573 lines
23 KiB
Python
573 lines
23 KiB
Python
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"""Revised simplex method for linear programming
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The *revised simplex* method uses the method described in [1]_, except
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that a factorization [2]_ of the basis matrix, rather than its inverse,
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is efficiently maintained and used to solve the linear systems at each
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iteration of the algorithm.
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.. versionadded:: 1.3.0
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References
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----------
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.. [1] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear
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programming." Athena Scientific 1 (1997): 997.
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.. [2] Bartels, Richard H. "A stabilization of the simplex method."
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Journal in Numerische Mathematik 16.5 (1971): 414-434.
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"""
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# Author: Matt Haberland
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import numpy as np
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from numpy.linalg import LinAlgError
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from scipy.linalg import solve
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from ._optimize import _check_unknown_options
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from ._bglu_dense import LU
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from ._bglu_dense import BGLU as BGLU
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from ._linprog_util import _postsolve
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from ._optimize import OptimizeResult
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def _phase_one(A, b, x0, callback, postsolve_args, maxiter, tol, disp,
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maxupdate, mast, pivot):
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"""
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The purpose of phase one is to find an initial basic feasible solution
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(BFS) to the original problem.
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Generates an auxiliary problem with a trivial BFS and an objective that
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minimizes infeasibility of the original problem. Solves the auxiliary
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problem using the main simplex routine (phase two). This either yields
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a BFS to the original problem or determines that the original problem is
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infeasible. If feasible, phase one detects redundant rows in the original
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constraint matrix and removes them, then chooses additional indices as
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necessary to complete a basis/BFS for the original problem.
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"""
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m, n = A.shape
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status = 0
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# generate auxiliary problem to get initial BFS
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A, b, c, basis, x, status = _generate_auxiliary_problem(A, b, x0, tol)
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if status == 6:
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residual = c.dot(x)
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iter_k = 0
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return x, basis, A, b, residual, status, iter_k
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# solve auxiliary problem
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phase_one_n = n
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iter_k = 0
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x, basis, status, iter_k = _phase_two(c, A, x, basis, callback,
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postsolve_args,
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maxiter, tol, disp,
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maxupdate, mast, pivot,
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iter_k, phase_one_n)
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# check for infeasibility
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residual = c.dot(x)
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if status == 0 and residual > tol:
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status = 2
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# drive artificial variables out of basis
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# TODO: test redundant row removal better
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# TODO: make solve more efficient with BGLU? This could take a while.
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keep_rows = np.ones(m, dtype=bool)
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for basis_column in basis[basis >= n]:
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B = A[:, basis]
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try:
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basis_finder = np.abs(solve(B, A)) # inefficient
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pertinent_row = np.argmax(basis_finder[:, basis_column])
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eligible_columns = np.ones(n, dtype=bool)
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eligible_columns[basis[basis < n]] = 0
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eligible_column_indices = np.where(eligible_columns)[0]
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index = np.argmax(basis_finder[:, :n]
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[pertinent_row, eligible_columns])
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new_basis_column = eligible_column_indices[index]
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if basis_finder[pertinent_row, new_basis_column] < tol:
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keep_rows[pertinent_row] = False
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else:
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basis[basis == basis_column] = new_basis_column
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except LinAlgError:
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status = 4
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# form solution to original problem
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A = A[keep_rows, :n]
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basis = basis[keep_rows]
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x = x[:n]
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m = A.shape[0]
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return x, basis, A, b, residual, status, iter_k
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def _get_more_basis_columns(A, basis):
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"""
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Called when the auxiliary problem terminates with artificial columns in
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the basis, which must be removed and replaced with non-artificial
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columns. Finds additional columns that do not make the matrix singular.
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"""
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m, n = A.shape
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# options for inclusion are those that aren't already in the basis
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a = np.arange(m+n)
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bl = np.zeros(len(a), dtype=bool)
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bl[basis] = 1
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options = a[~bl]
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options = options[options < n] # and they have to be non-artificial
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# form basis matrix
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B = np.zeros((m, m))
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B[:, 0:len(basis)] = A[:, basis]
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if (basis.size > 0 and
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np.linalg.matrix_rank(B[:, :len(basis)]) < len(basis)):
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raise Exception("Basis has dependent columns")
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rank = 0 # just enter the loop
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for i in range(n): # somewhat arbitrary, but we need another way out
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# permute the options, and take as many as needed
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new_basis = np.random.permutation(options)[:m-len(basis)]
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B[:, len(basis):] = A[:, new_basis] # update the basis matrix
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rank = np.linalg.matrix_rank(B) # check the rank
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if rank == m:
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break
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return np.concatenate((basis, new_basis))
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def _generate_auxiliary_problem(A, b, x0, tol):
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"""
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Modifies original problem to create an auxiliary problem with a trivial
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initial basic feasible solution and an objective that minimizes
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infeasibility in the original problem.
