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""" An interior-point method for linear programming. """ # Author: Matt Haberland
c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None): """ Given user inputs for a linear programming problem, return the objective vector, upper bound constraints, equality constraints, and simple bounds in a preferred format.
Parameters ---------- c : array_like Coefficients of the linear objective function to be minimized. A_ub : array_like, optional 2-D array which, when matrix-multiplied by ``x``, gives the values of the upper-bound inequality constraints at ``x``. b_ub : array_like, optional 1-D array of values representing the upper-bound of each inequality constraint (row) in ``A_ub``. A_eq : array_like, optional 2-D array which, when matrix-multiplied by ``x``, gives the values of the equality constraints at ``x``. b_eq : array_like, optional 1-D array of values representing the RHS of each equality constraint (row) in ``A_eq``. bounds : sequence, optional ``(min, max)`` pairs for each element in ``x``, defining the bounds on that parameter. Use None for one of ``min`` or ``max`` when there is no bound in that direction. By default bounds are ``(0, None)`` (non-negative) If a sequence containing a single tuple is provided, then ``min`` and ``max`` will be applied to all variables in the problem.
Returns ------- c : 1-D array Coefficients of the linear objective function to be minimized. A_ub : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the upper-bound inequality constraints at ``x``. b_ub : 1-D array 1-D array of values representing the upper-bound of each inequality constraint (row) in ``A_ub``. A_eq : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the equality constraints at ``x``. b_eq : 1-D array 1-D array of values representing the RHS of each equality constraint (row) in ``A_eq``. bounds : sequence of tuples ``(min, max)`` pairs for each element in ``x``, defining the bounds on that parameter. Use None for each of ``min`` or ``max`` when there is no bound in that direction. By default bounds are ``(0, None)`` (non-negative)
"""
try: if c is None: raise TypeError try: c = np.asarray(c, dtype=float).copy().squeeze() except BaseException: # typically a ValueError and shouldn't be, IMO raise TypeError if c.size == 1: c = c.reshape((-1)) n_x = len(c) if n_x == 0 or len(c.shape) != 1: raise ValueError( "Invalid input for linprog: c should be a 1D array; it must " "not have more than one non-singleton dimension") if not(np.isfinite(c).all()): raise ValueError( "Invalid input for linprog: c must not contain values " "inf, nan, or None") except TypeError: raise TypeError( "Invalid input for linprog: c must be a 1D array of numerical " "coefficients")
try: try: if sps.issparse(A_eq) or sps.issparse(A_ub): A_ub = sps.coo_matrix( (0, n_x), dtype=float) if A_ub is None else sps.coo_matrix( A_ub, dtype=float).copy() else: A_ub = np.zeros( (0, n_x), dtype=float) if A_ub is None else np.asarray( A_ub, dtype=float).copy() except BaseException: raise TypeError n_ub = A_ub.shape[0] if len(A_ub.shape) != 2 or A_ub.shape[1] != len(c): raise ValueError( "Invalid input for linprog: A_ub must have exactly two " "dimensions, and the number of columns in A_ub must be " "equal to the size of c ") if (sps.issparse(A_ub) and not np.isfinite(A_ub.data).all() or not sps.issparse(A_ub) and not np.isfinite(A_ub).all()): raise ValueError( "Invalid input for linprog: A_ub must not contain values " "inf, nan, or None") except TypeError: raise TypeError( "Invalid input for linprog: A_ub must be a numerical 2D array " "with each row representing an upper bound inequality constraint")
try: try: b_ub = np.array( [], dtype=float) if b_ub is None else np.asarray( b_ub, dtype=float).copy().squeeze() except BaseException: raise TypeError if b_ub.size == 1: b_ub = b_ub.reshape((-1)) if len(b_ub.shape) != 1: raise ValueError( "Invalid input for linprog: b_ub should be a 1D array; it " "must not have more than one non-singleton dimension") if len(b_ub) != n_ub: raise ValueError( "Invalid input for linprog: The number of rows in A_ub must " "be equal to the number of values in b_ub") if not(np.isfinite(b_ub).all()): raise ValueError( "Invalid input for linprog: b_ub must not contain values " "inf, nan, or None") except TypeError: raise TypeError( "Invalid input for linprog: b_ub must be a 1D array of " "numerical values, each representing the upper bound of an " "inequality constraint (row) in A_ub")
try: try: if sps.issparse(A_eq) or sps.issparse(A_ub): A_eq = sps.coo_matrix( (0, n_x), dtype=float) if A_eq is None else sps.coo_matrix( A_eq, dtype=float).copy() else: A_eq = np.zeros( (0, n_x), dtype=float) if A_eq is None else np.asarray( A_eq, dtype=float).copy() except BaseException: raise TypeError n_eq = A_eq.shape[0] if len(A_eq.shape) != 2 or A_eq.shape[1] != len(c): raise ValueError( "Invalid input for linprog: A_eq must have exactly two " "dimensions, and the number of columns in A_eq must be " "equal to the size of c ")
if (sps.issparse(A_eq) and not np.isfinite(A_eq.data).all() or not sps.issparse(A_eq) and not np.isfinite(A_eq).all()): raise ValueError( "Invalid input for linprog: A_eq must not contain values " "inf, nan, or None") except TypeError: raise TypeError( "Invalid input for linprog: A_eq must be a 2D array with each " "row representing an equality constraint")
try: try: b_eq = np.array( [], dtype=float) if b_eq is None else np.asarray( b_eq, dtype=float).copy().squeeze() except BaseException: raise TypeError if b_eq.size == 1: b_eq = b_eq.reshape((-1)) if len(b_eq.shape) != 1: raise ValueError( "Invalid input for linprog: b_eq should be a 1D array; it " "must not have more than one non-singleton dimension") if len(b_eq) != n_eq: raise ValueError( "Invalid input for linprog: the number of rows in A_eq " "must be equal to the number of values in b_eq") if not(np.isfinite(b_eq).all()): raise ValueError( "Invalid input for linprog: b_eq must not contain values " "inf, nan, or None") except TypeError: raise TypeError( "Invalid input for linprog: b_eq must be a 1D array of " "numerical values, each representing the right hand side of an " "equality constraints (row) in A_eq")
# "If a sequence containing a single tuple is provided, then min and max # will be applied to all variables in the problem." # linprog doesn't treat this right: it didn't accept a list with one tuple # in it try: if isinstance(bounds, str): raise TypeError if bounds is None or len(bounds) == 0: bounds = [(0, None)] * n_x elif len(bounds) == 1: b = bounds[0] if len(b) != 2: raise ValueError( "Invalid input for linprog: exactly one lower bound and " "one upper bound must be specified for each element of x") bounds = [b] * n_x elif len(bounds) == n_x: try: len(bounds[0]) except BaseException: bounds = [(bounds[0], bounds[1])] * n_x for i, b in enumerate(bounds): if len(b) != 2: raise ValueError( "Invalid input for linprog, bound " + str(i) + " " + str(b) + ": exactly one lower bound and one upper bound must " "be specified for each element of x") elif (len(bounds) == 2 and np.isreal(bounds[0]) and np.isreal(bounds[1])): bounds = [(bounds[0], bounds[1])] * n_x else: raise ValueError( "Invalid input for linprog: exactly one lower bound and one " "upper bound must be specified for each element of x")
clean_bounds = [] # also creates a copy so user's object isn't changed for i, b in enumerate(bounds): if b[0] is not None and b[1] is not None and b[0] > b[1]: raise ValueError( "Invalid input for linprog, bound " + str(i) + " " + str(b) + ": a lower bound must be less than or equal to the " "corresponding upper bound") if b[0] == np.inf: raise ValueError( "Invalid input for linprog, bound " + str(i) + " " + str(b) + ": infinity is not a valid lower bound") if b[1] == -np.inf: raise ValueError( "Invalid input for linprog, bound " + str(i) + " " + str(b) + ": negative infinity is not a valid upper bound") lb = float(b[0]) if b[0] is not None and b[0] != -np.inf else None ub = float(b[1]) if b[1] is not None and b[1] != np.inf else None clean_bounds.append((lb, ub)) bounds = clean_bounds except ValueError as e: if "could not convert string to float" in e.args[0]: raise TypeError else: raise e except TypeError as e: print(e) raise TypeError( "Invalid input for linprog: bounds must be a sequence of " "(min,max) pairs, each defining bounds on an element of x ")
return c, A_ub, b_ub, A_eq, b_eq, bounds
""" Given inputs for a linear programming problem in preferred format, presolve the problem: identify trivial infeasibilities, redundancies, and unboundedness, tighten bounds where possible, and eliminate fixed variables.
