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_constraints.py
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"""Constraints definition for minimize.""" import numpy as np from ._hessian_update_strategy import BFGS from ._differentiable_functions import ( VectorFunction, LinearVectorFunction, IdentityVectorFunction) from .optimize import OptimizeWarning from warnings import warn from numpy.testing import suppress_warnings from scipy.sparse import issparse def _arr_to_scalar(x): # If x is a numpy array, return x.item(). This will # fail if the array has more than one element. return x.item() if isinstance(x, np.ndarray) else x class NonlinearConstraint(object): """Nonlinear constraint on the variables. The constraint has the general inequality form:: lb <= fun(x) <= ub Here the vector of independent variables x is passed as ndarray of shape (n,) and ``fun`` returns a vector with m components. It is possible to use equal bounds to represent an equality constraint or infinite bounds to represent a one-sided constraint. Parameters ---------- fun : callable The function defining the constraint. The signature is ``fun(x) -> array_like, shape (m,)``. lb, ub : array_like Lower and upper bounds on the constraint. Each array must have the shape (m,) or be a scalar, in the latter case a bound will be the same for all components of the constraint. Use ``np.inf`` with an appropriate sign to specify a one-sided constraint. Set components of `lb` and `ub` equal to represent an equality constraint. Note that you can mix constraints of different types: interval, one-sided or equality, by setting different components of `lb` and `ub` as necessary. jac : {callable, '2-point', '3-point', 'cs'}, optional Method of computing the Jacobian matrix (an m-by-n matrix, where element (i, j) is the partial derivative of f[i] with respect to x[j]). The keywords {'2-point', '3-point', 'cs'} select a finite difference scheme for the numerical estimation. A callable must have the following signature: ``jac(x) -> {ndarray, sparse matrix}, shape (m, n)``. Default is '2-point'. hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy, None}, optional Method for computing the Hessian matrix. The keywords {'2-point', '3-point', 'cs'} select a finite difference scheme for numerical estimation. Alternatively, objects implementing `HessianUpdateStrategy` interface can be used to approximate the Hessian. Currently available implementations are: - `BFGS` (default option) - `SR1` A callable must return the Hessian matrix of ``dot(fun, v)`` and must have the following signature: ``hess(x, v) -> {LinearOperator, sparse matrix, array_like}, shape (n, n)``. Here ``v`` is ndarray with shape (m,) containing Lagrange multipliers. keep_feasible : array_like of bool, optional Whether to keep the constraint components feasible throughout iterations. A single value set this property for all components. Default is False. Has no effect for equality constraints. finite_diff_rel_step: None or array_like, optional Relative step size for the finite difference approximation. Default is None, which will select a reasonable value automatically depending on a finite difference scheme. finite_diff_jac_sparsity: {None, array_like, sparse matrix}, optional Defines the sparsity structure of the Jacobian matrix for finite difference estimation, its shape must be (m, n). If the Jacobian has only few non-zero elements in *each* row, providing the sparsity structure will greatly speed up the computations. A zero entry means that a corresponding element in the Jacobian is identically zero. If provided, forces the use of 'lsmr' trust-region solver. If None (default) then dense differencing will be used. Notes ----- Finite difference schemes {'2-point', '3-point', 'cs'} may be used for approximating either the Jacobian or the Hessian. We, however, do not allow its use for approximating both simultaneously. Hence whenever