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Conceptually, this is done by stacking an identity matrix on the right of
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the original constraint matrix, adding artificial variables to correspond
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with each of these new columns, and generating a cost vector that is all
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zeros except for ones corresponding with each of the new variables.
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A initial basic feasible solution is trivial: all variables are zero
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except for the artificial variables, which are set equal to the
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corresponding element of the right hand side `b`.
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Runnning the simplex method on this auxiliary problem drives all of the
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artificial variables - and thus the cost - to zero if the original problem
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is feasible. The original problem is declared infeasible otherwise.
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Much of the complexity below is to improve efficiency by using singleton
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columns in the original problem where possible, thus generating artificial
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variables only as necessary, and using an initial 'guess' basic feasible
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solution.
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"""
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status = 0
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m, n = A.shape
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if x0 is not None:
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x = x0
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else:
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x = np.zeros(n)
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r = b - A@x # residual; this must be all zeros for feasibility
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A[r < 0] = -A[r < 0] # express problem with RHS positive for trivial BFS
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b[r < 0] = -b[r < 0] # to the auxiliary problem
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r[r < 0] *= -1
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# Rows which we will need to find a trivial way to zero.
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# This should just be the rows where there is a nonzero residual.
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# But then we would not necessarily have a column singleton in every row.
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# This makes it difficult to find an initial basis.
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if x0 is None:
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nonzero_constraints = np.arange(m)
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else:
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nonzero_constraints = np.where(r > tol)[0]
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# these are (at least some of) the initial basis columns
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basis = np.where(np.abs(x) > tol)[0]
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if len(nonzero_constraints) == 0 and len(basis) <= m: # already a BFS
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c = np.zeros(n)
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basis = _get_more_basis_columns(A, basis)
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return A, b, c, basis, x, status
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elif (len(nonzero_constraints) > m - len(basis) or
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np.any(x < 0)): # can't get trivial BFS
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c = np.zeros(n)
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status = 6
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return A, b, c, basis, x, status
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# chooses existing columns appropriate for inclusion in initial basis
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cols, rows = _select_singleton_columns(A, r)
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# find the rows we need to zero that we _can_ zero with column singletons
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i_tofix = np.isin(rows, nonzero_constraints)
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# these columns can't already be in the basis, though
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# we are going to add them to the basis and change the corresponding x val
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i_notinbasis = np.logical_not(np.isin(cols, basis))
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i_fix_without_aux = np.logical_and(i_tofix, i_notinbasis)
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rows = rows[i_fix_without_aux]
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cols = cols[i_fix_without_aux]
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# indices of the rows we can only zero with auxiliary variable
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# these rows will get a one in each auxiliary column
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arows = nonzero_constraints[np.logical_not(
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np.isin(nonzero_constraints, rows))]
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n_aux = len(arows)
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acols = n + np.arange(n_aux) # indices of auxiliary columns
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basis_ng = np.concatenate((cols, acols)) # basis columns not from guess
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basis_ng_rows = np.concatenate((rows, arows)) # rows we need to zero
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# add auxiliary singleton columns
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A = np.hstack((A, np.zeros((m, n_aux))))
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A[arows, acols] = 1
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# generate initial BFS
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x = np.concatenate((x, np.zeros(n_aux)))
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x[basis_ng] = r[basis_ng_rows]/A[basis_ng_rows, basis_ng]
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# generate costs to minimize infeasibility
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c = np.zeros(n_aux + n)
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c[acols] = 1
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# basis columns correspond with nonzeros in guess, those with column
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# singletons we used to zero remaining constraints, and any additional
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# columns to get a full set (m columns)
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basis = np.concatenate((basis, basis_ng))
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basis = _get_more_basis_columns(A, basis) # add columns as needed
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return A, b, c, basis, x, status
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def _select_singleton_columns(A, b):
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"""
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Finds singleton columns for which the singleton entry is of the same sign
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as the right-hand side; these columns are eligible for inclusion in an
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initial basis. Determines the rows in which the singleton entries are
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located. For each of these rows, returns the indices of the one singleton
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column and its corresponding row.