Parameters ---------- c : 1-D array Coefficients of the linear objective function to be minimized. A_ub : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the upper-bound inequality constraints at ``x``. b_ub : 1-D array 1-D array of values representing the upper-bound of each inequality constraint (row) in ``A_ub``. A_eq : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the equality constraints at ``x``. b_eq : 1-D array 1-D array of values representing the RHS of each equality constraint (row) in ``A_eq``. bounds : sequence of tuples ``(min, max)`` pairs for each element in ``x``, defining the bounds on that parameter. Use None for each of ``min`` or ``max`` when there is no bound in that direction.
Returns ------- c : 1-D array Coefficients of the linear objective function to be minimized. c0 : 1-D array Constant term in objective function due to fixed (and eliminated) variables. A_ub : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the upper-bound inequality constraints at ``x``. Unnecessary rows/columns have been removed. b_ub : 1-D array 1-D array of values representing the upper-bound of each inequality constraint (row) in ``A_ub``. Unnecessary elements have been removed. A_eq : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the equality constraints at ``x``. Unnecessary rows/columns have been removed. b_eq : 1-D array 1-D array of values representing the RHS of each equality constraint (row) in ``A_eq``. Unnecessary elements have been removed. bounds : sequence of tuples ``(min, max)`` pairs for each element in ``x``, defining the bounds on that parameter. Use None for each of ``min`` or ``max`` when there is no bound in that direction. Bounds have been tightened where possible. x : 1-D array Solution vector (when the solution is trivial and can be determined in presolve) undo: list of tuples (index, value) pairs that record the original index and fixed value for each variable removed from the problem complete: bool Whether the solution is complete (solved or determined to be infeasible or unbounded in presolve) 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
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. .. [5] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear programming." Mathematical Programming 71.2 (1995): 221-245.
""" # ideas from Reference [5] by Andersen and Andersen # however, unlike the reference, this is performed before converting # problem to standard form # There are a few advantages: # * artificial variables have not been added, so matrices are smaller # * bounds have not been converted to constraints yet. (It is better to # do that after presolve because presolve may adjust the simple bounds.) # There are many improvements that can be made, namely: # * implement remaining checks from [5] # * loop presolve until no additional changes are made # * implement additional efficiency improvements in redundancy removal [2]
tol = 1e-9 # tolerance for equality. should this be exposed to user?
undo = [] # record of variables eliminated from problem # constant term in cost function may be added if variables are eliminated c0 = 0 complete = False # complete is True if detected infeasible/unbounded x = np.zeros(c.shape) # this is solution vector if completed in presolve
status = 0 # all OK unless determined otherwise message = ""
# Standard form for bounds (from _clean_inputs) is list of tuples # but numpy array is more convenient here # In retrospect, numpy array should have been the standard bounds = np.array(bounds) lb = bounds[:, 0] ub = bounds[:, 1] lb[np.equal(lb, None)] = -np.inf ub[np.equal(ub, None)] = np.inf bounds = bounds.astype(float) lb = lb.astype(float) ub = ub.astype(float)
m_eq, n = A_eq.shape m_ub, n = A_ub.shape
if (sps.issparse(A_eq)): A_eq = A_eq.tolil() A_ub = A_ub.tolil()
def where(A): return A.nonzero()
vstack = sps.vstack else: where = np.where vstack = np.vstack
# zero row in equality constraints zero_row = np.array(np.sum(A_eq != 0, axis=1) == 0).flatten() if np.any(zero_row): if np.any( np.logical_and( zero_row, np.abs(b_eq) > tol)): # test_zero_row_1 # infeasible if RHS is not zero status = 2 message = ("The problem is (trivially) infeasible due to a row " "of zeros in the equality constraint matrix with a " "nonzero corresponding constraint value.") complete = True return (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message) else: # test_zero_row_2 # if RHS is zero, we can eliminate this equation entirely A_eq = A_eq[np.logical_not(zero_row), :] b_eq = b_eq[np.logical_not(zero_row)]
# zero row in inequality constraints zero_row = np.array(np.sum(A_ub != 0, axis=1) == 0).flatten() if np.any(zero_row): if np.any(np.logical_and(zero_row, b_ub < -tol)): # test_zero_row_1 # infeasible if RHS is less than zero (because LHS is zero) status = 2 message = ("The problem is (trivially) infeasible due to a row " "of zeros in the equality constraint matrix with a " "nonzero corresponding constraint value.") complete = True return (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message) else: # test_zero_row_2 # if LHS is >= 0, we can eliminate this constraint entirely A_ub = A_ub[np.logical_not(zero_row), :] b_ub = b_ub[np.logical_not(zero_row)]
# zero column in (both) constraints # this indicates that a variable isn't constrained and can be removed A = vstack((A_eq, A_ub)) if A.shape[0] > 0: zero_col = np.array(np.sum(A != 0, axis=0) == 0).flatten() # variable will be at upper or lower bound, depending on objective x[np.logical_and(zero_col, c < 0)] = ub[ np.logical_and(zero_col, c < 0)] x[np.logical_and(zero_col, c > 0)] = lb[ np.logical_and(zero_col, c > 0)] if np.any(np.isinf(x)): # if an unconstrained variable has no bound status = 3 message = ("If feasible, the problem is (trivially) unbounded " "due to a zero column in the constraint matrices. If " "you wish to check whether the problem is infeasible, " "turn presolve off.") complete = True return (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message) # variables will equal upper/lower bounds will be removed later lb[np.logical_and(zero_col, c < 0)] = ub[ np.logical_and(zero_col, c < 0)] ub[np.logical_and(zero_col, c > 0)] = lb[ np.logical_and(zero_col, c > 0)]
# row singleton in equality constraints # this