EVOLUTION-MANAGER

Edit File: _minimize.py

""" Unified interfaces to minimization algorithms. Functions --------- - minimize : minimization of a function of several variables. - minimize_scalar : minimization of a function of one variable. """ from __future__ import division, print_function, absolute_import __all__ = ['minimize', 'minimize_scalar'] from warnings import warn import numpy as np from scipy._lib.six import callable # unconstrained minimization from .optimize import (_minimize_neldermead, _minimize_powell, _minimize_cg, _minimize_bfgs, _minimize_newtoncg, _minimize_scalar_brent, _minimize_scalar_bounded, _minimize_scalar_golden, MemoizeJac) from ._trustregion_dogleg import _minimize_dogleg from ._trustregion_ncg import _minimize_trust_ncg from ._trustregion_krylov import _minimize_trust_krylov from ._trustregion_exact import _minimize_trustregion_exact from ._trustregion_constr import _minimize_trustregion_constr # constrained minimization from .lbfgsb import _minimize_lbfgsb from .tnc import _minimize_tnc from .cobyla import _minimize_cobyla from .slsqp import _minimize_slsqp from ._constraints import (old_bound_to_new, new_bounds_to_old, old_constraint_to_new, new_constraint_to_old, NonlinearConstraint, LinearConstraint, Bounds) MINIMIZE_METHODS = ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg', 'l-bfgs-b', 'tnc', 'cobyla', 'slsqp', 'trust-constr', 'dogleg', 'trust-ncg', 'trust-exact', 'trust-krylov'] def minimize(fun, x0, args=(), method=None, jac=None, hess=None, hessp=None, bounds=None, constraints=(), tol=None, callback=None, options=None): """Minimization of scalar function of one or more variables. Parameters ---------- fun : callable The objective function to be minimized. ``fun(x, *args) -> float`` where x is an 1-D array with shape (n,) and `args` is a tuple of the fixed parameters needed to completely specify the function. x0 : ndarray, shape (n,) Initial guess. Array of real elements of size (n,), where 'n' is the number of independent variables. args : tuple, optional Extra arguments passed to the objective function and its derivatives (`fun`, `jac` and `hess` functions). method : str or callable, optional Type of solver. Should be one of - 'Nelder-Mead' :ref:`(see here) <optimize.minimize-neldermead>` - 'Powell' :ref:`(see here) <optimize.minimize-powell>` - 'CG' :ref:`(see here) <optimize.minimize-cg>` - 'BFGS' :ref:`(see here) <optimize.minimize-bfgs>` - 'Newton-CG' :ref:`(see here) <optimize.minimize-newtoncg>` - 'L-BFGS-B' :ref:`(see here) <optimize.minimize-lbfgsb>` - 'TNC' :ref:`(see here) <optimize.minimize-tnc>` - 'COBYLA' :ref:`(see here) <optimize.minimize-cobyla>` - 'SLSQP' :ref:`(see here) <optimize.minimize-slsqp>` - 'trust-constr':ref:`(see here) <optimize.minimize-trustconstr>` - 'dogleg' :ref:`(see here) <optimize.minimize-dogleg>` - 'trust-ncg' :ref:`(see here) <optimize.minimize-trustncg>` - 'trust-exact' :ref:`(see here) <optimize.minimize-trustexact>` - 'trust-krylov' :ref:`(see here) <optimize.minimize-trustkrylov>` - custom - a callable object (added in version 0.14.0), see below for description. If not given, chosen to be one of ``BFGS``, ``L-BFGS-B``, ``SLSQP``, depending if the problem has constraints or bounds. jac : {callable, '2-point', '3-point', 'cs', bool}, optional Method for computing the gradient vector. Only for CG, BFGS, Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg, trust-krylov, trust-exact and trust-constr. If it is a callable, it should be a function that returns the gradient vector: ``jac(x, *args) -> array_like, shape (n,)`` where x is an array with shape (n,) and `args` is a tuple with the fixed parameters. Alternatively, the keywords {'2-point', '3-point', 'cs'} select a finite difference scheme for numerical estimation of the gradient. Options '3-point' and 'cs' are available only to 'trust-constr'. If `jac` is a Boolean and is True, `fun` is assumed to return the gradient along with the objective function. If False, the gradient will be estimated using '2-point' finite difference estimation. hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy}, optional Method for computing the Hessian matrix. Only for Newton-CG, dogleg, trust-ncg, trust-krylov, trust-exact and trust-constr. If it is callable, it should return the Hessian matrix: ``hess(x, *args) -> {LinearOperator, spmatrix, array}, (n, n)`` where x is a (n,) ndarray and `args` is a tuple with the fixed parameters. LinearOperator and sparse