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""" Routines for removing redundant (linearly dependent) equations from linear programming equality constraints. """ # Author: Matt Haberland
""" Counts the number of nonzeros in each row of input array A. Nonzeros are defined as any element with absolute value greater than tol = 1e-13. This value should probably be an input to the function.
Parameters ---------- A : 2-D array An array representing a matrix
Returns ------- rowcount : 1-D array Number of nonzeros in each row of A
""" tol = 1e-13 return np.array((abs(A) > tol).sum(axis=1)).flatten()
""" Returns the index of the densest row of A. Ignores rows that are not eligible for consideration.
Parameters ---------- A : 2-D array An array representing a matrix eligibleRows : 1-D logical array Values indicate whether the corresponding row of A is eligible to be considered
Returns ------- i_densest : int Index of the densest row in A eligible for consideration
""" rowCounts = _row_count(A) return np.argmax(rowCounts * eligibleRows)
""" Eliminates trivial equations from system of equations defined by Ax = b and identifies trivial infeasibilities
Parameters ---------- A : 2-D array An array representing the left-hand side of a system of equations b : 1-D array An array representing the right-hand side of a system of equations
Returns ------- A : 2-D array An array representing the left-hand side of a system of equations b : 1-D array An array representing the right-hand side of a system of equations status: int An integer indicating the status of the removal operation 0: No infeasibility identified 2: Trivially infeasible message : str A string descriptor of the exit status of the optimization.
""" status = 0 message = "" i_zero = _row_count(A) == 0 A = A[np.logical_not(i_zero), :] if not(np.allclose(b[i_zero], 0)): status = 2 message = "There is a zero row in A_eq with a nonzero corresponding " \ "entry in b_eq. The problem is infeasible." b = b[np.logical_not(i_zero)] return A, b, status, message
LU, p = plu
u = scipy.linalg.solve_triangular(LU, v[perm_r], lower=True, unit_diagonal=True) LU[:j+1, j] = u[:j+1] l = u[j+1:] piv = LU[j, j] LU[j+1:, j] += (l/piv) return LU, p
""" Eliminates redundant equations from system of equations defined by Ax = b and identifies infeasibilities.
Parameters ---------- A : 2-D sparse matrix An matrix representing the left-hand side of a system of equations rhs : 1-D array An array representing the right-hand side of a system of equations
Returns ---------- A : 2-D sparse matrix A matrix representing the left-hand side of a system of equations rhs : 1-D array An array representing the right-hand side of a system of equations status: int An integer indicating the status of the system 0: No infeasibility identified 2: Trivially infeasible message : str A string descriptor of the exit status of the optimization.
References ---------- .. [2] Andersen, Erling D. "Finding all linearly dependent rows in large-scale linear programming." Optimization Methods and Software 6.3 (1995): 219-227.
""" tolapiv = 1e-8 tolprimal = 1e-8 status = 0 message = "" inconsistent = ("There is a linear combination of rows of A_eq that " "results in zero, suggesting a redundant constraint. " "However the same linear combination of b_eq is " "nonzero, suggesting that the constraints conflict " "and the problem is infeasible.") A, rhs, status, message = _remove_zero_rows(A, rhs)
if status != 0: return A, rhs, status, message
m, n = A.shape
v = list(range(m)) # Artificial column indices. b = list(v) # Basis column indices. # This is better as a list than a set because column order of basis matrix # needs to be consistent. k = set(range(m, m+n)) # Structural column indices. d = [] # Indices of dependent rows lu = None perm_r = None
A_orig = A A = np.hstack((np.eye(m), A)) e = np.zeros(m)
# Implements basic algorithm from [2] # Uses some of the suggested improvements (removing zero rows and # Bartels-Golub update idea). # Removing column singletons would be easy, but it is not as important # because the procedure is performed only on the equality constraint # matrix from the original problem - not on the canonical form matrix, # which would have many more column singletons due to slack variables # from the inequality constraints. # The thoughts on "crashing" the initial basis sound useful, but the # description of the procedure seems to assume a lot of familiarity with # the subject; it is not very explicit. I already went through enough # trouble getting the basic algorithm working, so I was not interested in # trying to decipher this, too. (Overall, the paper is fraught with # mistakes and ambiguities - which is strange, because the rest of # Andersen's papers are quite good.)