fixes a variable and removes the constraint singleton_row = np.array(np.sum(A_eq != 0, axis=1) == 1).flatten() rows = where(singleton_row)[0] cols = where(A_eq[rows, :])[1] if len(rows) > 0: for row, col in zip(rows, cols): val = b_eq[row] / A_eq[row, col] if not lb[col] - tol <= val <= ub[col] + tol: # infeasible if fixed value is not within bounds status = 2 message = ("The problem is (trivially) infeasible because a " "singleton row in the equality constraints is " "inconsistent with the bounds.") complete = True return (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message) else: # sets upper and lower bounds at that fixed value - variable # will be removed later lb[col] = val ub[col] = val A_eq = A_eq[np.logical_not(singleton_row), :] b_eq = b_eq[np.logical_not(singleton_row)]
# row singleton in inequality constraints # this indicates a simple bound and the constraint can be removed # simple bounds may be adjusted here # After all of the simple bound information is combined here, get_Abc will # turn the simple bounds into constraints singleton_row = np.array(np.sum(A_ub != 0, axis=1) == 1).flatten() cols = where(A_ub[singleton_row, :])[1] rows = where(singleton_row)[0] if len(rows) > 0: for row, col in zip(rows, cols): val = b_ub[row] / A_ub[row, col] if A_ub[row, col] > 0: # upper bound if val < lb[col] - tol: # infeasible complete = True elif val < ub[col]: # new upper bound ub[col] = val else: # lower bound if val > ub[col] + tol: # infeasible complete = True elif val > lb[col]: # new lower bound lb[col] = val if complete: status = 2 message = ("The problem is (trivially) infeasible because a " "singleton row in the upper bound constraints is " "inconsistent with the bounds.") return (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message) A_ub = A_ub[np.logical_not(singleton_row), :] b_ub = b_ub[np.logical_not(singleton_row)]
# identical bounds indicate that variable can be removed i_f = np.abs(lb - ub) < tol # indices of "fixed" variables i_nf = np.logical_not(i_f) # indices of "not fixed" variables
# test_bounds_equal_but_infeasible if np.all(i_f): # if bounds define solution, check for consistency residual = b_eq - A_eq.dot(lb) slack = b_ub - A_ub.dot(lb) if ((A_ub.size > 0 and np.any(slack < 0)) or (A_eq.size > 0 and not np.allclose(residual, 0))): status = 2 message = ("The problem is (trivially) infeasible because the " "bounds fix all variables to values inconsistent with " "the constraints") complete = True return (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message)
ub_mod = ub lb_mod = lb if np.any(i_f): c0 += c[i_f].dot(lb[i_f]) b_eq = b_eq - A_eq[:, i_f].dot(lb[i_f]) b_ub = b_ub - A_ub[:, i_f].dot(lb[i_f]) c = c[i_nf] x = x[i_nf] A_eq = A_eq[:, i_nf] A_ub = A_ub[:, i_nf] # record of variables to be added back in undo = [np.where(i_f)[0], lb[i_f]] # don't remove these entries from bounds; they'll be used later. # but we _also_ need a version of the bounds with these removed lb_mod = lb[i_nf] ub_mod = ub[i_nf]
# no constraints indicates that problem is trivial if A_eq.size == 0 and A_ub.size == 0: b_eq = np.array([]) b_ub = np.array([]) # test_empty_constraint_1 if c.size == 0: status = 0 message = ("The solution was determined in presolve as there are " "no non-trivial constraints.") elif (np.any(np.logical_and(c < 0, ub == np.inf)) or np.any(np.logical_and(c > 0, lb == -np.inf))): # test_no_constraints() status = 3 message = ("If feasible, the problem is (trivially) unbounded " "because there are no constraints and at least one " "element of c is negative. If you wish to check " "whether the problem is infeasible, turn presolve " "off.") else: # test_empty_constraint_2 status = 0 message = ("The solution was determined in presolve as there are " "no non-trivial constraints.") complete = True x[c < 0] = ub_mod[c < 0] x[c > 0] = lb_mod[c > 0] # if this is not the last step of presolve, should convert bounds back # to array and return here
# *sigh* - convert bounds back to their standard form (list of tuples) # again, in retrospect, numpy array would be standard form lb[np.equal(lb, -np.inf)] = None ub[np.equal(ub, np.inf)] = None bounds = np.hstack((lb[:, np.newaxis], ub[:, np.newaxis])) bounds = bounds.tolist() for i, row in enumerate(bounds): for j, col in enumerate(row): if str( col) == "nan": # comparing col to float("nan") and # np.nan doesn't work. should use np.isnan bounds[i][j] = None
# remove redundant (linearly dependent) rows from equality constraints n_rows_A = A_eq.shape[0] redundancy_warning = ("A_eq does not appear to be of full row rank. To " "improve performance, check the problem formulation " "for redundant equality constraints.") if (sps.issparse(A_eq)): if rr and A_eq.size > 0: # TODO: Fast sparse rank check? A_eq, b_eq, status, message = _remove_redundancy_sparse(A_eq, b_eq) if A_eq.shape[0] < n_rows_A: warn(redundancy_warning, OptimizeWarning) if status != 0: complete = True return (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message)
# This is a wild guess for which redundancy removal algorithm will be # faster. More testing would be good. small_nullspace = 5 if rr and A_eq.size > 0: try: # TODO: instead use results of first SVD in _remove_redundancy rank = np.linalg.matrix_rank(A_eq) except: # oh well, we'll have to go with _remove_redundancy_dense rank = 0 if rr and A_eq.size > 0 and rank < A_eq.shape[0]: warn(redundancy_warning, OptimizeWarning) dim_row_nullspace = A_eq.shape[0]-rank if dim_row_nullspace <= small_nullspace: A_eq, b_eq, status, message = _remove_redundancy(A_eq, b_eq) if dim_row_nullspace > small_nullspace or status == 4: A_eq, b_eq, status, message = _remove_redundancy_dense(A_eq, b_eq) if A_eq.shape[0] < rank: 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.") status = 4 if status != 0: complete = True return (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message)
c, c0=0, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, undo=[]): """ Given a linear programming problem of the form:
minimize: c^T * x
subject to: A_ub * x <= b_ub A_eq * x == b_eq bounds[i][0] < x_i < bounds[i][1]
return the problem in standard form: minimize: c'^T * x'
subject to: A * x' == b 0 < x' < oo
by adding slack variables and making variable substitutions as necessary.