matrix returns are allowed only for 'trust-constr' method. Alternatively, the keywords {'2-point', '3-point', 'cs'} select a finite difference scheme for numerical estimation. Or, objects implementing `HessianUpdateStrategy` interface can be used to approximate the Hessian. Available quasi-Newton methods implementing this interface are: - `BFGS`; - `SR1`. Whenever the gradient is estimated via finite-differences, the Hessian cannot be estimated with options {'2-point', '3-point', 'cs'} and needs to be estimated using one of the quasi-Newton strategies. Finite-difference options {'2-point', '3-point', 'cs'} and `HessianUpdateStrategy` are available only for 'trust-constr' method. hessp : callable, optional Hessian of objective function times an arbitrary vector p. Only for Newton-CG, trust-ncg, trust-krylov, trust-constr. Only one of `hessp` or `hess` needs to be given. If `hess` is provided, then `hessp` will be ignored. `hessp` must compute the Hessian times an arbitrary vector: ``hessp(x, p, *args) -> ndarray shape (n,)`` where x is a (n,) ndarray, p is an arbitrary vector with dimension (n,) and `args` is a tuple with the fixed parameters. bounds : sequence or `Bounds`, optional Bounds on variables for L-BFGS-B, TNC, SLSQP and trust-constr methods. There are two ways to specify the bounds: 1. Instance of `Bounds` class. 2. Sequence of ``(min, max)`` pairs for each element in `x`. None is used to specify no bound. constraints : {Constraint, dict} or List of {Constraint, dict}, optional Constraints definition (only for COBYLA, SLSQP and trust-constr). Constraints for 'trust-constr' are defined as a single object or a list of objects specifying constraints to the optimization problem. Available constraints are: - `LinearConstraint` - `NonlinearConstraint` Constraints for COBYLA, SLSQP are defined as a list of dictionaries. Each dictionary with fields: type : str Constraint type: 'eq' for equality, 'ineq' for inequality. fun : callable The function defining the constraint. jac : callable, optional The Jacobian of `fun` (only for SLSQP). args : sequence, optional Extra arguments to be passed to the function and Jacobian. Equality constraint means that the constraint function result is to be zero whereas inequality means that it is to be non-negative. Note that COBYLA only supports inequality constraints. tol : float, optional Tolerance for termination. For detailed control, use solver-specific options. options : dict, optional A dictionary of solver options. All methods accept the following generic options: maxiter : int Maximum number of iterations to perform. Depending on the method each iteration may use several function evaluations. disp : bool Set to True to print convergence messages. For method-specific options, see :func:`show_options()`. callback : callable, optional Called after each iteration. For 'trust-constr' it is a callable with the signature: ``callback(xk, OptimizeResult state) -> bool`` where ``xk`` is the current parameter vector. and ``state`` is an `OptimizeResult` object, with the same fields as the ones from the return. If callback returns True the algorithm execution is terminated. For all the other methods, the signature is: ``callback(xk)`` where ``xk`` is the current parameter vector. Returns ------- res : OptimizeResult The optimization result represented as a ``OptimizeResult`` object. Important attributes are: ``x`` the solution array, ``success`` a Boolean flag indicating if the optimizer exited successfully and ``message`` which describes the cause of the termination. See `OptimizeResult` for a description of other attributes. See also -------- minimize_scalar : Interface to minimization algorithms for scalar univariate functions 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 *BFGS*. **Unconstrained minimization** Method :ref:`Nelder-Mead <optimize.minimize-neldermead>` uses the Simplex algorithm [1]_, [2]_. This algorithm is robust in many applications. However, if numerical computation of derivative can be trusted, other algorithms using the first and/or second derivatives information might be preferred for their better performance in general. Method :ref:`Powell <optimize.minimize-powell>` is a modification of Powell's method [3]_, [4]_ which is a conjugate direction method. It performs sequential one-dimensional minimizations along each vector of the directions set (`direc` field in `options` and `info`), which is updated at each iteration of