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# TNC Python interface # @(#) $Jeannot: tnc.py,v 1.11 2005/01/28 18:27:31 js Exp $ # Copyright (c) 2004-2005, Jean-Sebastien Roy (js@jeannot.org) # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the # "Software"), to deal in the Software without restriction, including # without limitation the rights to use, copy, modify, merge, publish, # distribute, sublicense, and/or sell copies of the Software, and to # permit persons to whom the Software is furnished to do so, subject to # the following conditions: # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF # MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. # IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY # CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, # TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE # SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. """ TNC: A Python interface to the TNC non-linear optimizer TNC is a non-linear optimizer. To use it, you must provide a function to minimize. The function must take one argument: the list of coordinates where to evaluate the function; and it must return either a tuple, whose first element is the value of the function, and whose second argument is the gradient of the function (as a list of values); or None, to abort the minimization. """ from scipy.optimize import moduleTNC from .optimize import (MemoizeJac, OptimizeResult, _check_unknown_options, _prepare_scalar_function) from ._constraints import old_bound_to_new from numpy import inf, array, zeros, asfarray __all__ = ['fmin_tnc'] MSG_NONE = 0 # No messages MSG_ITER = 1 # One line per iteration MSG_INFO = 2 # Informational messages MSG_VERS = 4 # Version info MSG_EXIT = 8 # Exit reasons MSG_ALL = MSG_ITER + MSG_INFO + MSG_VERS + MSG_EXIT MSGS = { MSG_NONE: "No messages", MSG_ITER: "One line per iteration", MSG_INFO: "Informational messages", MSG_VERS: "Version info", MSG_EXIT: "Exit reasons", MSG_ALL: "All messages" } INFEASIBLE = -1 # Infeasible (lower bound > upper bound) LOCALMINIMUM = 0 # Local minimum reached (|pg| ~= 0) FCONVERGED = 1 # Converged (|f_n-f_(n-1)| ~= 0) XCONVERGED = 2 # Converged (|x_n-x_(n-1)| ~= 0) MAXFUN = 3 # Max. number of function evaluations reached LSFAIL = 4 # Linear search failed CONSTANT = 5 # All lower bounds are equal to the upper bounds NOPROGRESS = 6 # Unable to progress USERABORT = 7 # User requested end of minimization RCSTRINGS = { INFEASIBLE: "Infeasible (lower bound > upper bound)", LOCALMINIMUM: "Local minimum reached (|pg| ~= 0)", FCONVERGED: "Converged (|f_n-f_(n-1)| ~= 0)", XCONVERGED: "Converged (|x_n-x_(n-1)| ~= 0)", MAXFUN: "Max. number of function evaluations reached", LSFAIL: "Linear search failed", CONSTANT: "All lower bounds are equal to the upper bounds", NOPROGRESS: "Unable to progress", USERABORT: "User requested end of minimization" } # Changes to interface made by Travis Oliphant, Apr. 2004 for inclusion in # SciPy def fmin_tnc(func, x0, fprime=None, args=(), approx_grad=0, bounds=None, epsilon=1e-8, scale=None, offset=None, messages=MSG_ALL, maxCGit=-1, maxfun=None, eta=-1, stepmx=0, accuracy=0, fmin=0, ftol=-1, xtol=-1, pgtol=-1, rescale=-1, disp=None, callback=None): """ Minimize a function with variables subject to bounds, using gradient information in a truncated Newton algorithm. This method wraps a C implementation of the algorithm. Parameters ---------- func : callable ``func(x, *args)`` Function to minimize. Must do one of: 1. Return f and g, where f is the value of the function and g its gradient (a list of floats). 2. Return the function value but supply gradient function separately as `fprime`. 3. Return the function value and set ``approx_grad=True``. If the function returns None, the minimization is aborted. x0 : array_like Initial estimate of minimum. fprime : callable ``fprime(x, *args)``, optional Gradient of `func`. If None, then either `func` must return the function value and the gradient (``f,g = func(x, *args)``) or `approx_grad` must be True. args : tuple, optional Arguments to pass to function. approx_grad : bool, optional If true, approximate the gradient numerically. bounds : list, optional (min, max) pairs for each element in x0, defining the bounds on that parameter. Use None or +/-inf for one of min or max when there is no bound in that direction. epsilon : float, optional Used if approx_grad is True. The stepsize in a finite difference approximation for fprime. scale : array_like, optional Scaling factors to apply to each variable. If None, the factors are up-low for interval bounded variables and 1+|x| for the others. Defaults to None. offset : array_like, optional Value to subtract from each variable. If None, the offsets are (up+low)/2 for interval bounded variables and x for the others. messages : int, optional Bit mask used to select messages display during minimization values defined in the MSGS dict. Defaults to MGS_ALL. disp : int, optional Integer interface to messages. 