U Gv fOI@sdZddgZddlZddlmZddlmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZmZmZmZmZdd lmZdd lmZmZd Ze e ejZd dZ d dd dd dddd dddddedfddZ!d ddd ddddeddf ddZ"ddZ#ddZ$dS)a  This module implements the Sequential Least Squares Programming optimization algorithm (SLSQP), originally developed by Dieter Kraft. See http://www.netlib.org/toms/733 Functions --------- .. autosummary:: :toctree: generated/ approx_jacobian fmin_slsqp approx_jacobian fmin_slsqpN)slsqp) zerosarraylinalgappendasfarray concatenatefinfosqrtvstackisfinite atleast_1d)OptimizeResult_check_unknown_options_prepare_scalar_function_clip_x_for_func _check_clip_x)approx_derivative)old_bound_to_new_arr_to_scalarzrestructuredtext encGst||d||d}t|S)a Approximate the Jacobian matrix of a callable function. Parameters ---------- x : array_like The state vector at which to compute the Jacobian matrix. func : callable f(x,*args) The vector-valued function. epsilon : float The perturbation used to determine the partial derivatives. args : sequence Additional arguments passed to func. Returns ------- An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length of the outputs of `func`, and ``lenx`` is the number of elements in `x`. Notes ----- The approximation is done using forward differences. 2-point)methodabs_stepargs)rnpZ atleast_2d)xfuncepsilonrjacr"cs|dk r |} | | | | dk||d}d}|tfdd|D7}|tfdd|D7}|rr|d||d f7}|r|d || d f7}t||f|||d |}|r|d |d |d|d|dfS|d SdS)a/ Minimize a function using Sequential Least Squares Programming Python interface function for the SLSQP Optimization subroutine originally implemented by Dieter Kraft. Parameters ---------- func : callable f(x,*args) Objective function. Must return a scalar. x0 : 1-D ndarray of float Initial guess for the independent variable(s). eqcons : list, optional A list of functions of length n such that eqcons[j](x,*args) == 0.0 in a successfully optimized problem. f_eqcons : callable f(x,*args), optional Returns a 1-D array in which each element must equal 0.0 in a successfully optimized problem. If f_eqcons is specified, eqcons is ignored. ieqcons : list, optional A list of functions of length n such that ieqcons[j](x,*args) >= 0.0 in a successfully optimized problem. f_ieqcons : callable f(x,*args), optional Returns a 1-D ndarray in which each element must be greater or equal to 0.0 in a successfully optimized problem. If f_ieqcons is specified, ieqcons is ignored. bounds : list, optional A list of tuples specifying the lower and upper bound for each independent variable [(xl0, xu0),(xl1, xu1),...] Infinite values will be interpreted as large floating values. fprime : callable `f(x,*args)`, optional A function that evaluates the partial derivatives of func. fprime_eqcons : callable `f(x,*args)`, optional A function of the form `f(x, *args)` that returns the m by n array of equality constraint normals. If not provided, the normals will be approximated. The array returned by fprime_eqcons should be sized as ( len(eqcons), len(x0) ). fprime_ieqcons : callable `f(x,*args)`, optional A function of the form `f(x, *args)` that returns the m by n array of inequality constraint normals. If not provided, the normals will be approximated. The array returned by fprime_ieqcons should be sized as ( len(ieqcons), len(x0) ). args : sequence, optional Additional arguments passed to func and fprime. iter : int, optional The maximum number of iterations. acc : float, optional Requested accuracy. iprint : int, optional The verbosity of fmin_slsqp : * iprint <= 0 : Silent operation * iprint == 1 : Print summary upon completion (default) * iprint >= 2 : Print status of each iterate and summary disp : int, optional Overrides the iprint interface (preferred). full_output : bool, optional If False, return only the minimizer of func (default). Otherwise, output final objective function and summary information. epsilon : float, optional The step size for finite-difference derivative estimates. callback : callable, optional Called after each iteration, as ``callback(x)``, where ``x`` is the current parameter vector. Returns ------- out : ndarray of float The final minimizer of func. fx : ndarray of float, if full_output is true The final value of the objective function. its : int, if full_output is true The number of