Parameters ---------- c : 1-D array Coefficients of the linear objective function to be minimized. Components corresponding with fixed variables have been eliminated. c0 : float Constant term in objective function due to fixed (and eliminated) variables. A_ub : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the upper-bound inequality constraints at ``x``. Unnecessary rows/columns have been removed. b_ub : 1-D array 1-D array of values representing the upper-bound of each inequality constraint (row) in ``A_ub``. Unnecessary elements have been removed. A_eq : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the equality constraints at ``x``. Unnecessary rows/columns have been removed. b_eq : 1-D array 1-D array of values representing the RHS of each equality constraint (row) in ``A_eq``. Unnecessary elements have been removed. bounds : sequence of tuples ``(min, max)`` pairs for each element in ``x``, defining the bounds on that parameter. Use None for each of ``min`` or ``max`` when there is no bound in that direction. Bounds have been tightened where possible. undo: list of tuples (`index`, `value`) pairs that record the original index and fixed value for each variable removed from the problem
Returns ------- A : 2-D array 2-D array which, when matrix-multiplied by x, gives the values of the equality constraints at x (for standard form problem). 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.
References ---------- .. [6] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear programming." Athena Scientific 1 (1997): 997.
"""
if sps.issparse(A_eq): sparse = True A_eq = sps.lil_matrix(A_eq) A_ub = sps.lil_matrix(A_ub)
def hstack(blocks): return sps.hstack(blocks, format="lil")
def vstack(blocks): return sps.vstack(blocks, format="lil")
zeros = sps.lil_matrix eye = sps.eye else: sparse = False hstack = np.hstack vstack = np.vstack zeros = np.zeros eye = np.eye
fixed_x = set() if len(undo) > 0: # these are indices of variables removed from the problem # however, their bounds are still part of the bounds list fixed_x = set(undo[0]) # they are needed elsewhere, but not here bounds = [bounds[i] for i in range(len(bounds)) if i not in fixed_x] # in retrospect, the standard form of bounds should have been an n x 2 # array. maybe change it someday.
# modify problem such that all variables have only non-negativity bounds
bounds = np.array(bounds) lbs = bounds[:, 0] ubs = bounds[:, 1] m_ub, n_ub = A_ub.shape
lb_none = np.equal(lbs, None) ub_none = np.equal(ubs, None) lb_some = np.logical_not(lb_none) ub_some = np.logical_not(ub_none)
# if preprocessing is on, lb == ub can't happen # if preprocessing is off, then it would be best to convert that # to an equality constraint, but it's tricky to make the other # required modifications from inside here.
# unbounded below: substitute xi = -xi' (unbounded above) l_nolb_someub = np.logical_and(lb_none, ub_some) i_nolb = np.where(l_nolb_someub)[0] lbs[l_nolb_someub], ubs[l_nolb_someub] = ( -ubs[l_nolb_someub], lbs[l_nolb_someub]) lb_none = np.equal(lbs, None) ub_none = np.equal(ubs, None) lb_some = np.logical_not(lb_none) ub_some = np.logical_not(ub_none) c[i_nolb] *= -1 if len(i_nolb) > 0: if A_ub.shape[0] > 0: # sometimes needed for sparse arrays... weird A_ub[:, i_nolb] *= -1 if A_eq.shape[0] > 0: A_eq[:, i_nolb] *= -1
# upper bound: add inequality constraint i_newub = np.where(ub_some)[0] ub_newub = ubs[ub_some] n_bounds = np.count_nonzero(ub_some) A_ub = vstack((A_ub, zeros((n_bounds, A_ub.shape[1])))) b_ub = np.concatenate((b_ub, np.zeros(n_bounds))) A_ub[range(m_ub, A_ub.shape[0]), i_newub] = 1 b_ub[m_ub:] = ub_newub
A1 = vstack((A_ub, A_eq)) b = np.concatenate((b_ub, b_eq)) c = np.concatenate((c, np.zeros((A_ub.shape[0],))))
# unbounded: substitute xi = xi+ + xi- l_free = np.logical_and(lb_none, ub_none) i_free = np.where(l_free)[0] n_free = len(i_free) A1 = hstack((A1, zeros((A1.shape[0], n_free)))) c = np.concatenate((c, np.zeros(n_free))) A1[:, range(n_ub, A1.shape[1])] = -A1[:, i_free] c[np.arange(n_ub, A1.shape[1])] = -c[i_free]
# add slack variables A2 = vstack([eye(A_ub.shape[0]), zeros((A_eq.shape[0], A_ub.shape[0]))]) A = hstack([A1, A2])
# lower bound: substitute xi = xi' + lb # now there is a constant term in objective i_shift = np.where(lb_some)[0] lb_shift = lbs[lb_some].astype(float) c0 += np.sum(lb_shift * c[i_shift]) if sparse: b = b.reshape(-1, 1) A = A.tocsc() b -= (A[:, i_shift] * sps.diags(lb_shift)).sum(axis=1) b = b.ravel() else: b -= (A[:, i_shift] * lb_shift).sum(axis=1)
return A, b, c, c0
x, c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, complete=False, undo=[], status=0, message="", tol=1e-8): """ Given solution x to presolved, standard form linear program x, add fixed variables back into the problem and undo the variable substitutions to get solution to original linear program. Also, calculate the objective function value, slack in original upper bound constraints, and residuals in original equality constraints.
Parameters ---------- x : 1-D array Solution vector to the standard-form problem. c : 1-D array Original coefficients of the linear objective function to be minimized. A_ub : 2-D array Original upper bound constraint matrix. b_ub : 1-D array Original upper bound constraint vector. A_eq : 2-D array Original equality constraint matrix. b_eq : 1-D array Original equality constraint vector. bounds : sequence of tuples Bounds, as modified in presolve complete : bool Whether the solution is was determined in presolve (``True`` if so) undo: list of tuples (`index`, `value`) pairs that record the original index and fixed value for each variable removed from the 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. tol : float Termination tolerance; see [1]_ Section 4.5.
Returns ------- x : 1-D array Solution vector to original linear programming problem fun: float optimal objective value for original problem slack: 1-D array The (non-negative) slack in the upper bound constraints, that is, ``b_ub - A_ub * x`` con : 1-D array The (nominally zero) residuals of the equality constraints, that is, ``b - A_eq * x`` 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.
