""" A top-level linear programming interface. Currently this interface solves linear programming problems via the Simplex and Interior-Point methods. .. versionadded:: 0.15.0 Functions --------- .. autosummary:: :toctree: generated/ linprog linprog_verbose_callback linprog_terse_callback """ from __future__ import division, print_function, absolute_import import numpy as np from .optimize import OptimizeResult from ._linprog_ip import _linprog_ip from ._linprog_simplex import _linprog_simplex from ._linprog_util import ( _parse_linprog, _presolve, _get_Abc, _postprocess ) __all__ = ['linprog', 'linprog_verbose_callback', 'linprog_terse_callback'] __docformat__ = "restructuredtext en" def linprog_verbose_callback(res): """ A sample callback function demonstrating the linprog callback interface. This callback produces detailed output to sys.stdout before each iteration and after the final iteration of the simplex algorithm. Parameters ---------- res : A `scipy.optimize.OptimizeResult` consisting of the following fields: x : 1D array The independent variable vector which optimizes the linear programming problem. fun : float Value of the objective function. success : bool True if the algorithm succeeded in finding an optimal solution. slack : 1D array 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. con : 1D array The (nominally zero) residuals of the equality constraints, that is, ``b - A_eq @ x`` phase : int The phase of the optimization being executed. In phase 1 a basic feasible solution is sought and the T has an additional row representing an alternate objective function. 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. """ x = res['x'] fun = res['fun'] success = res['success'] phase = res['phase'] status = res['status'] nit = res['nit'] message = res['message'] complete = res['complete'] saved_printoptions = np.get_printoptions() np.set_printoptions(linewidth=500, formatter={'float': lambda x: "{0: 12.4f}".format(x)}) if status: print('--------- Simplex Early Exit -------\n'.format(nit)) print('The simplex method exited early with status {0:d}'.format(status)) print(message) elif complete: print('--------- Simplex Complete --------\n') print('Iterations required: {}'.format(nit)) else: print('--------- Iteration {0:d} ---------\n'.format(nit)) if nit > 0: if phase == 1: print('Current Pseudo-Objective Value:') else: print('Current Objective Value:') print('f = ', fun) print() print('Current Solution Vector:') print('x = ', x) print() np.set_printoptions(**saved_printoptions) def linprog_terse_callback(res): """ A sample callback function demonstrating the linprog callback interface. This callback produces brief output to sys.stdout before each iteration and after the final iteration of the simplex algorithm. Parameters ---------- res : A `scipy.optimize.OptimizeResult` consisting of the following fields: x : 1D array The independent variable vector which optimizes the linear programming problem. fun : float Value of the objective function. success : bool True if the algorithm succeeded in finding an optimal solution. slack : 1D array 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. con : 1D array The (nominally zero) residuals of the equality constraints, that is, ``b - A_eq @ x`` phase : int The phase of the optimization being executed. In phase 1 a basic feasible solution is sought and the T has an additional row representing an alternate objective function. 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. """ nit = res['nit'] x = res['x'] if nit == 0: print("Iter: X:") print("{0: <5d} ".format(nit), end="") print(x) def linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='simplex', callback=None, options=None): """ Minimize a linear objective function subject to linear equality and inequality constraints. Linear Programming is intended to solve the following problem form: Minimize:: c @ x Subject to:: A_ub @ x <= b_ub A_eq @ x == b_eq lb <= x <= ub where ``lb = 0`` and ``ub = None`` unless set in ``bounds``. Parameters ---------- c : 1D array Coefficients of the linear objective function to be minimized. A_ub : 2D array, optional 2D array such that ``A_ub @ x`` gives the values of the upper-bound inequality constraints at ``x``. b_ub : 1D array, optional 1D array of values representing the upper-bound of each inequality constraint (row) in ``A_ub``. A_eq : 2D, optional 2D array such that ``A_eq @ x`` gives the values of the equality constraints at ``x``. b_eq : 1D array, optional 1D 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. method : str, optional Type of solver. :ref:`'simplex' ` and :ref:`'interior-point' ` are supported. callback : callable, optional (simplex only) If a callback function is provided, it will be called within each iteration of the simplex algorithm. The callback must require a `scipy.optimize.OptimizeResult` consisting of the following fields: x : 1D array The independent variable vector which optimizes the linear programming problem. fun : float Value of the objective function. success : bool True if the algorithm succeeded in finding an optimal solution. slack : 1D array The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, the corresponding constraint is active. con : 1D array The (nominally zero) residuals of the equality constraints that is, ``b - A_eq @ x`` phase : int The phase of the optimization being executed. In phase 1 a basic feasible solution is sought and the T has an additional row representing an alternate objective function. 