the main minimization loop. The function need not be differentiable, and no derivatives are taken. Method :ref:`CG <optimize.minimize-cg>` uses a nonlinear conjugate gradient algorithm by Polak and Ribiere, a variant of the Fletcher-Reeves method described in [5]_ pp. 120-122. Only the first derivatives are used. Method :ref:`BFGS <optimize.minimize-bfgs>` uses the quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [5]_ pp. 136. It uses the first derivatives only. BFGS has proven good performance even for non-smooth optimizations. This method also returns an approximation of the Hessian inverse, stored as `hess_inv` in the OptimizeResult object. Method :ref:`Newton-CG <optimize.minimize-newtoncg>` uses a Newton-CG algorithm [5]_ pp. 168 (also known as the truncated Newton method). It uses a CG method to the compute the search direction. See also *TNC* method for a box-constrained minimization with a similar algorithm. Suitable for large-scale problems. Method :ref:`dogleg <optimize.minimize-dogleg>` uses the dog-leg trust-region algorithm [5]_ for unconstrained minimization. This algorithm requires the gradient and Hessian; furthermore the Hessian is required to be positive definite. Method :ref:`trust-ncg <optimize.minimize-trustncg>` uses the Newton conjugate gradient trust-region algorithm [5]_ for unconstrained minimization. This algorithm requires the gradient and either the Hessian or a function that computes the product of the Hessian with a given vector. Suitable for large-scale problems. Method :ref:`trust-krylov <optimize.minimize-trustkrylov>` uses the Newton GLTR trust-region algorithm [14]_, [15]_ for unconstrained minimization. This algorithm requires the gradient and either the Hessian or a function that computes the product of the Hessian with a given vector. Suitable for large-scale problems. On indefinite problems it requires usually less iterations than the `trust-ncg` method and is recommended for medium and large-scale problems. Method :ref:`trust-exact <optimize.minimize-trustexact>` is a trust-region method for unconstrained minimization in which quadratic subproblems are solved almost exactly [13]_. This algorithm requires the gradient and the Hessian (which is *not* required to be positive definite). It is, in many situations, the Newton method to converge in fewer iteraction and the most recommended for small and medium-size problems. **Bound-Constrained minimization** Method :ref:`L-BFGS-B <optimize.minimize-lbfgsb>` uses the L-BFGS-B algorithm [6]_, [7]_ for bound constrained minimization. Method :ref:`TNC <optimize.minimize-tnc>` uses a truncated Newton algorithm [5]_, [8]_ to minimize a function with variables subject to bounds. This algorithm uses gradient information; it is also called Newton Conjugate-Gradient. It differs from the *Newton-CG* method described above as it wraps a C implementation and allows each variable to be given upper and lower bounds. **Constrained Minimization** Method :ref:`COBYLA <optimize.minimize-cobyla>` uses the Constrained Optimization BY Linear Approximation (COBYLA) method [9]_, [10]_, [11]_. The algorithm is based on linear approximations to the objective function and each constraint. The method wraps a FORTRAN implementation of the algorithm. The constraints functions 'fun' may return either a single number or an array or list of numbers. Method :ref:`SLSQP <optimize.minimize-slsqp>` uses Sequential Least SQuares Programming to minimize a function of several variables with any combination of bounds, equality and inequality constraints. The method wraps the SLSQP Optimization subroutine originally implemented by Dieter Kraft [12]_. Note that the wrapper handles infinite values in bounds by converting them into large floating values. Method :ref:`trust-constr <optimize.minimize-trustconstr>` is a trust-region algorithm for constrained optimization. It swiches between two implementations depending on the problem definition. It is the most versatile constrained minimization algorithm implemented in SciPy and the most appropriate for large-scale problems. For equality constrained problems it is an implementation of Byrd-Omojokun Trust-Region SQP method described in [17]_ and in [5]_, p. 549. When inequality constraints are imposed as well, it swiches to the trust-region interior point method described in [16]_. This interior point algorithm, in turn, solves inequality constraints by introducing slack variables and solving