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"""
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# find indices of all singleton columns and corresponding row indicies
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column_indices = np.nonzero(np.sum(np.abs(A) != 0, axis=0) == 1)[0]
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columns = A[:, column_indices] # array of singleton columns
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row_indices = np.zeros(len(column_indices), dtype=int)
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nonzero_rows, nonzero_columns = np.nonzero(columns)
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row_indices[nonzero_columns] = nonzero_rows # corresponding row indicies
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# keep only singletons with entries that have same sign as RHS
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# this is necessary because all elements of BFS must be non-negative
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same_sign = A[row_indices, column_indices]*b[row_indices] >= 0
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column_indices = column_indices[same_sign][::-1]
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row_indices = row_indices[same_sign][::-1]
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# Reversing the order so that steps below select rightmost columns
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# for initial basis, which will tend to be slack variables. (If the
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# guess corresponds with a basic feasible solution but a constraint
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# is not satisfied with the corresponding slack variable zero, the slack
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# variable must be basic.)
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# for each row, keep rightmost singleton column with an entry in that row
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unique_row_indices, first_columns = np.unique(row_indices,
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return_index=True)
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return column_indices[first_columns], unique_row_indices
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def _find_nonzero_rows(A, tol):
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"""
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Returns logical array indicating the locations of rows with at least
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one nonzero element.
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"""
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return np.any(np.abs(A) > tol, axis=1)
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def _select_enter_pivot(c_hat, bl, a, rule="bland", tol=1e-12):
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"""
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Selects a pivot to enter the basis. Currently Bland's rule - the smallest
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index that has a negative reduced cost - is the default.
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"""
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if rule.lower() == "mrc": # index with minimum reduced cost
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return a[~bl][np.argmin(c_hat)]
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else: # smallest index w/ negative reduced cost
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return a[~bl][c_hat < -tol][0]
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def _display_iter(phase, iteration, slack, con, fun):
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"""
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Print indicators of optimization status to the console.
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"""
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header = True if not iteration % 20 else False
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if header:
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print("Phase",
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"Iteration",
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"Minimum Slack ",
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"Constraint Residual",
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"Objective ")
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# :<X.Y left aligns Y digits in X digit spaces
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fmt = '{0:<6}{1:<10}{2:<20.13}{3:<20.13}{4:<20.13}'
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try:
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slack = np.min(slack)
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except ValueError:
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slack = "NA"
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print(fmt.format(phase, iteration, slack, np.linalg.norm(con), fun))
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def _display_and_callback(phase_one_n, x, postsolve_args, status,
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iteration, disp, callback):
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if phase_one_n is not None:
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phase = 1
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x_postsolve = x[:phase_one_n]
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else:
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phase = 2
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x_postsolve = x
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x_o, fun, slack, con = _postsolve(x_postsolve,
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postsolve_args)
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if callback is not None:
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res = OptimizeResult({'x': x_o, 'fun': fun, 'slack': slack,
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'con': con, 'nit': iteration,
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'phase': phase, 'complete': False,
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'status': status, 'message': "",
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'success': False})
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callback(res)
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if disp:
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_display_iter(phase, iteration, slack, con, fun)
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def _phase_two(c, A, x, b, callback, postsolve_args, maxiter, tol, disp,
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maxupdate, mast, pivot, iteration=0, phase_one_n=None):
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"""
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The heart of the simplex method. Beginning with a basic feasible solution,
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moves to adjacent basic feasible solutions successively lower reduced cost.
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Terminates when there are no basic feasible solutions with lower reduced
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cost or if the problem is determined to be unbounded.
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This implementation follows the revised simplex method based on LU
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decomposition. Rather than maintaining a tableau or an inverse of the
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basis matrix, we keep a factorization of the basis matrix that allows
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efficient solution of linear systems while avoiding stability issues
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associated with inverted matrices.