B = A[:, b] for i in v:
e[i] = 1 if i > 0: e[i-1] = 0
try: # fails for i==0 and any time it gets ill-conditioned j = b[i-1] lu = bg_update_dense(lu, perm_r, A[:, j], i-1) except: lu = scipy.linalg.lu_factor(B) LU, p = lu perm_r = list(range(m)) for i1, i2 in enumerate(p): perm_r[i1], perm_r[i2] = perm_r[i2], perm_r[i1]
pi = scipy.linalg.lu_solve(lu, e, trans=1)
# not efficient, but this is not the time sink... js = np.array(list(k-set(b))) batch = 50 dependent = True
# This is a tiny bit faster than looping over columns indivually, # like for j in js: if abs(A[:,j].transpose().dot(pi)) > tolapiv: for j_index in range(0, len(js), batch): j_indices = js[np.arange(j_index, min(j_index+batch, len(js)))]
c = abs(A[:, j_indices].transpose().dot(pi)) if (c > tolapiv).any(): j = js[j_index + np.argmax(c)] # very independent column B[:, i] = A[:, j] b[i] = j dependent = False break if dependent: bibar = pi.T.dot(rhs.reshape(-1, 1)) bnorm = np.linalg.norm(rhs) if abs(bibar)/(1+bnorm) > tolprimal: # inconsistent status = 2 message = inconsistent return A_orig, rhs, status, message else: # dependent d.append(i)
keep = set(range(m)) keep = list(keep - set(d)) return A_orig[keep, :], rhs[keep], status, message
""" Eliminates redundant equations from system of equations defined by Ax = b and identifies infeasibilities.
Parameters ---------- A : 2-D sparse matrix An matrix representing the left-hand side of a system of equations rhs : 1-D array An array representing the right-hand side of a system of equations
Returns ------- A : 2-D sparse matrix A matrix representing the left-hand side of a system of equations rhs : 1-D array An array representing the right-hand side of a system of equations status: int An integer indicating the status of the system 0: No infeasibility identified 2: Trivially infeasible message : str A string descriptor of the exit status of the optimization.
References ---------- .. [2] Andersen, Erling D. "Finding all linearly dependent rows in large-scale linear programming." Optimization Methods and Software 6.3 (1995): 219-227.
"""
tolapiv = 1e-8 tolprimal = 1e-8 status = 0 message = "" inconsistent = ("There is a linear combination of rows of A_eq that " "results in zero, suggesting a redundant constraint. " "However the same linear combination of b_eq is " "nonzero, suggesting that the constraints conflict " "and the problem is infeasible.") A, rhs, status, message = _remove_zero_rows(A, rhs)
if status != 0: return A, rhs, status, message
m, n = A.shape
v = list(range(m)) # Artificial column indices. b = list(v) # Basis column indices. # This is better as a list than a set because column order of basis matrix # needs to be consistent. k = set(range(m, m+n)) # Structural column indices. d = [] # Indices of dependent rows
A_orig = A A = scipy.sparse.hstack((scipy.sparse.eye(m), A)).tocsc() e = np.zeros(m)
# Implements basic algorithm from [2] # Uses only one of the suggested improvements (removing zero rows). # Removing column singletons would be easy, but it is not as important # because the procedure is performed only on the equality constraint # matrix from the original problem - not on the canonical form matrix, # which would have many more column singletons due to slack variables # from the inequality constraints. # The thoughts on "crashing" the initial basis sound useful, but the # description of the procedure seems to assume a lot of familiarity with # the subject; it is not very explicit. I already went through enough # trouble getting the basic algorithm working, so I was not interested in # trying to decipher this, too. (Overall, the paper is fraught with # mistakes and ambiguities - which is strange, because the rest of # Andersen's papers are quite good.) # I tried and tried and tried to improve performance using the # Bartels-Golub update. It works, but it's only practical if the LU # factorization can be specialized as described, and that is not possible # until the Scipy SuperLU interface permits control over column # permutation - see issue #7700.
for i in v: B = A[:, b]
e[i] = 1 if i > 0: e[i-1] = 0
pi = scipy.sparse.linalg.spsolve(B.transpose(), e).reshape(-1, 1)
js = list(k-set(b)) # not efficient, but this is not the time sink...