a sequence of equality-constrained barrier problems for progressively smaller values of the barrier parameter. The previously described equality constrained SQP method is used to solve the subproblems with increasing levels of accuracy as the iterate gets closer to a solution. **Finite-Difference Options** For Method :ref:`trust-constr <optimize.minimize-trustconstr>` the gradient and the Hessian may be approximated using three finite-difference schemes: {'2-point', '3-point', 'cs'}. The scheme 'cs' is, potentially, the most accurate but it requires the function to correctly handles complex inputs and to be differentiable in the complex plane. The scheme '3-point' is more accurate than '2-point' but requires twice as much operations. **Custom minimizers** It may be useful to pass a custom minimization method, for example when using a frontend to this method such as `scipy.optimize.basinhopping` or a different library. You can simply pass a callable as the ``method`` parameter. The callable is called as ``method(fun, x0, args, **kwargs, **options)`` where ``kwargs`` corresponds to any other parameters passed to `minimize` (such as `callback`, `hess`, etc.), except the `options` dict, which has its contents also passed as `method` parameters pair by pair. Also, if `jac` has been passed as a bool type, `jac` and `fun` are mangled so that `fun` returns just the function values and `jac` is converted to a function returning the Jacobian. The method shall return an `OptimizeResult` object. The provided `method` callable must be able to accept (and possibly ignore) arbitrary parameters; the set of parameters accepted by `minimize` may expand in future versions and then these parameters will be passed to the method. You can find an example in the scipy.optimize tutorial. .. versionadded:: 0.11.0 References ---------- .. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function Minimization. The Computer Journal 7: 308-13. .. [2] Wright M H. 1996. Direct search methods: Once scorned, now respectable, in Numerical Analysis 1995: Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis (Eds. D F Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK. 191-208. .. [3] Powell, M J D. 1964. An efficient method for finding the minimum of a function of several variables without calculating derivatives. The Computer Journal 7: 155-162. .. [4] Press W, S A Teukolsky, W T Vetterling and B P Flannery. Numerical Recipes (any edition), Cambridge University Press. .. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization. Springer New York. .. [6] Byrd, R H and P Lu and J. Nocedal. 1995. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM Journal on Scientific and Statistical Computing 16 (5): 1190-1208. .. [7] Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization. ACM Transactions on Mathematical Software 23 (4): 550-560. .. [8] Nash, S G. Newton-Type Minimization Via the Lanczos Method. 1984. SIAM Journal of Numerical Analysis 21: 770-778. .. [9] Powell, M J D. A direct search optimization method that models the objective and constraint functions by linear interpolation. 1994. Advances in Optimization and Numerical Analysis, eds. S. Gomez and J-P Hennart, Kluwer Academic (Dordrecht), 51-67. .. [10] Powell M J D. Direct search algorithms for optimization calculations. 1998. Acta Numerica 7: 287-336. .. [11] Powell M J D. A view of algorithms for optimization without derivatives. 2007.Cambridge University Technical Report DAMTP 2007/NA03 .. [12] Kraft, D. A software package for sequential quadratic programming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace Center -- Institute for Flight Mechanics, Koln, Germany. .. [13] Conn, A. R., Gould, N. I., and Toint, P. L. Trust region methods. 2000. Siam. pp. 169-200. .. [14] F. Lenders, C. Kirches, A. Potschka: "trlib: A vector-free implementation of the GLTR method for iterative solution of the trust region problem", https://arxiv.org/abs/1611.04718 .. [15] N. Gould, S. Lucidi, M. Roma, P. Toint: "Solving the Trust-Region Subproblem using the Lanczos Method", SIAM J. Optim., 9(2), 504--525, (1999). .. [16] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. 1999. An interior point algorithm for large-scale nonlinear programming. SIAM Journal on Optimization 9.4: 877-900. .. [17] Lalee, Marucha, Jorge Nocedal, and Todd Plantega. 