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"""
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m, n = A.shape
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status = 0
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a = np.arange(n) # indices of columns of A
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ab = np.arange(m) # indices of columns of B
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if maxupdate:
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# basis matrix factorization object; similar to B = A[:, b]
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B = BGLU(A, b, maxupdate, mast)
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||
|
|
else:
|
||
|
|
B = LU(A, b)
|
||
|
|
|
||
|
|
for iteration in range(iteration, maxiter):
|
||
|
|
|
||
|
|
if disp or callback is not None:
|
||
|
|
_display_and_callback(phase_one_n, x, postsolve_args, status,
|
||
|
|
iteration, disp, callback)
|
||
|
|
|
||
|
|
bl = np.zeros(len(a), dtype=bool)
|
||
|
|
bl[b] = 1
|
||
|
|
|
||
|
|
xb = x[b] # basic variables
|
||
|
|
cb = c[b] # basic costs
|
||
|
|
|
||
|
|
try:
|
||
|
|
v = B.solve(cb, transposed=True) # similar to v = solve(B.T, cb)
|
||
|
|
except LinAlgError:
|
||
|
|
status = 4
|
||
|
|
break
|
||
|
|
|
||
|
|
# TODO: cythonize?
|
||
|
|
c_hat = c - v.dot(A) # reduced cost
|
||
|
|
c_hat = c_hat[~bl]
|
||
|
|
# Above is much faster than:
|
||
|
|
# N = A[:, ~bl] # slow!
|
||
|
|
# c_hat = c[~bl] - v.T.dot(N)
|
||
|
|
# Can we perform the multiplication only on the nonbasic columns?
|
||
|
|
|
||
|
|
if np.all(c_hat >= -tol): # all reduced costs positive -> terminate
|
||
|
|
break
|
||
|
|
|
||
|
|
j = _select_enter_pivot(c_hat, bl, a, rule=pivot, tol=tol)
|
||
|
|
u = B.solve(A[:, j]) # similar to u = solve(B, A[:, j])
|
||
|
|
|
||
|
|
i = u > tol # if none of the u are positive, unbounded
|
||
|
|
if not np.any(i):
|
||
|
|
status = 3
|
||
|
|
break
|
||
|
|
|
||
|
|
th = xb[i]/u[i]
|
||
|
|
l = np.argmin(th) # implicitly selects smallest subscript
|
||
|
|
th_star = th[l] # step size
|
||
|
|
|
||
|
|
x[b] = x[b] - th_star*u # take step
|
||
|
|
x[j] = th_star
|
||
|
|
B.update(ab[i][l], j) # modify basis
|
||
|
|
b = B.b # similar to b[ab[i][l]] =
|
||
|
|
|
||
|
|
else:
|
||
|
|
# If the end of the for loop is reached (without a break statement),
|
||
|
|
# then another step has been taken, so the iteration counter should
|
||
|
|
# increment, info should be displayed, and callback should be called.
|
||
|
|
iteration += 1
|
||
|
|
status = 1
|
||
|
|
if disp or callback is not None:
|
||
|
|
_display_and_callback(phase_one_n, x, postsolve_args, status,
|
||
|
|
iteration, disp, callback)
|
||
|
|
|
||
|
|
return x, b, status, iteration
|
||
|
|
|
||
|
|
|
||
|
|
def _linprog_rs(c, c0, A, b, x0, callback, postsolve_args,
|
||
|
|
maxiter=5000, tol=1e-12, disp=False,
|
||
|
|
maxupdate=10, mast=False, pivot="mrc",
|
||
|
|
**unknown_options):
|
||
|
|
"""
|
||
|
|
Solve the following linear programming problem via a two-phase
|
||
|
|
revised simplex algorithm.::
|
||
|
|
|
||
|
|
minimize: c @ x
|
||
|
|
|
||
|
|
subject to: A @ x == b
|
||
|
|
0 <= x < oo
|
||
|
|
|
||
|
|
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. (Currently unused.)
|
||
|
|
A : 2-D array
|
||
|
|
2-D array which, when matrix-multiplied by ``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_eq``.
|
||
|
|
x0 : 1-D array, optional
|
||
|
|
Starting values of the independent variables, which will be refined by
|
||
|
|
the optimization algorithm. For the revised simplex method, these must
|
||
|
|
correspond with a basic feasible solution.
|
||
|
|
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 ``c @ x``.
|
||
|
|
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.
|
||
|
|
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.