""" # note that all the inputs are the ORIGINAL, unmodified versions # no rows, columns have been removed # the only exception is bounds; it has been modified # we need these modified values to undo the variable substitutions # in retrospect, perhaps this could have been simplified if the "undo" # variable also contained information for undoing variable substitutions
n_x = len(c)
# we don't have to undo variable substitutions for fixed variables that # were removed from the problem no_adjust = set()
# if there were variables removed from the problem, add them back into the # solution vector if len(undo) > 0: no_adjust = set(undo[0]) x = x.tolist() for i, val in zip(undo[0], undo[1]): x.insert(i, val) x = np.array(x)
# now undo variable substitutions # if "complete", problem was solved in presolve; don't do anything here if not complete and bounds is not None: # bounds are never none, probably n_unbounded = 0 for i, b in enumerate(bounds): if i in no_adjust: continue lb, ub = b if lb is None and ub is None: n_unbounded += 1 x[i] = x[i] - x[n_x + n_unbounded - 1] else: if lb is None: x[i] = ub - x[i] else: x[i] += lb
n_x = len(c) x = x[:n_x] # all the rest of the variables were artificial fun = x.dot(c) slack = b_ub - A_ub.dot(x) # report slack for ORIGINAL UB constraints # report residuals of ORIGINAL EQ constraints con = b_eq - A_eq.dot(x)
# Patch for bug #8664. Detecting this sort of issue earlier # (via abnormalities in the indicators) would be better. bounds = np.array(bounds) # again, this should have been the standard form lb = bounds[:, 0] ub = bounds[:, 1] lb[np.equal(lb, None)] = -np.inf ub[np.equal(ub, None)] = np.inf tol = np.sqrt(tol) # Somewhat arbitrary, but status 5 is very unusual if status == 0 and ((slack < -tol).any() or (np.abs(con) > tol).any() or (x < lb - tol).any() or (x > ub + tol).any()): status = 4 message = ("The solution does not satisfy the constraints, yet " "no errors were raised and there is no certificate of " "infeasibility or unboundedness. This is known to occur " "if the `presolve` option is False and the problem is " "infeasible. If you uncounter this under different " "circumstances, please submit a bug report. Otherwise, " "please enable presolve.") elif status == 0 and (np.isnan(x).any() or np.isnan(fun) or np.isnan(slack).any() or np.isnan(con).any()): status = 4 message = ("Numerical difficulties were encountered but no errors " "were raised. This is known to occur if the 'presolve' " "option is False, 'sparse' is True, and A_eq includes " "redundant rows. If you encounter this under different " "circumstances, please submit a bug report. Otherwise, " "remove linearly dependent equations from your equality " "constraints or enable presolve.")
return x, fun, slack, con, status, message
""" Given solver options, return a handle to the appropriate linear system solver.
Parameters ---------- sparse : bool True if the system to be solved is sparse. This is typically set True when the original ``A_ub`` and ``A_eq`` arrays are sparse. lstsq : bool True if the system is ill-conditioned and/or (nearly) singular and thus a more robust least-squares solver is desired. This is sometimes needed as the solution is approached. sym_pos : bool True if the system matrix is symmetric positive definite Sometimes this needs to be set false as the solution is approached, even when the system should be symmetric positive definite, due to numerical difficulties. cholesky : bool True if the system is to be solved by Cholesky, rather than LU, decomposition. This is typically faster unless the problem is very small or prone to numerical difficulties.
Returns ------- solve : function Handle to the appropriate solver function
""" if sparse: if lstsq or not(sym_pos): def solve(M, r, sym_pos=False): return sps.linalg.lsqr(M, r)[0] else: # this is not currently used; it is replaced by splu solve # TODO: expose use of this as an option def solve(M, r): return sps.linalg.spsolve(M, r, permc_spec="MMD_AT_PLUS_A")
else: if lstsq: # sometimes necessary as solution is approached def solve(M, r): return sp.linalg.lstsq(M, r)[0] elif cholesky: solve = sp.linalg.cho_solve else: # this seems to cache the matrix factorization, so solving # with multiple right hand sides is much faster def solve(M, r, sym_pos=sym_pos): return sp.linalg.solve(M, r, sym_pos=sym_pos)
return solve
A, b, c, x, y, z, tau, kappa, gamma, eta, sparse=False, lstsq=False, sym_pos=True, cholesky=True, pc=True, ip=False, permc_spec='MMD_AT_PLUS_A'): """ Given standard form problem defined by ``A``, ``b``, and ``c``; current variable estimates ``x``, ``y``, ``z``, ``tau``, and ``kappa``; algorithmic parameters ``gamma and ``eta; and options ``sparse``, ``lstsq``, ``sym_pos``, ``cholesky``, ``pc`` (predictor-corrector), and ``ip`` (initial point improvement), get the search direction for increments to the variable estimates.
Parameters ---------- As defined in [1], except: sparse : bool True if the system to be solved is sparse. This is typically set True when the original ``A_ub`` and ``A_eq`` arrays are sparse. lstsq : bool True if the system is ill-conditioned and/or (nearly) singular and thus a more robust least-squares solver is desired. This is sometimes needed as the solution is approached. sym_pos : bool True if the system matrix is symmetric positive definite Sometimes this needs to be set false as the solution is approached, even when the system should be symmetric positive definite, due to numerical difficulties. cholesky : bool True if the system is to be solved by Cholesky, rather than LU, decomposition. This is typically faster unless the problem is very small or prone to numerical difficulties. pc : bool True if the predictor-corrector method of Mehrota is to be used. This is almost always (if not always) beneficial. Even though it requires the solution of an additional linear system, the factorization is typically (implicitly) reused so solution is efficient, and the number of algorithm iterations is typically reduced. ip : bool True if the improved initial point suggestion due to [1] section 4.3 is desired. It's unclear whether this is beneficial. permc_spec : str (default = 'MMD_AT_PLUS_A') (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = True``.) A matrix is factorized in each iteration of the algorithm. This option specifies how to permute the columns of the matrix for sparsity preservation. Acceptable values are:
- ``NATURAL``: natural ordering. - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. - ``COLAMD``: approximate minimum degree column ordering.
This option can impact the convergence of the interior point algorithm; test different values to determine which performs best for your problem. For more information, refer to ``scipy.sparse.linalg.splu``.
Returns ------- Search directions as defined in [1]
References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232.