1998. On the implementation of an algorithm for large-scale equality constrained optimization. SIAM Journal on Optimization 8.3: 682-706. Examples -------- Let us consider the problem of minimizing the Rosenbrock function. This function (and its respective derivatives) is implemented in `rosen` (resp. `rosen_der`, `rosen_hess`) in the `scipy.optimize`. >>> from scipy.optimize import minimize, rosen, rosen_der A simple application of the *Nelder-Mead* method is: >>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2] >>> res = minimize(rosen, x0, method='Nelder-Mead', tol=1e-6) >>> res.x array([ 1., 1., 1., 1., 1.]) Now using the *BFGS* algorithm, using the first derivative and a few options: >>> res = minimize(rosen, x0, method='BFGS', jac=rosen_der, ... options={'gtol': 1e-6, 'disp': True}) Optimization terminated successfully. Current function value: 0.000000 Iterations: 26 Function evaluations: 31 Gradient evaluations: 31 >>> res.x array([ 1., 1., 1., 1., 1.]) >>> print(res.message) Optimization terminated successfully. >>> res.hess_inv array([[ 0.00749589, 0.01255155, 0.02396251, 0.04750988, 0.09495377], # may vary [ 0.01255155, 0.02510441, 0.04794055, 0.09502834, 0.18996269], [ 0.02396251, 0.04794055, 0.09631614, 0.19092151, 0.38165151], [ 0.04750988, 0.09502834, 0.19092151, 0.38341252, 0.7664427 ], [ 0.09495377, 0.18996269, 0.38165151, 0.7664427, 1.53713523]]) Next, consider a minimization problem with several constraints (namely Example 16.4 from [5]_). The objective function is: >>> fun = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2 There are three constraints defined as: >>> cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2}, ... {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6}, ... {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2}) And variables must be positive, hence the following bounds: >>> bnds = ((0, None), (0, None)) The optimization problem is solved using the SLSQP method as: >>> res = minimize(fun, (2, 0), method='SLSQP', bounds=bnds, ... constraints=cons) It should converge to the theoretical solution (1.4 ,1.7). """ x0 = np.asarray(x0) if x0.dtype.kind in np.typecodes["AllInteger"]: x0 = np.asarray(x0, dtype=float) if not isinstance(args, tuple): args = (args,) if method is None: # Select automatically if constraints: method = 'SLSQP' elif bounds is not None: method = 'L-BFGS-B' else: method = 'BFGS' if callable(method): meth = "_custom" else: meth = method.lower() if options is None: options = {} # check if optional parameters are supported by the selected method # - jac if meth in ('nelder-mead', 'powell', 'cobyla') and bool(jac): warn('Method %s does not use gradient information (jac).' % method, RuntimeWarning) # - hess if meth not in ('newton-cg', 'dogleg', 'trust-ncg', 'trust-constr', 'trust-krylov', 'trust-exact', '_custom') and hess is not None: warn('Method %s does not use Hessian information (hess).' % method, RuntimeWarning) # - hessp if meth not in ('newton-cg', 'dogleg', 'trust-ncg', 'trust-constr', 'trust-krylov', '_custom') \ and hessp is not None: warn('Method %s does not use Hessian-vector product ' 'information (hessp).' % method, RuntimeWarning) # - constraints or bounds if (meth in ('nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg', 'dogleg', 'trust-ncg') and (bounds is not None or np.any(constraints))): warn('Method %s cannot handle constraints nor bounds.' % method, RuntimeWarning) if meth in ('l-bfgs-b', 'tnc') and np.any(constraints): warn('Method %s cannot handle constraints.' % method, RuntimeWarning) if meth == 'cobyla' and bounds is not None: warn('Method %s cannot handle bounds.' % method, RuntimeWarning) # - callback if (meth in ('cobyla',) and callback is not None): warn('Method %s does not support callback.' % method, RuntimeWarning) # - return_all if (meth in ('l-bfgs-b', 'tnc', 'cobyla', 'slsqp') and options.get('return_all', False)): warn('Method %s does not support the return_all option.' % method, RuntimeWarning) # check gradient vector if meth == 'trust-constr': if type(jac) is bool: if jac: fun = MemoizeJac(fun) jac = fun.derivative else: jac = '2-point' elif jac is None: jac = '2-point' elif not callable(jac) and jac not in ('2-point', '3-point', 'cs'): raise ValueError("Unsupported jac definition.") else: if jac in ('2-point', '3-point', 'cs'): if jac in ('3-point', 'cs'): warn("Only 'trust-constr' method accept %s " "options for 'jac'. Using '2-point' instead." % jac) jac = None elif not callable(jac): if bool(jac): fun = MemoizeJac(fun) jac = fun.derivative else: jac = None # set default