|
||
|
|
|
||
|
|
Options
|
||
|
|
-------
|
||
|
|
maxiter : int
|
||
|
|
The maximum number of iterations to perform in either phase.
|
||
|
|
tol : float
|
||
|
|
The tolerance which determines when a solution is "close enough" to
|
||
|
|
zero in Phase 1 to be considered a basic feasible solution or close
|
||
|
|
enough to positive to serve as an optimal solution.
|
||
|
|
disp : bool
|
||
|
|
Set to ``True`` if indicators of optimization status are to be printed
|
||
|
|
to the console each iteration.
|
||
|
|
maxupdate : int
|
||
|
|
The maximum number of updates performed on the LU factorization.
|
||
|
|
After this many updates is reached, the basis matrix is factorized
|
||
|
|
from scratch.
|
||
|
|
mast : bool
|
||
|
|
Minimize Amortized Solve Time. If enabled, the average time to solve
|
||
|
|
a linear system using the basis factorization is measured. Typically,
|
||
|
|
the average solve time will decrease with each successive solve after
|
||
|
|
initial factorization, as factorization takes much more time than the
|
||
|
|
solve operation (and updates). Eventually, however, the updated
|
||
|
|
factorization becomes sufficiently complex that the average solve time
|
||
|
|
begins to increase. When this is detected, the basis is refactorized
|
||
|
|
from scratch. Enable this option to maximize speed at the risk of
|
||
|
|
nondeterministic behavior. Ignored if ``maxupdate`` is 0.
|
||
|
|
pivot : "mrc" or "bland"
|
||
|
|
Pivot rule: Minimum Reduced Cost (default) or Bland's rule. Choose
|
||
|
|
Bland's rule if iteration limit is reached and cycling is suspected.
|
||
|
|
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 : Numerical difficulties encountered
|
||
|
|
5 : No constraints; turn presolve on
|
||
|
|
6 : Guess x0 cannot be converted to a basic feasible solution
|
||
|
|
|
||
|
|
message : str
|
||
|
|
A string descriptor of the exit status of the optimization.
|
||
|
|
iteration : int
|
||
|
|
The number of iterations taken to solve the problem.
|
||
|
|
"""
|
||
|
|
|
||
|
|
_check_unknown_options(unknown_options)
|
||
|
|
|
||
|
|
messages = ["Optimization terminated successfully.",
|
||
|
|
"Iteration limit reached.",
|
||
|
|
"The problem appears infeasible, as the phase one auxiliary "
|
||
|
|
"problem terminated successfully with a residual of {0:.1e}, "
|
||
|
|
"greater than the tolerance {1} required for the solution to "
|
||
|
|
"be considered feasible. Consider increasing the tolerance to "
|
||
|
|
"be greater than {0:.1e}. If this tolerance is unnaceptably "
|
||
|
|
"large, the problem is likely infeasible.",
|
||
|
|
"The problem is unbounded, as the simplex algorithm found "
|
||
|
|
"a basic feasible solution from which there is a direction "
|
||
|
|
"with negative reduced cost in which all decision variables "
|
||
|
|
"increase.",
|
||
|
|
"Numerical difficulties encountered; consider trying "
|
||
|
|
"method='interior-point'.",
|
||
|
|
"Problems with no constraints are trivially solved; please "
|
||
|
|
"turn presolve on.",
|
||
|
|
"The guess x0 cannot be converted to a basic feasible "
|
||
|
|
"solution. "
|
||
|
|
]
|
||
|
|
|
||
|
|
if A.size == 0: # address test_unbounded_below_no_presolve_corrected
|
||
|
|
return np.zeros(c.shape), 5, messages[5], 0
|
||
|
|
|
||
|
|
x, basis, A, b, residual, status, iteration = (
|
||
|
|
_phase_one(A, b, x0, callback, postsolve_args,
|
||
|
|
maxiter, tol, disp, maxupdate, mast, pivot))
|
||
|
|
|
||
|
|
if status == 0:
|
||
|
|
x, basis, status, iteration = _phase_two(c, A, x, basis, callback,
|
||
|
|
postsolve_args,
|
||
|
|
maxiter, tol, disp,
|
||
|
|
maxupdate, mast, pivot,
|
||
|
|
iteration)
|
||
|
|
|
||
|
|
return x, status, messages[status].format(residual, tol), iteration
|