iterations. imode : int, if full_output is true The exit mode from the optimizer (see below). smode : string, if full_output is true Message describing the exit mode from the optimizer. See also -------- minimize: Interface to minimization algorithms for multivariate functions. See the 'SLSQP' `method` in particular. Notes ----- Exit modes are defined as follows :: -1 : Gradient evaluation required (g & a) 0 : Optimization terminated successfully 1 : Function evaluation required (f & c) 2 : More equality constraints than independent variables 3 : More than 3*n iterations in LSQ subproblem 4 : Inequality constraints incompatible 5 : Singular matrix E in LSQ subproblem 6 : Singular matrix C in LSQ subproblem 7 : Rank-deficient equality constraint subproblem HFTI 8 : Positive directional derivative for linesearch 9 : Iteration limit reached Examples -------- Examples are given :ref:`in the tutorial `. Nr)maxiterftoliprintdispepscallbackr"c3s|]}d|dVqdS)eqtypefunrNr".0crr"r# szfmin_slsqp..c3s|]}d|dVqdS)ineqr,Nr"r/r2r"r#r3sr+)r-r.r!rr4)r!bounds constraintsrr.nitstatusmessage)tuple_minimize_slsqp)rx0ZeqconsZf_eqconsZieqconsZ f_ieqconsr5ZfprimeZ fprime_eqconsZfprime_ieqconsriteraccr'r(Z full_outputr r*optsconsresr"r2r#rDs8p  "Fc C! st| |d}|}| | s d}t||dks@t|dkrPtj tjfnt|tddt|t r~|f}ddd}t |D]*\}}z|d }Wnt k r}zt d||W5d}~XYntt k r}zt d|W5d}~XYnHtk r2}zt d |W5d}~XYnX|dkrNtd |dd |krdtd ||d }|dkrfdd}||d }|||d ||dddf7<qdddddddddddd }tttfdd|d D}tttfd!d|d"D}||}td|g}t}|d}||||}d#|||d||d|d$d$|||||d$|||d|d$d$|d#|d#|d}|} t|}!t| }"|dkst|dkr,tj|td%}#tj|td%}$|#tj|$tjntd&d|Dt}%|%jd|krXtd'tjd(d)&|%dddf|%dddfk}&W5QRX|&rtd*d+d,d-|&D|%dddf|%dddf}#}$t|%}'tj|#|'dddf<tj|$|'dddf<t ||| d.}(t!|(j"})t!|(j#}*tdt$}+t|t}t|t$},d}-tdt}.tdt}/tdt}0tdt}1tdt}2tdt}3tdt}4tdt}5tdt}6tdt}7tdt$}8tdt$}9tdt$}:tdt$};tdt$}|d$kr(t%d/d0|)}?t&|*d1}@t'|}At(||||||}Bt)|||#|$|?|A|@|B||,|+|!|"|.|/|0|1|2|3|4|5|6|7|8|9|:|;|<||=|> |+dkr|)}?t'|}A|+d2krt&|*d1}@t(||||||}B|,|-kr2| dk r | t*|d$kr2t%d3|,|(j+|?t,-|@ft.|+dkrDqPt$|,}-q\|dkrt%|t$|+d4t/|+d5t%d6|?t%d7|,t%d8|(j+t%d9|(j0t1|?|@dd2t$|,|(j+|(j0t$|+|t$|+|+dkd: S);a Minimize a scalar function of one or more variables using Sequential Least Squares Programming (SLSQP). Options ------- ftol : float Precision goal for the value of f in the stopping criterion. eps : float Step size used for numerical approximation of the Jacobian. disp : bool Set to True to print convergence messages. If False, `verbosity` is ignored and set to 0. maxiter : int Maximum number of iterations. finite_diff_rel_step : None or array_like, optional If `jac in ['2-point', '3-point', 'cs']` the relative step size to use for numerical approximation of `jac`. The absolute step size is computed as ``h = rel_step * sign(x) * max(1, abs(x))``, possibly adjusted to fit into the bounds. For ``method='3-point'`` the sign of `h` is ignored. If None (default) then step is selected automatically. rrNr")r+r4r-z"Constraint %d has no type defined.z/Constraints must be defined using a dictionary.z#Constraint's type must be a string.zUnknown constraint type '%s'.r.z&Constraint %d has no function defined.r!csfdd}|S)Ncs>t|}dkr&t||dSt|d|dSdS)N)rz3-pointcs)rrZrel_stepr5r)rrrr5)rr)rr)r finite_diff_rel_stepr.r! new_boundsr"r#cjac%s  z3_minimize_slsqp..cjac_factory..cjacr")r.rE)r rCr!rD)r.r# cjac_factory$s z%_minimize_slsqp..cjac_factoryr)r.r!rz$Gradient evaluation required (g & a)z$Optimization terminated successfullyz$Function evaluation required (f & c)z4More equality constraints than independent variablesz*More than 3*n iterations in LSQ subproblemz#Inequality constraints incompatiblez#Singular matrix E in LSQ subproblemz#Singular matrix C in LSQ subproblemz2Rank-deficient equality constraint subproblem HFTIz.Positive directional derivative for linesearchzIteration limit reached) rr cs&g|]}t|df|dqSr.rrr/rr"r# Gsz#_minimize_slsqp..r+cs&g|]}t|df|dqSrPrQr/rRr"r#rSIsr4rIrH)ZdtypecSs g|]\}}t|t|fqSr")r)r0lur"r"r#rSbszDSLSQP Error: the length of bounds is not compatible with that of x0.ignore)invalidz"SLSQP Error: lb > ub in bounds %s.z, css|]}t|VqdS)N)str)r0br"r"r#r3msz"_minimize_slsqp..)r!rr rCr5z%5s %5s %16s %16s)ZNITZFCZOBJFUNZGNORMgrGz%5i %5i % 16.6E % 16.6Ez (Exit mode )z# Current function value:z Iterations:z! 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