tolerances if tol is not None: options = dict(options) if meth == 'nelder-mead': options.setdefault('xatol', tol) options.setdefault('fatol', tol) if meth in ('newton-cg', 'powell', 'tnc'): options.setdefault('xtol', tol) if meth in ('powell', 'l-bfgs-b', 'tnc', 'slsqp'): options.setdefault('ftol', tol) if meth in ('bfgs', 'cg', 'l-bfgs-b', 'tnc', 'dogleg', 'trust-ncg', 'trust-exact', 'trust-krylov'): options.setdefault('gtol', tol) if meth in ('cobyla', '_custom'): options.setdefault('tol', tol) if meth == 'trust-constr': options.setdefault('xtol', tol) options.setdefault('gtol', tol) options.setdefault('barrier_tol', tol) if meth == '_custom': # custom method called before bounds and constraints are 'standardised' # custom method should be able to accept whatever bounds/constraints # are provided to it. return method(fun, x0, args=args, jac=jac, hess=hess, hessp=hessp, bounds=bounds, constraints=constraints, callback=callback, **options) if bounds is not None: bounds = standardize_bounds(bounds, x0, meth) if constraints is not None: constraints = standardize_constraints(constraints, x0, meth) if meth == 'nelder-mead': return _minimize_neldermead(fun, x0, args, callback, **options) elif meth == 'powell': return _minimize_powell(fun, x0, args, callback, **options) elif meth == 'cg': return _minimize_cg(fun, x0, args, jac, callback, **options) elif meth == 'bfgs': return _minimize_bfgs(fun, x0, args, jac, callback, **options) elif meth == 'newton-cg': return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback, **options) elif meth == 'l-bfgs-b': return _minimize_lbfgsb(fun, x0, args, jac, bounds, callback=callback, **options) elif meth == 'tnc': return _minimize_tnc(fun, x0, args, jac, bounds, callback=callback, **options) elif meth == 'cobyla': return _minimize_cobyla(fun, x0, args, constraints, **options) elif meth == 'slsqp': return _minimize_slsqp(fun, x0, args, jac, bounds, constraints, callback=callback, **options) elif meth == 'trust-constr': return _minimize_trustregion_constr(fun, x0, args, jac, hess, hessp, bounds, constraints, callback=callback, **options) elif meth == 'dogleg': return _minimize_dogleg(fun, x0, args, jac, hess, callback=callback, **options) elif meth == 'trust-ncg': return _minimize_trust_ncg(fun, x0, args, jac, hess, hessp, callback=callback, **options) elif meth == 'trust-krylov': return _minimize_trust_krylov(fun, x0, args, jac, hess, hessp, callback=callback, **options) elif meth == 'trust-exact': return _minimize_trustregion_exact(fun, x0, args, jac, hess, callback=callback, **options) else: raise ValueError('Unknown solver %s' % method) def minimize_scalar(fun, bracket=None, bounds=None, args=(), method='brent', tol=None, options=None): """Minimization of scalar function of one variable. Parameters ---------- fun : callable Objective function. Scalar function, must return a scalar. bracket : sequence, optional For methods 'brent' and 'golden', `bracket` defines the bracketing interval and can either have three items ``(a, b, c)`` so that ``a < b < c`` and ``fun(b) < fun(a), fun(c)`` or two items ``a`` and ``c`` which are assumed to be a starting interval for a downhill bracket search (see `bracket`); it doesn't always mean that the obtained solution will satisfy ``a <= x <= c``. bounds : sequence, optional For method 'bounded', `bounds` is mandatory and must have two items corresponding to the optimization bounds. args : tuple, optional Extra arguments passed to the objective function. method : str or callable, optional Type of solver. Should be one of: - 'Brent' :ref:`(see here) <optimize.minimize_scalar-brent>` - 'Bounded' :ref:`(see here) <optimize.minimize_scalar-bounded>` - 'Golden' :ref:`(see here) <optimize.minimize_scalar-golden>` - custom - a callable object (added in version 0.14.0), see below tol : float, optional Tolerance for termination. For detailed control, use solver-specific options. options : dict, optional A dictionary of solver options. maxiter : int Maximum number of iterations to perform. disp : bool Set to True to print convergence messages. See :func:`show_options()` for solver-specific options. Returns ------- res : OptimizeResult The optimization result represented as a ``OptimizeResult`` object. Important attributes are: ``x`` the solution array, ``success`` a Boolean flag indicating if the optimizer exited successfully and ``message`` which describes the cause of the termination. See `OptimizeResult` for a description of