the Jacobian is estimated via finite-differences, we require the Hessian to be estimated using one of the quasi-Newton strategies. The scheme 'cs' is potentially the most accurate, but requires the function to correctly handles complex inputs and be analytically continuable to the complex plane. The scheme '3-point' is more accurate than '2-point' but requires twice as many operations. Examples -------- Constrain ``x[0] < sin(x[1]) + 1.9`` >>> from scipy.optimize import NonlinearConstraint >>> con = lambda x: x[0] - np.sin(x[1]) >>> nlc = NonlinearConstraint(con, -np.inf, 1.9) """ def __init__(self, fun, lb, ub, jac='2-point', hess=BFGS(), keep_feasible=False, finite_diff_rel_step=None, finite_diff_jac_sparsity=None): self.fun = fun self.lb = lb self.ub = ub self.finite_diff_rel_step = finite_diff_rel_step self.finite_diff_jac_sparsity = finite_diff_jac_sparsity self.jac = jac self.hess = hess self.keep_feasible = keep_feasible class LinearConstraint(object): """Linear constraint on the variables. The constraint has the general inequality form:: lb <= A.dot(x) <= ub Here the vector of independent variables x is passed as ndarray of shape (n,) and the matrix A has shape (m, n). It is possible to use equal bounds to represent an equality constraint or infinite bounds to represent a one-sided constraint. Parameters ---------- A : {array_like, sparse matrix}, shape (m, n) Matrix defining the constraint. lb, ub : array_like Lower and upper bounds on the constraint. Each array must have the shape (m,) or be a scalar, in the latter case a bound will be the same for all components of the constraint. Use ``np.inf`` with an appropriate sign to specify a one-sided constraint. Set components of `lb` and `ub` equal to represent an equality constraint. Note that you can mix constraints of different types: interval, one-sided or equality, by setting different components of `lb` and `ub` as necessary. keep_feasible : array_like of bool, optional Whether to keep the constraint components feasible throughout iterations. A single value set this property for all components. Default is False. Has no effect for equality constraints. """ def __init__(self, A, lb, ub, keep_feasible=False): self.A = A self.lb = lb self.ub = ub self.keep_feasible = keep_feasible class Bounds(object): """Bounds constraint on the variables. The constraint has the general inequality form:: lb <= x <= ub It is possible to use equal bounds to represent an equality constraint or infinite bounds to represent a one-sided constraint. Parameters ---------- lb, ub : array_like, optional Lower and upper bounds on independent variables. Each array must have the same size as x or be a scalar, in which case a bound will be the same for all the variables. Set components of `lb` and `ub` equal to fix a variable. Use ``np.inf`` with an appropriate sign to disable bounds on all or some variables. Note that you can mix constraints of different types: interval, one-sided or equality, by setting different components of `lb` and `ub` as necessary. keep_feasible : array_like of bool, optional Whether to keep the constraint components feasible throughout iterations. A single value set this property for all components. Default is False. Has no effect for equality constraints. """ def __init__(self, lb, ub, keep_feasible=False): self.lb = lb self.ub = ub self.keep_feasible = keep_feasible def __repr__(self): if np.any(self.keep_feasible): return "{}({!r}, {!r}, keep_feasible={!r})".format(type(self).__name__, self.lb, self.ub, self.keep_feasible) else: return "{}({!r}, {!r})".format(type(self).