(v) - r1)
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return u, v
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|
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def _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, alpha0):
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"""
|
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An implementation of [4] equation 8.21
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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
|
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# same step is taken in primal and dual spaces
|
|
# alpha0 is basically beta3 from [4] Table 8.1, but instead of beta3
|
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# the value 1 is used in Mehrota corrector and initial point correction
|
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i_x = d_x < 0
|
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i_z = d_z < 0
|
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alpha_x = alpha0 * np.min(x[i_x] / -d_x[i_x]) if np.any(i_x) else 1
|
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alpha_tau = alpha0 * tau / -d_tau if d_tau < 0 else 1
|
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alpha_z = alpha0 * np.min(z[i_z] / -d_z[i_z]) if np.any(i_z) else 1
|
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alpha_kappa = alpha0 * kappa / -d_kappa if d_kappa < 0 else 1
|
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alpha = np.min([1, alpha_x, alpha_tau, alpha_z, alpha_kappa])
|
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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, postsolve_args):
|
|
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 : 2-D array
|
|
2-D array such that ``A @ x``, gives the values of the equality
|
|
constraints at ``x``.
|
|
b : 1-D array
|
|
1-D array of values representing the RHS of each equality constraint
|
|
(row) in ``A`` (for standard form problem).
|
|
c : 1-D 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 : 1-D 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 : 1-D 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 : 1-D 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.
|
|
postsolve_args : tuple
|
|
Data needed by _postsolve to convert the solution to the standard-form
|
|
problem into the solution to the original problem.
|
|
|
|
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, postsolve_args)
|
|
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, postsolve_args)
|
|
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, A, b, callback, postsolve_args, maxiter=1000, tol=1e-8,
|
|
disp=False, alpha0=.99995, beta=0.1, 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
|
|
|
|
User-facing documentation is in _linprog_doc.py.
|
|
|
|
Parameters
|
|
----------
|
|
c : 1-D 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 : 2-D array
|
|
2-D array such that ``A @ x``, gives the values of the equality
|
|
constraints at ``x``.
|
|
b : 1-D array
|
|
1-D array of values representing the right hand side of each equality
|
|
constraint (row) in ``A``.
|
|
callback : callable, optional
|
|
Callback function to be executed once per iteration.
|
|
postsolve_args : tuple
|
|
Data needed by _postsolve to convert the solution to the standard-form
|
|
problem into the solution to the original problem.
|
|
|
|
Options
|
|
-------
|
|
maxiter : int (default = 1000)
|
|
The maximum number of iterations of the algorithm.
|
|
tol : float (default = 1e-8)
|
|
Termination tolerance to be used for all termination criteria;
|
|
see [4]_ Section 4.5.
|
|
disp : bool (default = False)
|
|
Set to ``True`` if indicators of optimization status are to be printed
|
|
to the console each iteration.
|
|
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``.
|
|
unknown_options : dict
|
|
Optional arguments not used by this particular solver. If
|
|
`unknown_options` is non-empty a warning is issued listing all
|
|
unused options.
|
|
|
|
Returns
|
|
-------
|
|
x : 1-D 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,
|
|
postsolve_args)
|
|
|
|
return x, status, message, iteration
|