other attributes. See also -------- minimize : Interface to minimization algorithms for scalar multivariate functions 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 *Brent*. Method :ref:`Brent <optimize.minimize_scalar-brent>` uses Brent's algorithm to find a local minimum. The algorithm uses inverse parabolic interpolation when possible to speed up convergence of the golden section method. Method :ref:`Golden <optimize.minimize_scalar-golden>` uses the golden section search technique. It uses analog of the bisection method to decrease the bracketed interval. It is usually preferable to use the *Brent* method. Method :ref:`Bounded <optimize.minimize_scalar-bounded>` can perform bounded minimization. It uses the Brent method to find a local minimum in the interval x1 < xopt < x2. **Custom minimizers** It may be useful to pass a custom minimization method, for example when using some library frontend to minimize_scalar. You can simply pass a callable as the ``method`` parameter. The callable is called as ``method(fun, args, **kwargs, **options)`` where ``kwargs`` corresponds to any other parameters passed to `minimize` (such as `bracket`, `tol`, etc.), except the `options` dict, which has its contents also passed as `method` parameters pair by pair. The method shall return an `OptimizeResult` object. The provided `method` callable must be able to accept (and possibly ignore) arbitrary parameters; the set of parameters accepted by `minimize` may expand in future versions and then these parameters will be passed to the method. You can find an example in the scipy.optimize tutorial. .. versionadded:: 0.11.0 Examples -------- Consider the problem of minimizing the following function. >>> def f(x): ... return (x - 2) * x * (x + 2)**2 Using the *Brent* method, we find the local minimum as: >>> from scipy.optimize import minimize_scalar >>> res = minimize_scalar(f) >>> res.x 1.28077640403 Using the *Bounded* method, we find a local minimum with specified bounds as: >>> res = minimize_scalar(f, bounds=(-3, -1), method='bounded') >>> res.x -2.0000002026 """ if not isinstance(args, tuple): args = (args,) if callable(method): meth = "_custom" else: meth = method.lower() if options is None: options = {} if tol is not None: options = dict(options) if meth == 'bounded' and 'xatol' not in options: warn("Method 'bounded' does not support relative tolerance in x; " "defaulting to absolute tolerance.", RuntimeWarning) options['xatol'] = tol elif meth == '_custom': options.setdefault('tol', tol) else: options.setdefault('xtol', tol) if meth == '_custom': return method(fun, args=args, bracket=bracket, bounds=bounds, **options) elif meth == 'brent': return _minimize_scalar_brent(fun, bracket, args, **options) elif meth == 'bounded': if bounds is None: raise ValueError('The `bounds` parameter is mandatory for ' 'method `bounded`.') # replace boolean "disp" option, if specified, by an integer value, as # expected by _minimize_scalar_bounded() disp = options.get('disp') if isinstance(disp, bool): options['disp'] = 2 * int(disp) return _minimize_scalar_bounded(fun, bounds, args, **options) elif meth == 'golden': return _minimize_scalar_golden(fun, bracket, args, **options) else: raise ValueError('Unknown solver %s' % method) def standardize_bounds(bounds, x0, meth): """Converts bounds to the form required by the solver.""" if meth == 'trust-constr': if not isinstance(bounds, Bounds): lb, ub = old_bound_to_new(bounds) bounds = Bounds(lb, ub) elif meth in ('l-bfgs-b', 'tnc', 'slsqp'): if isinstance(bounds, Bounds): bounds = new_bounds_to_old(bounds.lb, bounds.ub, x0.shape[0]) return bounds def standardize_constraints(constraints, x0, meth): """Converts constraints to the form required by the solver.""" all_constraint_types = (NonlinearConstraint, LinearConstraint, dict) new_constraint_types = all_constraint_types[:-1] if isinstance(constraints, all_constraint_types): constraints = [constraints] constraints = list(constraints) # ensure it's a mutable sequence if meth == 'trust-constr': for i, con in enumerate(constraints): if not isinstance(con, new_constraint_types): constraints[i] = old_constraint_to_new(i, con) else: # iterate over copy, changing original for i, con in enumerate(list(constraints)): if isinstance(con, new_constraint_types): old_constraints = new_constraint_to_old(con, x0) constraints[i] = old_constraints[0] constraints.extend(old_constraints[1:]) # appends 1 if present return constraints