# Due to overhead, it tends to be faster (for problems tested) to # compute the full matrix-vector product rather than individual # vector-vector products (with the chance of terminating as soon # as any are nonzero). For very large matrices, it might be worth # it to compute, say, 100 or 1000 at a time and stop when a nonzero # is found.
c = (np.abs(A[:, js].transpose().dot(pi)) > tolapiv).nonzero()[0] if len(c) > 0: # independent j = js[c[0]] # in a previous commit, the previous line was changed to choose # index j corresponding with the maximum dot product. # While this avoided issues with almost # singular matrices, it slowed the routine in most NETLIB tests. # I think this is because these columns were denser than the # first column with nonzero dot product (c[0]). # It would be nice to have a heuristic that balances sparsity with # high dot product, but I don't think it's worth the time to # develop one right now. Bartels-Golub update is a much higher # priority. b[i] = j # replace artificial column else: bibar = pi.T.dot(rhs.reshape(-1, 1)) bnorm = np.linalg.norm(rhs) if abs(bibar)/(1 + bnorm) > tolprimal: status = 2 message = inconsistent return A_orig, rhs, status, message else: # dependent d.append(i)
keep = set(range(m)) keep = list(keep - set(d)) return A_orig[keep, :], rhs[keep], status, message
""" Eliminates redundant equations from system of equations defined by Ax = b and identifies infeasibilities.
Parameters ---------- A : 2-D array An array representing the left-hand side of a system of equations b : 1-D array An array representing the right-hand side of a system of equations
Returns ------- A : 2-D array An array representing the left-hand side of a system of equations b : 1-D array An array representing the right-hand side of a system of equations status: int An integer indicating the status of the system 0: No infeasibility identified 2: Trivially infeasible message : str A string descriptor of the exit status of the optimization.
References ---------- .. [2] Andersen, Erling D. "Finding all linearly dependent rows in large-scale linear programming." Optimization Methods and Software 6.3 (1995): 219-227.
"""
A, b, status, message = _remove_zero_rows(A, b)
if status != 0: return A, b, status, message
U, s, Vh = svd(A) eps = np.finfo(float).eps tol = s.max() * max(A.shape) * eps
m, n = A.shape s_min = s[-1] if m <= n else 0
# this algorithm is faster than that of [2] when the nullspace is small # but it could probably be improvement by randomized algorithms and with # a sparse implementation. # it relies on repeated singular value decomposition to find linearly # dependent rows (as identified by columns of U that correspond with zero # singular values). Unfortunately, only one row can be removed per # decomposition (I tried otherwise; doing so can cause problems.) # It would be nice if we could do truncated SVD like sp.sparse.linalg.svds # but that function is unreliable at finding singular values near zero. # Finding max eigenvalue L of A A^T, then largest eigenvalue (and # associated eigenvector) of -A A^T + L I (I is identity) via power # iteration would also work in theory, but is only efficient if the # smallest nonzero eigenvalue of A A^T is close to the largest nonzero # eigenvalue.
while abs(s_min) < tol: v = U[:, -1] # TODO: return these so user can eliminate from problem? # rows need to be represented in significant amount eligibleRows = np.abs(v) > tol * 10e6 if not np.any(eligibleRows) or np.any(np.abs(v.dot(A)) > tol): status = 4 message = ("Due to numerical issues, redundant equality " "constraints could not be removed automatically. " "Try providing your constraint matrices as sparse " "matrices to activate sparse presolve, try turning " "off redundancy removal, or try turning off presolve " "altogether.") break if np.any(np.abs(v.dot(b)) > tol): status = 2 message = ("There is a linear combination of rows of A_eq that " "results in zero, suggesting a redundant constraint. " "However the same linear combination of b_eq is " "nonzero, suggesting that the constraints conflict " "and the problem is infeasible.") break
i_remove = _get_densest(A, eligibleRows) A = np.delete(A, i_remove, axis=0) b = np.delete(b, i_remove) U, s, Vh = svd(A) m, n = A.shape s_min = s[-1] if m <= n else 0
return A, b, status, message |