__name__, self.lb, self.ub) class PreparedConstraint(object): """Constraint prepared from a user defined constraint. On creation it will check whether a constraint definition is valid and the initial point is feasible. If created successfully, it will contain the attributes listed below. Parameters ---------- constraint : {NonlinearConstraint, LinearConstraint`, Bounds} Constraint to check and prepare. x0 : array_like Initial vector of independent variables. sparse_jacobian : bool or None, optional If bool, then the Jacobian of the constraint will be converted to the corresponded format if necessary. If None (default), such conversion is not made. finite_diff_bounds : 2-tuple, optional Lower and upper bounds on the independent variables for the finite difference approximation, if applicable. Defaults to no bounds. Attributes ---------- fun : {VectorFunction, LinearVectorFunction, IdentityVectorFunction} Function defining the constraint wrapped by one of the convenience classes. bounds : 2-tuple Contains lower and upper bounds for the constraints --- lb and ub. These are converted to ndarray and have a size equal to the number of the constraints. keep_feasible : ndarray Array indicating which components must be kept feasible with a size equal to the number of the constraints. """ def __init__(self, constraint, x0, sparse_jacobian=None, finite_diff_bounds=(-np.inf, np.inf)): if isinstance(constraint, NonlinearConstraint): fun = VectorFunction(constraint.fun, x0, constraint.jac, constraint.hess, constraint.finite_diff_rel_step, constraint.finite_diff_jac_sparsity, finite_diff_bounds, sparse_jacobian) elif isinstance(constraint, LinearConstraint): fun = LinearVectorFunction(constraint.A, x0, sparse_jacobian) elif isinstance(constraint, Bounds): fun = IdentityVectorFunction(x0, sparse_jacobian) else: raise ValueError("`constraint` of an unknown type is passed.") m = fun.m lb = np.asarray(constraint.lb, dtype=float) ub = np.asarray(constraint.ub, dtype=float) if lb.ndim == 0: lb = np.resize(lb, m) if ub.ndim == 0: ub = np.resize(ub, m) keep_feasible = np.asarray(constraint.keep_feasible, dtype=bool) if keep_feasible.ndim == 0: keep_feasible = np.resize(keep_feasible, m) if keep_feasible.shape != (m,): raise ValueError("`keep_feasible` has a wrong shape.") mask = keep_feasible & (lb != ub) f0 = fun.f if np.any(f0[mask] < lb[mask]) or np.any(f0[mask] > ub[mask]): raise ValueError("`x0` is infeasible with respect to some " "inequality constraint with `keep_feasible` " "set to True.") self.fun = fun self.bounds = (lb, ub) self.keep_feasible = keep_feasible def violation(self, x): """How much the constraint is exceeded by. Parameters ---------- x : array-like Vector of independent variables Returns ------- excess : array-like How much the constraint is exceeded by, for each of the constraints specified by `PreparedConstraint.fun`. """ with suppress_warnings() as sup: sup.filter(UserWarning) ev = self.fun.fun(np.asarray(x)) excess_lb = np.maximum(self.bounds[0] - ev, 0) excess_ub = np.maximum(ev - self.bounds[1], 0) return excess_lb + excess_ub def new_bounds_to_old(lb, ub, n): """Convert the new bounds representation to the old one. The new representation is a tuple (lb, ub) and the old one is a list containing n tuples, ith containing lower and upper bound on a ith variable. If any of the entries in lb/ub are -np.inf/np.inf they are replaced by None. """ lb = np.asarray(lb) ub = np.asarray(ub) if lb.ndim == 0: lb = np.resize(lb, n) if ub.ndim == 0: ub = np.resize(ub, n) lb = [float(x) if x > -np.inf else None for x in lb] ub = [float(x) if x < np.inf else None for x in ub] return list(zip(lb, ub)) def old_bound_to_new(bounds): """Convert the old bounds representation to the new one. The new representation is a tuple (lb, ub) and the old one is a list containing n tuples, ith containing lower and upper bound on a ith variable. If any of the entries in lb/ub are None they are replaced by -np.inf/np.inf. """ lb, ub = zip(*bounds) # Convert occurrences of None to -inf or inf, and replace occurrences of # any numpy array x with x.item(). Then wrap the results in numpy arrays. lb = np.array([float(_arr_to_scalar(x)) if x is not None else -np.inf for x in lb]) ub = np.array([float(_arr_to_scalar(x)) if