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. options : dict, optional A dictionary of solver options. All methods accept the following generic options: maxiter : int Maximum number of iterations to perform. disp : bool Set to True to print convergence messages. For method-specific options, see :func:`show_options('linprog')`. Returns ------- res : OptimizeResult A :class:`scipy.optimize.OptimizeResult` consisting of the fields: x : 1D array The independent variable vector which optimizes the linear programming problem. fun : float Value of the objective function. slack : 1D array 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. con : 1D array The (nominally zero) residuals of the equality constraints, that is, ``b - A_eq @ x`` 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. See Also -------- show_options : Additional options accepted by the solvers Notes ----- This section describes the available solvers that can be selected by the 'method' parameter. The default method is :ref:`Simplex `. :ref:`Interior point ` is also available. Method *simplex* uses the simplex algorithm (as it relates to linear programming, NOT the Nelder-Mead simplex) [1]_, [2]_. This algorithm should be reasonably reliable and fast for small problems. .. versionadded:: 0.15.0 Method *interior-point* uses the primal-dual path following algorithm as outlined in [4]_. This algorithm is intended to provide a faster and more reliable alternative to *simplex*, especially for large, sparse problems. Note, however, that the solution returned may be slightly less accurate than that of the simplex method and may not correspond with a vertex of the polytope defined by the constraints. Before applying either method a presolve procedure based on [8]_ 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 prescribed 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 [8]_ 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 [5]_ 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. References ---------- .. [1] Dantzig, George B., Linear programming and extensions. Rand Corporation Research Study Princeton Univ. Press, Princeton, NJ, 1963 .. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to Mathematical Programming", McGraw-Hill, Chapter 4. .. [3] Bland, Robert G. New finite pivoting rules for the simplex method. Mathematics of Operations Research (2), 1977: pp. 103-107. .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232. .. [5] Andersen, Erling D. "Finding all linearly dependent rows in large-scale linear programming." Optimization Methods and Software 6.3 (1995): 219-227. .. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear Programming based on Newton's Method." Unpublished Course Notes, March 2004. Available 2/25/2017 at https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf .. [7] 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 .. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear programming." Mathematical Programming 71.2 (1995): 221-245. .. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear programming." Athena Scientific 1 (1997): 997. .. [10] Andersen, Erling D., et al. Implementation of interior point methods for large scale linear programming. HEC/Universite de Geneve, 1996. Examples -------- Consider the following problem: Minimize:: f = -1x[0] + 4x[1] Subject to:: -3x[0] + 1x[1] <= 6 1x[0] + 2x[1] <= 4 x[1] >= -3 -inf <= x[0] <= inf This problem deviates from the standard linear programming problem. In standard form, linear programming problems assume the variables x are non-negative. Since the problem variables don't have the standard bounds of ``(0, None)``, the variable bounds must be set using ``bounds`` explicitly. There are two upper-bound constraints, which can be expressed as dot(A_ub, x) <= b_ub The input for this problem is as follows: >>> c = [-1, 4] >>> A = [[-3, 1], [1, 2]] >>> b = [6, 4] >>> x0_bounds = (None, None) >>> x1_bounds = (-3, None) >>> from scipy.optimize import linprog >>> res = linprog(c, A_ub=A, b_ub=b, bounds=(x0_bounds, x1_bounds), ... options={"disp": True}) Optimization terminated successfully. Current function value: -22.000000 Iterations: 5 # may vary >>> print(res) con: array([], dtype=float64) fun: -22.0 message: 'Optimization terminated successfully.' nit: 5 # may vary slack: array([39., 0.]) # may vary status: 0 success: True x: array([10., -3.]) """ meth = method.lower() default_tol = 1e-12 if meth == 'simplex' else 1e-9 c, A_ub, b_ub, A_eq, b_eq, bounds, solver_options = _parse_linprog( c, A_ub, b_ub, A_eq, b_eq, bounds, options) tol = solver_options.get('tol', default_tol) iteration = 0 complete = False # will become True if solved in presolve undo = [] # 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 solver_options.pop('presolve', True): rr = solver_options.pop('rr', 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, tol) if not complete: A, b, c, c0 = _get_Abc(c, c0, A_ub, b_ub, A_eq, b_eq, bounds, undo) T_o = (c_o, A_ub_o, b_ub_o, A_eq_o, b_eq_o, bounds, undo) if meth == 'simplex': x, status, message, iteration = _linprog_simplex( c, c0=c0, A=A, b=b, callback=callback, _T_o=T_o, **solver_options) elif meth == 'interior-point': x, status, message, iteration = _linprog_ip( c, c0=c0, A=A, b=b, callback=callback, **solver_options) else: raise ValueError('Unknown solver %s' % method) # Eliminate artificial variables, re-introduce presolved variables, etc... # need modified bounds here to translate variables appropriately disp = solver_options.get('disp', False) 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, iteration, disp) sol = { 'x': x, 'fun': fun, 'slack': slack, 'con': con, 'status': status, 'message': message, 'nit': iteration, 'success': status == 0} return OptimizeResult(sol)