""" Unified interfaces to minimization algorithms. Functions --------- - minimize : minimization of a function of several variables. - minimize_scalar : minimization of a function of one variable. """ __all__ = ['minimize', 'minimize_scalar'] from warnings import warn from numpy import any # 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 anneal import _minimize_anneal # contrained minimization from lbfgsb import _minimize_lbfgsb from tnc import _minimize_tnc from cobyla import _minimize_cobyla from slsqp import _minimize_slsqp def minimize(fun, x0, args=(), method='BFGS', jac=None, hess=None, hessp=None, bounds=None, constraints=(), tol=None, callback=None, options=None): """ Minimization of scalar function of one or more variables. .. versionadded:: 0.11.0 Parameters ---------- fun : callable Objective function. x0 : ndarray Initial guess. args : tuple, optional Extra arguments passed to the objective function and its derivatives (Jacobian, Hessian). method : str, optional Type of solver. Should be one of - 'Nelder-Mead' - 'Powell' - 'CG' - 'BFGS' - 'Newton-CG' - 'Anneal' - 'L-BFGS-B' - 'TNC' - 'COBYLA' - 'SLSQP' jac : bool or callable, optional Jacobian of objective function. Only for CG, BFGS, Newton-CG. If `jac` is a Boolean and is True, `fun` is assumed to return the value of Jacobian along with the objective function. If False, the Jacobian will be estimated numerically. `jac` can also be a callable returning the Jacobian of the objective. In this case, it must accept the same arguments as `fun`. hess, hessp : callable, optional Hessian of objective function or Hessian of objective function times an arbitrary vector p. Only for Newton-CG. Only one of `hessp` or `hess` needs to be given. If `hess` is provided, then `hessp` will be ignored. If neither `hess` nor `hessp` is provided, then the hessian product will be approximated using finite differences on `jac`. `hessp` must compute the Hessian times an arbitrary vector. bounds : sequence, optional Bounds for variables (only for L-BFGS-B, TNC, COBYLA and SLSQP). ``(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. constraints : dict or sequence of dict, optional Constraints definition (only for COBYLA and SLSQP). Each constraint is defined in a 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. disp : bool Set to True to print convergence messages. For method-specific options, see `show_options('minimize', method)`. callback : callable, optional Called after each iteration, as ``callback(xk)``, where ``xk`` is the current parameter vector. Returns ------- res : Result The optimization result represented as a ``Result`` 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 `Result` for a description of other attributes. See also -------- minimize_scalar: Interface to minimization algorithms for scalar univariate functions. Notes ----- This section describes the available solvers that can be selected by the 'method' parameter. The default method is *BFGS*. **Unconstrained minimization** Method *Nelder-Mead* uses the Simplex algorithm [1]_, [2]_. This algorithm has been successful in many applications but other algorithms using the first and/or second derivatives information might be preferred for their better performances and robustness in general. Method *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 *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 *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 Method *Newton-CG* 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. Method *Anneal* uses simulated annealing, which is a probabilistic metaheuristic algorithm for global optimization. It uses no derivative information from the function being optimized. **Constrained minimization** Method *L-BFGS-B* uses the L-BFGS-B algorithm [6]_, [7]_ for bound constrained minimization. Method *TNC* uses a truncated Newton algorithm [5]_, [8]_ to minimize a function with variables subject to bounds. This algorithm is 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. Method *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. Method *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]_. 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. 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') >>> res.x [ 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: 52 Function evaluations: 64 Gradient evaluations: 64 >>> res.x [ 1. 1. 1. 1. 1.] >>> print res.message Optimization terminated successfully. >>> res.hess [[ 0.00749589 0.01255155 0.02396251 0.04750988 0.09495377] [ 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). """ 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', 'anneal', 'cobyla'] and bool(jac): warn('Method %s does not use gradient information (jac).' % method, RuntimeWarning) # - hess if meth != 'newton-cg' and hess is not None: warn('Method %s does not use Hessian information (hess).' % method, RuntimeWarning) # - constraints or bounds if (meth in ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg'] and (bounds is not None or any(constraints))): warn('Method %s cannot handle constraints nor bounds.' % method, RuntimeWarning) if meth in ['l-bfgs-b', 'tnc'] and any(constraints): warn('Method %s cannot handle constraints.' % method, RuntimeWarning) if meth is 'cobyla' and bounds is not None: warn('Method %s cannot handle bounds.' % method, RuntimeWarning) # - callback if (meth in ['anneal', 'l-bfgs-b', 'tnc', 'cobyla', 'slsqp'] and callback is not None): warn('Method %s does not support callback.' % method, RuntimeWarning) # - return_all if (meth in ['anneal', 'l-bfgs-b', 'tnc', 'cobyla', 'slsqp'] and options.get('return_all', False)): warn('Method %s does not support the return_all option.' % method, RuntimeWarning) # fun also returns the jacobian if 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 in ['nelder-mead', 'newton-cg', 'powell', 'tnc']: options.setdefault('xtol', tol) if meth in ['nelder-mead', 'powell', 'anneal', 'l-bfgs-b', 'tnc', 'slsqp']: options.setdefault('ftol', tol) if meth in ['bfgs', 'cg', 'l-bfgs-b', 'tnc']: options.setdefault('gtol', tol) if meth in ['cobyla']: options.setdefault('tol', tol) 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 == 'anneal': return _minimize_anneal(fun, x0, args, **options) elif meth == 'l-bfgs-b': return _minimize_lbfgsb(fun, x0, args, jac, bounds, **options) elif meth == 'tnc': return _minimize_tnc(fun, x0, args, jac, bounds, **options) elif meth == 'cobyla': return _minimize_cobyla(fun, x0, args, constraints, **options) elif meth == 'slsqp': return _minimize_slsqp(fun, x0, args, jac, bounds, constraints, **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. .. versionadded:: 0.11.0 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, optional Type of solver. Should be one of - 'Brent' - 'Bounded' - 'Golden' tol : float, optional Tolerance for termination. For detailed control, use solver-specific options. options : dict, optional A dictionary of solver options. xtol : float Relative error in solution `xopt` acceptable for convergence. maxiter : int Maximum number of iterations to perform. disp : bool Set to True to print convergence messages. Returns ------- res : Result The optimization result represented as a ``Result`` 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 `Result` for a description of other attributes. See also -------- minimize: Interface to minimization algorithms for scalar multivariate functions. Notes ----- This section describes the available solvers that can be selected by the 'method' parameter. The default method is *Brent*. Method *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 *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 *Bounded* can perform bounded minimization. It uses the Brent method to find a local minimum in the interval x1 < xopt < x2. 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 """ meth = method.lower() if options is None: options = {} if tol is not None: options = dict(options) options.setdefault('xtol', tol) if 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`.') 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)