x is not None else np.inf for x in ub]) return lb, ub def strict_bounds(lb, ub, keep_feasible, n_vars): """Remove bounds which are not asked to be kept feasible.""" strict_lb = np.resize(lb, n_vars).astype(float) strict_ub = np.resize(ub, n_vars).astype(float) keep_feasible = np.resize(keep_feasible, n_vars) strict_lb[~keep_feasible] = -np.inf strict_ub[~keep_feasible] = np.inf return strict_lb, strict_ub def new_constraint_to_old(con, x0): """ Converts new-style constraint objects to old-style constraint dictionaries. """ if isinstance(con, NonlinearConstraint): if (con.finite_diff_jac_sparsity is not None or con.finite_diff_rel_step is not None or not isinstance(con.hess, BFGS) or # misses user specified BFGS con.keep_feasible): warn("Constraint options `finite_diff_jac_sparsity`, " "`finite_diff_rel_step`, `keep_feasible`, and `hess`" "are ignored by this method.", OptimizeWarning) fun = con.fun if callable(con.jac): jac = con.jac else: jac = None else: # LinearConstraint if con.keep_feasible: warn("Constraint option `keep_feasible` is ignored by this " "method.", OptimizeWarning) A = con.A if issparse(A): A = A.todense() fun = lambda x: np.dot(A, x) jac = lambda x: A # FIXME: when bugs in VectorFunction/LinearVectorFunction are worked out, # use pcon.fun.fun and pcon.fun.jac. Until then, get fun/jac above. pcon = PreparedConstraint(con, x0) lb, ub = pcon.bounds i_eq = lb == ub i_bound_below = np.logical_xor(lb != -np.inf, i_eq) i_bound_above = np.logical_xor(ub != np.inf, i_eq) i_unbounded = np.logical_and(lb == -np.inf, ub == np.inf) if np.any(i_unbounded): warn("At least one constraint is unbounded above and below. Such " "constraints are ignored.", OptimizeWarning) ceq = [] if np.any(i_eq): def f_eq(x): y = np.array(fun(x)).flatten() return y[i_eq] - lb[i_eq] ceq = [{"type": "eq", "fun": f_eq}] if jac is not None: def j_eq(x): dy = jac(x) if issparse(dy): dy = dy.todense() dy = np.atleast_2d(dy) return dy[i_eq, :] ceq[0]["jac"] = j_eq cineq = [] n_bound_below = np.sum(i_bound_below) n_bound_above = np.sum(i_bound_above) if n_bound_below + n_bound_above: def f_ineq(x): y = np.zeros(n_bound_below + n_bound_above) y_all = np.array(fun(x)).flatten() y[:n_bound_below] = y_all[i_bound_below] - lb[i_bound_below] y[n_bound_below:] = -(y_all[i_bound_above] - ub[i_bound_above]) return y cineq = [{"type": "ineq", "fun": f_ineq}] if jac is not None: def j_ineq(x): dy = np.zeros((n_bound_below + n_bound_above, len(x0))) dy_all = jac(x) if issparse(dy_all): dy_all = dy_all.todense() dy_all = np.atleast_2d(dy_all) dy[:n_bound_below, :] = dy_all[i_bound_below] dy[n_bound_below:, :] = -dy_all[i_bound_above] return dy cineq[0]["jac"] = j_ineq old_constraints = ceq + cineq if len(old_constraints) > 1: warn("Equality and inequality constraints are specified in the same " "element of the constraint list. For efficient use with this " "method, equality and inequality constraints should be specified " "in separate elements of the constraint list. ", OptimizeWarning) return old_constraints def old_constraint_to_new(ic, con): """ Converts old-style constraint dictionaries to new-style constraint objects. """ # check type try: ctype = con['type'].lower() except KeyError: raise KeyError('Constraint %d has no type defined.' % ic) except TypeError: raise TypeError('Constraints must be a sequence of dictionaries.') except AttributeError: raise TypeError("Constraint's type must be a string.") else: if ctype not in ['eq', 'ineq']: raise ValueError("Unknown constraint type '%s'." % con['type']) if 'fun' not in con: raise ValueError('Constraint %d has no function defined.' % ic) lb = 0 if ctype == 'eq': ub = 0 else: ub = np.inf jac = '2-point' if 'args' in con: args = con['args'] fun = lambda x: con['fun'](x, *args) if 'jac' in con: jac = lambda x: con['jac'](x, *args) else: fun = con['fun'] if 'jac' in con: jac = con['jac'] return NonlinearConstraint(fun, lb, ub, jac)