"""
if A.shape[0] == 0: # If there are no constraints, some solvers fail (understandably) # rather than returning empty solution. This gets the job done. sparse, lstsq, sym_pos, cholesky = False, False, True, False solve = _get_solver(sparse, lstsq, sym_pos, cholesky) n_x = len(x)
# [1] Equation 8.8 r_P = b * tau - A.dot(x) r_D = c * tau - A.T.dot(y) - z r_G = c.dot(x) - b.transpose().dot(y) + kappa mu = (x.dot(z) + tau * kappa) / (n_x + 1)
# Assemble M from [1] Equation 8.31 Dinv = x / z splu = False if sparse and not lstsq: # sparse requires Dinv to be diag matrix M = A.dot(sps.diags(Dinv, 0, format="csc").dot(A.T)) try: # TODO: should use linalg.factorized instead, but I don't have # umfpack and therefore cannot test its performance solve = sps.linalg.splu(M, permc_spec=permc_spec).solve splu = True except: lstsq = True solve = _get_solver(sparse, lstsq, sym_pos, cholesky) else: # dense does not; use broadcasting M = A.dot(Dinv.reshape(-1, 1) * A.T)
# For some small problems, calling sp.linalg.solve w/ sym_pos = True # may be faster. I am pretty certain it caches the factorization for # multiple uses and checks the incoming matrix to see if it's the same as # the one it already factorized. (I can't explain the speed otherwise.) if cholesky: try: L = sp.linalg.cho_factor(M) except: cholesky = False solve = _get_solver(sparse, lstsq, sym_pos, cholesky)
# pc: "predictor-corrector" [1] Section 4.1 # In development this option could be turned off # but it always seems to improve performance substantially n_corrections = 1 if pc else 0
i = 0 alpha, d_x, d_z, d_tau, d_kappa = 0, 0, 0, 0, 0 while i <= n_corrections: # Reference [1] Eq. 8.6 rhatp = eta(gamma) * r_P rhatd = eta(gamma) * r_D rhatg = np.array(eta(gamma) * r_G).reshape((1,))
# Reference [1] Eq. 8.7 rhatxs = gamma * mu - x * z rhattk = np.array(gamma * mu - tau * kappa).reshape((1,))
if i == 1: if ip: # if the correction is to get "initial point" # Reference [1] Eq. 8.23 rhatxs = ((1 - alpha) * gamma * mu - x * z - alpha**2 * d_x * d_z) rhattk = np.array( (1 - alpha) * gamma * mu - tau * kappa - alpha**2 * d_tau * d_kappa).reshape( (1, )) else: # if the correction is for "predictor-corrector" # Reference [1] Eq. 8.13 rhatxs -= d_x * d_z rhattk -= d_tau * d_kappa
# sometimes numerical difficulties arise as the solution is approached # this loop tries to solve the equations using a sequence of functions # for solve. For dense systems, the order is: # 1. scipy.linalg.cho_factor/scipy.linalg.cho_solve, # 2. scipy.linalg.solve w/ sym_pos = True, # 3. scipy.linalg.solve w/ sym_pos = False, and if all else fails # 4. scipy.linalg.lstsq # For sparse systems, the order is: # 1. scipy.sparse.linalg.splu # 2. scipy.sparse.linalg.lsqr # TODO: if umfpack is installed, use factorized instead of splu. # Can't do that now because factorized doesn't pass permc_spec # to splu if umfpack isn't installed. Also, umfpack not tested. solved = False while(not solved): try: solve_this = L if cholesky else M # [1] Equation 8.28 p, q = _sym_solve(Dinv, solve_this, A, c, b, solve, splu) # [1] Equation 8.29 u, v = _sym_solve(Dinv, solve_this, A, rhatd - (1 / x) * rhatxs, rhatp, solve, splu) if np.any(np.isnan(p)) or np.any(np.isnan(q)): raise LinAlgError solved = True except (LinAlgError, ValueError) as e: # Usually this doesn't happen. If it does, it happens when # there are redundant constraints or when approaching the # solution. If so, change solver. cholesky = False if not lstsq: if sym_pos: warn( "Solving system with option 'sym_pos':True " "failed. It is normal for this to happen " "occasionally, especially as the solution is " "approached. However, if you see this frequently, " "consider setting option 'sym_pos' to False.", OptimizeWarning) sym_pos = False else: warn( "Solving system with option 'sym_pos':False " "failed. This may happen occasionally, " "especially as the solution is " "approached. However, if you see this frequently, " "your problem may be numerically challenging. " "If you cannot improve the formulation, consider " "setting 'lstsq' to True. Consider also setting " "`presolve` to True, if it is not already.", OptimizeWarning) lstsq = True else: raise e solve = _get_solver(sparse, lstsq, sym_pos) # [1] Results after 8.29 d_tau = ((rhatg + 1 / tau * rhattk - (-c.dot(u) + b.dot(v))) / (1 / tau * kappa + (-c.dot(p) + b.dot(q)))) d_x = u + p * d_tau d_y = v + q * d_tau
# [1] Relations between after 8.25 and 8.26 d_z = (1 / x) * (rhatxs - z * d_x) d_kappa = 1 / tau * (rhattk - kappa * d_tau)
# [1] 8.12 and "Let alpha be the maximal possible step..." before 8.23 alpha = _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, 1) if ip: # initial point - see [1] 4.4 gamma = 10 else: # predictor-corrector, [1] definition after 8.12 beta1 = 0.1 # [1] pg. 220 (Table 8.1) gamma = (1 - alpha)**2 * min(beta1, (1 - alpha)) i += 1
return d_x, d_y, d_z, d_tau, d_kappa
""" An implementation of [1] equation 8.31 and 8.32
References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232.
""" # [1] 8.31 r = r2 + A.dot(Dinv * r1) if splu: v = solve(r) else: v = solve(M, r) # [1] 8.32 u = Dinv * (A.T.dot(v) - r1) return u, v
""" An implementation of [1] equation 8.21
References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232.
""" # [1] 4.3 Equation 8.21, ignoring 8.20 requirement # same step is taken in primal and dual spaces # alpha0 is basically beta3 from [1] Table 8.1, but instead of beta3 # the value 1 is used in Mehrota corrector and initial point correction i_x = d_x < 0 i_z = d_z < 0 alpha_x = alpha0 * np.min(x[i_x] / -d_x[i_x]) if np.any(i_x) else 1 alpha_tau = alpha0 * tau / -d_tau if d_tau < 0 else 1 alpha_z = alpha0 * np.min(z[i_z] / -d_z[i_z]) if np.any(i_z) else 1 alpha_kappa = alpha0 * kappa / -d_kappa if d_kappa < 0 else 1 alpha = np.min([1, alpha_x, alpha_tau, alpha_z, alpha_kappa]) return alpha
""" 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]
""" An implementation of [1] Equation 8.9
References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232.
""" 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
""" Return the starting point from [1] 4.4
References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232.
""" 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
""" Implementation of several equations from [1] used as indicators of the status of optimization.
References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232.
"""
# residuals for termination are relative to initial values x0, y0, z0, tau0, kappa0 = _get_blind_start(A.shape)
# See [1], 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 [1], 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
""" Print indicators of optimization status to the console.
Parameters ---------- rho_p : float The (normalized) primal feasibility, see [1] 4.5 rho_d : float The (normalized) dual feasibility, see [1] 4.5 rho_g : float The (normalized) duality gap, see [1] 4.5 alpha : float The step size, see [1] 4.3 rho_mu : float The (normalized) path parameter, see [1] 4.5 obj : float The objective function value of the current iterate header : bool True if a header is to be printed
References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232.