0 = no message, 5 = all messages maxCGit : int, optional Maximum number of hessian*vector evaluations per main iteration. If maxCGit == 0, the direction chosen is -gradient if maxCGit < 0, maxCGit is set to max(1,min(50,n/2)). Defaults to -1. maxfun : int, optional Maximum number of function evaluation. If None, maxfun is set to max(100, 10*len(x0)). Defaults to None. eta : float, optional Severity of the line search. If < 0 or > 1, set to 0.25. Defaults to -1. stepmx : float, optional Maximum step for the line search. May be increased during call. If too small, it will be set to 10.0. Defaults to 0. accuracy : float, optional Relative precision for finite difference calculations. If <= machine_precision, set to sqrt(machine_precision). Defaults to 0. fmin : float, optional Minimum function value estimate. Defaults to 0. ftol : float, optional Precision goal for the value of f in the stopping criterion. If ftol < 0.0, ftol is set to 0.0 defaults to -1. xtol : float, optional Precision goal for the value of x in the stopping criterion (after applying x scaling factors). If xtol < 0.0, xtol is set to sqrt(machine_precision). Defaults to -1. pgtol : float, optional Precision goal for the value of the projected gradient in the stopping criterion (after applying x scaling factors). If pgtol < 0.0, pgtol is set to 1e-2 * sqrt(accuracy). Setting it to 0.0 is not recommended. Defaults to -1. rescale : float, optional Scaling factor (in log10) used to trigger f value rescaling. If 0, rescale at each iteration. If a large value, never rescale. If < 0, rescale is set to 1.3. callback : callable, optional Called after each iteration, as callback(xk), where xk is the current parameter vector. Returns ------- x : ndarray The solution. nfeval : int The number of function evaluations. rc : int Return code, see below See also -------- minimize: Interface to minimization algorithms for multivariate functions. See the 'TNC' `method` in particular. Notes ----- The underlying algorithm is truncated Newton, also called Newton Conjugate-Gradient. This method differs from scipy.optimize.fmin_ncg in that 1. it wraps a C implementation of the algorithm 2. it allows each variable to be given an upper and lower bound. The algorithm incorporates the bound constraints by determining the descent direction as in an unconstrained truncated Newton, but never taking a step-size large enough to leave the space of feasible x's. The algorithm keeps track of a set of currently active constraints, and ignores them when computing the minimum allowable step size. (The x's associated with the active constraint are kept fixed.) If the maximum allowable step size is zero then a new constraint is added. At the end of each iteration one of the constraints may be deemed no longer active and removed. A constraint is considered no longer active is if it is currently active but the gradient for that variable points inward from the constraint. The specific constraint removed is the one associated with the variable of largest index whose constraint is no longer active. Return codes are defined as follows:: -1 : Infeasible (lower bound > upper bound) 0 : Local minimum reached (|pg| ~= 0) 1 : Converged (|f_n-f_(n-1)| ~= 0) 2 : Converged (|x_n-x_(n-1)| ~= 0) 3 : Max. number of function evaluations reached 4 : Linear search failed 5 : All lower bounds are equal to the upper bounds 6 : Unable to progress 7 : User requested end of minimization References ---------- Wright S., Nocedal J. (2006), 'Numerical Optimization' Nash S.G. (1984), "Newton-Type Minimization Via the Lanczos Method", SIAM Journal of Numerical Analysis 21, pp. 770-778 """ # handle fprime/approx_grad if approx_grad: fun = func jac = None elif fprime is None: fun = MemoizeJac(func) jac = fun.derivative else: fun = func jac = fprime if disp is not None: # disp takes precedence over messages mesg_num = disp else: mesg_num = {0:MSG_NONE, 1:MSG_ITER, 2:MSG_INFO, 3:MSG_VERS, 4:MSG_EXIT, 5:MSG_ALL}.get(messages, MSG_ALL) # build options opts = {'eps': epsilon, 'scale': scale, 'offset': offset, 'mesg_num': mesg_num, 'maxCGit': maxCGit, 'maxfun': maxfun, 'eta': eta, 'stepmx': stepmx, 'accuracy': accuracy, 'minfev': fmin, 'ftol': ftol, 'xtol': xtol, 'gtol': pgtol, 'rescale': rescale, 'disp': False} res = _minimize_tnc(fun, x0, args, jac, bounds, callback=callback, **opts) return res['x'], res['nfev'], res['status'] def _minimize_tnc(fun, x0, args=(), jac=None, bounds=None, eps=1e-8, scale=None, offset=None, mesg_num=None, maxCGit=-1, maxiter=None, eta=-1, stepmx=0, accuracy=0, minfev=0, ftol=-1, xtol=-1, gtol=-1, rescale=-1, disp=False, callback=None, finite_diff_rel_step=None, maxfun=None, **unknown_options): """ Minimize a scalar function of one or more variables using a truncated Newton (TNC) algorithm. Options ------- eps : float or ndarray If `jac is None` the absolute step size used for numerical approximation of the jacobian via forward differences. scale : list of floats Scaling factors to apply to each variable. If None, the factors are up-low for interval bounded variables and 1+|x] fo the others. Defaults to None. offset : float Value to subtract from each variable. If None, the offsets are (up+low)/2 for interval bounded variables and x for the others. disp : bool Set to True to print convergence messages. maxCGit : int Maximum number of hessian*vector evaluations per main iteration. If maxCGit == 0, the direction chosen is -gradient if maxCGit < 0, maxCGit is set to max(1,min(50,n/2)). Defaults to -1. maxiter : int, optional Maximum number of function evaluations. This keyword is deprecated in favor of `maxfun`. Only if `maxfun` is None is this keyword used. eta : float Severity of the line search. If < 0 or > 1, set to 0.25. Defaults to -1. stepmx : float Maximum step for the line search. May be increased during call. If too small, it will be set to 10.0. Defaults to 0. accuracy : float Relative precision for finite difference calculations. If <= machine_precision, set to sqrt(machine_precision). Defaults to 0. minfev : float Minimum function value estimate. Defaults to 0. ftol : float Precision goal for the value of f in the stopping criterion. If ftol < 0.0, ftol is set to 0.0 defaults to -1. xtol : float Precision goal for the value of x in the stopping criterion (after applying x scaling factors). If xtol < 0.0, xtol is set to sqrt(machine_precision). Defaults to -1. gtol : float Precision goal for the value of the projected gradient in the stopping criterion (after applying x scaling factors). If gtol < 0.0, gtol is set to 1e-2 * sqrt(accuracy). Setting it to 0.0 is not recommended. Defaults to -1. rescale : float Scaling factor (in log10) used to trigger f value rescaling. If 0, rescale at each iteration. If a large value, never rescale. If < 0, rescale is set to 1.3. finite_diff_rel_step : None or array_like, optional If `jac in ['2-point', '3-point', 'cs']` the relative step size to use for numerical approximation of the jacobian. The absolute step size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``, possibly adjusted to fit into the bounds. For ``method='3-point'`` the sign of `h` is ignored. If None (default) then step is selected automatically. maxfun : int Maximum number of function evaluations. If None, `maxfun` is set to max(100, 10*len(x0)). Defaults to None. """ _check_unknown_options(unknown_options) fmin = minfev pgtol = gtol x0 = asfarray(x0).flatten() n = len(x0) if bounds is None: bounds = [(None,None)] * n if len(bounds) != n: raise ValueError('length of x0 != length of bounds') new_bounds = old_bound_to_new(bounds) if mesg_num is not None: messages = {0:MSG_NONE, 1:MSG_ITER, 2:MSG_INFO, 3:MSG_VERS, 4:MSG_EXIT, 5:MSG_ALL}.get(mesg_num, MSG_ALL) elif disp: messages = MSG_ALL else: messages = MSG_NONE sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, epsilon=eps, finite_diff_rel_step=finite_diff_rel_step, bounds=new_bounds) func_and_grad = sf.fun_and_grad """ low, up : the bounds (lists of floats) if low is None, the lower bounds are removed. if up is None, the upper bounds are removed. low and up defaults to None """ low = zeros(n) up = zeros(n) for i in range(n): if bounds[i] is None: l, u = -inf, inf else: l,u = bounds[i] if l is None: low[i] = -inf else: low[i] = l if u is None: up[i] = inf else: up[i] = u if scale is None: scale = array([]) if offset is None: offset = array([]) if maxfun is None: if maxiter is not None: maxfun = maxiter else: maxfun = max(100, 10*len(x0)) rc, nf, nit, x = moduleTNC.minimize(func_and_grad, x0, low, up, scale, offset, messages, maxCGit, maxfun, eta, stepmx, accuracy, fmin, ftol, xtol, pgtol, rescale, callback) funv, jacv = func_and_grad(x) return OptimizeResult(x=x, fun=funv, jac=jacv, nfev=sf.nfev, nit=nit, status=rc, message=RCSTRINGS[rc], success=(-1 < rc < 3)) if __name__ == '__main__': # Examples for TNC def example(): print("Example") # A function to minimize def function(x): f = pow(x[0],2.0)+pow(abs(x[1]),3.0) g = [0,0] g[0] = 2.0*x[0] g[1] = 3.0*pow(abs(x[1]),2.0) if x[1] < 0: g[1] = -g[1] return f, g # Optimizer call x, nf, rc = fmin_tnc(function, [-7, 3], bounds=([-10, 1], [10, 10])) print("After", nf, "function evaluations, TNC returned:", RCSTRINGS[rc]) print("x =", x) print("exact value = [0, 1]") print() example()