""" 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( rho_p, rho_d, rho_g, alpha, rho_mu, obj))
sparse, lstsq, sym_pos, cholesky, pc, ip, permc_spec): r""" Solve a linear programming problem in standard form:
minimize: c'^T * x'
subject to: A * x' == b 0 < x' < oo
using the interior point method of [1].
Parameters ---------- A : 2-D array 2-D array which, when matrix-multiplied by ``x``, gives the values of the equality constraints at ``x`` (for standard form problem). 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 [1] Table 8.1 beta : float The desired reduction of the path parameter :math:`\mu` (see [3]_) 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 [1]_ 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 [1]_ Section 4.3 is desired. It's unclear whether this is beneficial. permc_spec : str (default = 'MMD_AT_PLUS_A') (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = True``.) A matrix is factorized in each iteration of the algorithm. This option specifies how to permute the columns of the matrix for sparsity preservation. Acceptable values are:
- ``NATURAL``: natural ordering. - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. - ``COLAMD``: approximate minimum degree column ordering.
This option can impact the convergence of the interior point algorithm; test different values to determine which performs best for your problem. For more information, refer to ``scipy.sparse.linalg.splu``.
Returns ------- 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 ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232. .. [3] 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
# [1] 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)
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 # [1] Section 4.4 gamma = 1
def eta(g): return 1 else: # gamma = 0 in predictor step according to [1] 4.1 # if predictor/corrector is off, use mean of complementarity [3] # 5.1 / [4] Below Figure 10-4 gamma = 0 if pc else beta * np.mean(z * x) # [1] Section 4.1
def eta(g=gamma): return 1 - g
try: # Solve [1] 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 # [1] 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: # [1] Section 4.3 alpha = _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, alpha0) # [1] 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
# [1] 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, float(rho_mu), obj)
# [1] 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: # [1] 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
if disp: print(message)
x_hat = x / tau # [1] Statement after Theorem 8.2 return x_hat, status, message, iteration
c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, callback=None, alpha0=.99995, beta=0.1, maxiter=1000, disp=False, tol=1e-8, sparse=False, lstsq=False, sym_pos=True, cholesky=None, pc=True, ip=False, presolve=True, permc_spec='MMD_AT_PLUS_A', rr=True, _sparse_presolve=False, **unknown_options): r""" Minimize a linear objective function subject to linear equality constraints, linear inequality constraints, and simple bounds using the interior point method of [1]_.
Linear programming is intended to solve problems of the following form::
Minimize: c^T * x
Subject to: A_ub * x <= b_ub A_eq * x == b_eq bounds[i][0] < x_i < bounds[i][1]
Parameters ---------- c : array_like Coefficients of the linear objective function to be minimized. A_ub : array_like, optional 2-D array which, when matrix-multiplied by ``x``, gives the values of the upper-bound inequality constraints at ``x``. b_ub : array_like, optional 1-D array of values representing the upper-bound of each inequality constraint (row) in ``A_ub``. A_eq : array_like, optional 2-D array which, when matrix-multiplied by ``x``, gives the values of the equality constraints at ``x``. b_eq : array_like, optional 1-D array of values representing the right hand side of each equality constraint (row) in ``A_eq``. bounds : sequence, optional ``(min, max)`` pairs for each element in ``x``, defining the bounds on that parameter. Use ``None`` for one of ``min`` or ``max`` when there is no bound in that direction. By default bounds are ``(0, None)`` (non-negative). If a sequence containing a single tuple is provided, then ``min`` and ``max`` will be applied to all variables in the problem.
Options ------- maxiter : int (default = 1000) The maximum number of iterations of the algorithm. disp : bool (default = False) Set to ``True`` if indicators of optimization status are to be printed to the console each iteration. tol : float (default = 1e-8) Termination tolerance to be used for all termination criteria; see [1]_ Section 4.5. alpha0 : float (default = 0.99995) The maximal step size for Mehrota's predictor-corrector search direction; see :math:`\beta_{3}` of [1]_ Table 8.1. beta : float (default = 0.1) The desired reduction of the path parameter :math:`\mu` (see [3]_) 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 moderate, dense 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 [1]_ Section 4.3 is desired. Whether this is beneficial or not depends on the problem. presolve : bool (default = True) Leave ``True`` if presolve routine should be run. The presolve routine is almost always useful because it can detect trivial infeasibilities and unboundedness, eliminate fixed variables, and remove redundancies. One circumstance in which it might be turned off (set ``False``) is when it detects that the problem is trivially unbounded; it is possible that that the problem is truly infeasibile but this has not been detected. rr : bool (default = True) Default ``True`` attempts to eliminate any redundant rows in ``A_eq``. Set ``False`` if ``A_eq`` is known to be of full row rank, or if you are looking for a potential speedup (at the expense of reliability). permc_spec : str (default = 'MMD_AT_PLUS_A') (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = True``.) A matrix is factorized in each iteration of the algorithm. This option specifies how to permute the columns of the matrix for sparsity preservation. Acceptable values are:
- ``NATURAL``: natural ordering. - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. - ``COLAMD``: approximate minimum degree column ordering.
This option can impact the convergence of the interior point algorithm; test different values to determine which performs best for your problem. For more information, refer to ``scipy.sparse.linalg.splu``.
Returns ------- A ``scipy.optimize.OptimizeResult`` consisting of the following fields:
x : ndarray The independent variable vector which optimizes the linear programming problem. fun : float The optimal value of the objective function con : float The residuals of the equality constraints (nominally zero). slack : ndarray The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, then the corresponding constraint is active. success : bool Returns True if the algorithm succeeded in finding an optimal solution. 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
nit : int The number of iterations performed. message : str A string descriptor of the exit status of the optimization.
Notes -----
This method implements the algorithm outlined in [1]_ with ideas from [5]_ and a structure inspired by the simpler methods of [3]_ and [4]_.
First, a presolve procedure based on [5]_ attempts to identify trivial infeasibilities, trivial unboundedness, and potential problem simplifications. Specifically, it checks for:
- rows of zeros in ``A_eq`` or ``A_ub``, representing trivial constraints; - columns of zeros in ``A_eq`` `and` ``A_ub``, representing unconstrained variables; - column singletons in ``A_eq``, representing fixed variables; and - column singletons in ``A_ub``, representing simple bounds.
If presolve reveals that the problem is unbounded (e.g. an unconstrained and unbounded variable has negative cost) or infeasible (e.g. a row of zeros in ``A_eq`` corresponds with a nonzero in ``b_eq``), the solver terminates with the appropriate status code. Note that presolve terminates as soon as any sign of unboundedness is detected; consequently, a problem may be reported as unbounded when in reality the problem is infeasible (but infeasibility has not been detected yet). Therefore, if the output message states that unboundedness is detected in presolve and it is necessary to know whether the problem is actually infeasible, set option ``presolve=False``.
If neither infeasibility nor unboundedness are detected in a single pass of the presolve check, bounds are tightened where possible and fixed variables are removed from the problem. Then, linearly dependent rows of the ``A_eq`` matrix are removed, (unless they represent an infeasibility) to avoid numerical difficulties in the primary solve routine. Note that rows that are nearly linearly dependent (within a prescibed tolerance) may also be removed, which can change the optimal solution in rare cases. If this is a concern, eliminate redundancy from your problem formulation and run with option ``rr=False`` or ``presolve=False``.
Several potential improvements can be made here: additional presolve checks outlined in [5]_ should be implemented, the presolve routine should be run multiple times (until no further simplifications can be made), and more of the efficiency improvements from [2]_ should be implemented in the redundancy removal routines.
After presolve, the problem is transformed to standard form by converting the (tightened) 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.
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 [1]_ 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 [1]_ Section 4.4.
A search direction is calculated using the predictor-corrector method (single correction) proposed by Mehrota and detailed in [1]_ Section 4.1. (A potential improvement would be to implement the method of multiple corrections described in [1]_ Section 4.2.) In practice, this is accomplished by solving the normal equations, [1]_ Section 5.1 Equations 8.31 and 8.32, derived from the Newton equations [1]_ Section 5 Equations 8.25 (compare to [1]_ 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 the default ``cholesky=True``, this is accomplished using ``scipy.linalg.cho_factor`` followed by forward/backward substitutions via ``scipy.linalg.cho_solve``. With ``cholesky=False`` and ``sym_pos=True``, Cholesky decomposition is performed instead by ``scipy.linalg.solve``. Based on speed tests, this also appears to retain the Cholesky decomposition of the matrix for later use, which is beneficial as the same system is solved four times with different right hand sides in each iteration of the algorithm.
In problems with redundancy (e.g. if presolve is turned off with option ``presolve=False``) or if the matrices become ill-conditioned (e.g. as the solution is approached and some decision variables approach zero), Cholesky decomposition can fail. Should this occur, successively more robust solvers (``scipy.linalg.solve`` with ``sym_pos=False`` then ``scipy.linalg.lstsq``) are tried, at the cost of computational efficiency. These solvers can be used from the outset by setting the options ``sym_pos=False`` and ``lstsq=True``, respectively.
Note that with the option ``sparse=True``, the normal equations are solved using ``scipy.sparse.linalg.spsolve``. Unfortunately, this uses the more expensive LU decomposition from the outset, but for large, sparse problems, the use of sparse linear algebra techniques improves the solve speed despite the use of LU rather than Cholesky decomposition. A simple improvement would be to use the sparse Cholesky decomposition of ``CHOLMOD`` via ``scikit-sparse`` when available.
Other potential improvements for combatting issues associated with dense columns in otherwise sparse problems are outlined in [1]_ Section 5.3 and [7]_ 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 [1]_ Section 4.1.) [1]_ Section 4.3 suggests improvements for choosing the step size.
The new point is tested according to the termination conditions of [1]_ 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.
If optimality is achieved, a postsolve procedure undoes transformations associated with presolve and converting to standard form. It then calculates the residuals (equality constraint violations, which should be very small) and slacks (difference between the left and right hand sides of the upper bound constraints) of the original problem, which are returned with the solution in an ``OptimizeResult`` object.
References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232. .. [2] Andersen, Erling D. "Finding all linearly dependent rows in large-scale linear programming." Optimization Methods and Software 6.3 (1995): 219-227. .. [3] 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 .. [4] Fourer, Robert. "Solving Linear Programs by Interior-Point Methods." Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at http://www.4er.org/CourseNotes/Book%20B/B-III.pdf .. [5] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear programming." Mathematical Programming 71.2 (1995): 221-245. .. [6] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear programming." Athena Scientific 1 (1997): 997. .. [7] 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)
if callback is not None: raise NotImplementedError("method 'interior-point' does not support " "callback functions.")
# This is an undocumented option for unit testing sparse presolve if _sparse_presolve and A_eq is not None: A_eq = sp.sparse.coo_matrix(A_eq) if _sparse_presolve and A_ub is not None: A_ub = sp.sparse.coo_matrix(A_ub)
# These should be warnings, not errors if not sparse and (sp.sparse.issparse(A_eq) or sp.sparse.issparse(A_ub)): sparse = True warn("Sparse constraint matrix detected; setting 'sparse':True.", OptimizeWarning)
if sparse and lstsq: warn("Invalid option combination 'sparse':True " "and 'lstsq':True; Sparse least squares is not recommended.", OptimizeWarning)
if sparse and not sym_pos: warn("Invalid option combination 'sparse':True " "and 'sym_pos':False; the effect is the same as sparse least " "squares, which is not recommended.", OptimizeWarning)
if sparse and cholesky: # Cholesky decomposition is not available for sparse problems warn("Invalid option combination 'sparse':True " "and 'cholesky':True; sparse Colesky decomposition is not " "available.", OptimizeWarning)
if lstsq and cholesky: warn("Invalid option combination 'lstsq':True " "and 'cholesky':True; option 'cholesky' has no effect when " "'lstsq' is set True.", OptimizeWarning)
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) 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 is None and sym_pos and not sparse and not lstsq
iteration = 0 complete = False # will become True if solved in presolve undo = []
# Convert lists to numpy arrays, etc... c, A_ub, b_ub, A_eq, b_eq, bounds = _clean_inputs( c, A_ub, b_ub, A_eq, b_eq, bounds)
# Keep the original arrays to calculate slack/residuals for original # problem. c_o, A_ub_o, b_ub_o, A_eq_o, b_eq_o = c.copy( ), A_ub.copy(), b_ub.copy(), A_eq.copy(), b_eq.copy()
# Solve trivial problem, eliminate variables, tighten bounds, etc... c0 = 0 # we might get a constant term in the objective if presolve is True: (c, c0, A_ub, b_ub, A_eq, b_eq, bounds, x, undo, complete, status, message) = _presolve(c, A_ub, b_ub, A_eq, b_eq, bounds, rr)
# If not solved in presolve, solve it if not complete: # Convert problem to standard form A, b, c, c0 = _get_Abc(c, c0, A_ub, b_ub, A_eq, b_eq, bounds, undo) # Solve the problem x, status, message, iteration = _ip_hsd(A, b, c, c0, alpha0, beta, maxiter, disp, tol, sparse, lstsq, sym_pos, cholesky, pc, ip, permc_spec)
# Eliminate artificial variables, re-introduce presolved variables, etc... # need modified bounds here to translate variables appropriately x, fun, slack, con, status, message = _postprocess( x, c_o, A_ub_o, b_ub_o, A_eq_o, b_eq_o, bounds, complete, undo, status, message, tol)
sol = { 'x': x, 'fun': fun, 'slack': slack, 'con': con, 'status': status, 'message': message, 'nit': iteration, "success": status == 0}
return OptimizeResult(sol) |