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_linprog.cpython-35.opt-1.pyc
�B�W�� � @ s d Z d d l m Z m Z m Z d d l Z d d l m Z m Z d d d g Z d Z d d � Z d d � Z d d d d � Z d d d � Z d d d d d d d d � Z d d d d d d d d d d d d � Z d d d d d d d d d d � Z d S)a" A top-level linear programming interface. Currently this interface only solves linear programming problems via the Simplex Method. .. versionadded:: 0.15.0 Functions --------- .. autosummary:: :toctree: generated/ linprog linprog_verbose_callback linprog_terse_callback � )�division�print_function�absolute_importN� )�OptimizeResult�_check_unknown_options�linprog�linprog_verbose_callback�linprog_terse_callbackzrestructuredtext enc K s~ | d } | d } | d \ } } | d } | d } | d } t j � } t j d d d d d d � i � | r� t d j | � � t d � nB | d k r� t d j | � � n t d j | | � � t d � | d k rmt d t | � d � | st d j | | � � t d | � t � t d � t d | � t � t d � t d | d � t � t j | � d S) a] A sample callback function demonstrating the linprog callback interface. This callback produces detailed output to sys.stdout before each iteration and after the final iteration of the simplex algorithm. Parameters ---------- xk : array_like The current solution vector. **kwargs : dict A dictionary containing the following parameters: tableau : array_like The current tableau of the simplex algorithm. Its structure is defined in _solve_simplex. phase : int The current Phase of the simplex algorithm (1 or 2) nit : int The current iteration number. pivot : tuple(int, int) The index of the tableau selected as the next pivot, or nan if no pivot exists basis : array(int) A list of the current basic variables. Each element contains the name of a basic variable and its value. complete : bool True if the simplex algorithm has completed (and this is the final call to callback), otherwise False. �tableau�nit�pivot�phase�basis�completeZ linewidthi� � formatter�floatc S s d j | � S)Nz {0: 12.4f})�format)�x� r � /_linprog.py�<lambda>C s z*linprog_verbose_callback.<locals>.<lambda>z3--------- Iteration Complete - Phase {0:d} ------- zTableau:r z3--------- Initial Tableau - Phase {0:d} ---------- z2--------- Iteration {0:d} - Phase {1:d} -------- � � z#Pivot Element: T[{0:.0f}, {1:.0f}] zBasic Variables:zCurrent Solution:zx = zCurrent Objective Value:zf = r N���r )r r )�npZget_printoptionsZset_printoptions�printr �str) �xk�kwargsr r �pivrow�pivcolr r r Zsaved_printoptionsr r r r s: c K sG | d } | d k r t d � t d j | � d d �t | � d S)az A sample callback function demonstrating the linprog callback interface. This callback produces brief output to sys.stdout before each iteration and after the final iteration of the simplex algorithm. Parameters ---------- xk : array_like The current solution vector. **kwargs : dict A dictionary containing the following parameters: tableau : array_like The current tableau of the simplex algorithm. Its structure is defined in _solve_simplex. vars : tuple(str, ...) Column headers for each column in tableau. "x[i]" for actual variables, "s[i]" for slack surplus variables, "a[i]" for artificial variables, and "RHS" for the constraint RHS vector. phase : int The current Phase of the simplex algorithm (1 or 2) nit : int The current iteration number. pivot : tuple(int, int) The index of the tableau selected as the next pivot, or nan if no pivot exists basics : list[tuple(int, float)] A list of the current basic variables. Each element contains the index of a basic variable and its value. complete : bool True if the simplex algorithm has completed (and this is the final call to callback), otherwise False. r r z Iter: X:z {0: <5d} �endr N)r r )r r r r r r r ] s $ g�-���q=Fc C s� t j j | d d d � f | k | d d d � f d d �} | j � d k ra d t j f S| r� d t j | j d k � d d f Sd t j j | | j � k � d d f S) a� Given a linear programming simplex tableau, determine the column of the variable to enter the basis. Parameters ---------- T : 2D ndarray The simplex tableau. tol : float Elements in the objective row larger than -tol will not be considered for pivoting. Nominally this value is zero, but numerical issues cause a tolerance about zero to be necessary. bland : bool If True, use Bland's rule for selection of the column (select the first column with a negative coefficient in the objective row, regardless of magnitude). Returns ------- status: bool True if a suitable pivot column was found, otherwise False. A return of False indicates that the linear programming simplex algorithm is complete. col: int The index of the column of the pivot element. If status is False, col will be returned as nan. r N�copyFr Tr r r r )r �ma� masked_where�count�nan�where�mask�min)�T�tol�blandr$ r r r � _pivot_col� s B $r. c C s� | d k r d } n d } t j j | d | � | f | k | d | � | f d d �} | j � d k r} d t j f St j j | d | � | f | k | d | � d f d d �} | | } d t j j | | j � k � d d f S) aI Given a linear programming simplex tableau, determine the row for the pivot operation. Parameters ---------- T : 2D ndarray The simplex tableau. pivcol : int The index of the pivot column. phase : int The phase of the simplex algorithm (1 or 2). tol : float Elements in the pivot column smaller than tol will not be considered for pivoting. Nominally this value is zero, but numerical issues cause a tolerance about zero to be necessary. Returns ------- status: bool True if a suitable pivot row was found, otherwise False. A return of False indicates that the linear programming problem is unbounded. row: int The index of the row of the pivot element. If status is False, row will be returned as nan. r � Nr# Fr Tr )r r$ r% r& r' r( r* )r+ r! r r, �kr$ Zmb�qr r r � _pivot_row� s C C r2 i� r/ c s | } d } | d k r, � j d d } n, | d k rL � j d d } n t d � � | d k r�x4� � f d d � t � j � D� D] � � � f d d � t � j d d � D� } t | � d k r� | d } | � � <� � | } � � d d � f | � � d d � f <xo t � j d � D]Z } | � k r/� | d d � f � � d d � f � | | f � | d d � f <q/W| d 7} q� Wt � d | � � d k r�t j � j d d d t j �} n@ t j t � j d d t � d | � � d � d t j �} x�| st � | | � \ } } | set j } t j � d } d } n- t � | | | � \ } � | s�d } d } | d k r d | d d � <� d | � d f | � d | � <| | d | � d � d | d | d � | f d � d | o| d k i � | s | | k r@d } d } q | � � <� � | } � � d d � f | � � d d � f <xo t � j d � D]Z } | � k r�� | d d � f � � d d � f � | | f � | d d � f <q�W| d 7} q W| | f S)a8 Solve a linear programming problem in "standard maximization form" using the Simplex Method. Minimize :math:`f = c^T x` subject to .. math:: Ax = b x_i >= 0 b_j >= 0 Parameters ---------- T : array_like A 2-D array representing the simplex T corresponding to the maximization problem. It should have the form: [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], . . . [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], [c[0], c[1], ..., c[n_total], 0]] for a Phase 2 problem, or the form: [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], . . . [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], [c[0], c[1], ..., c[n_total], 0], [c'[0], c'[1], ..., c'[n_total], 0]] for a Phase 1 problem (a Problem in which a basic feasible solution is sought prior to maximizing the actual objective. T is modified in place by _solve_simplex. n : int The number of true variables in the problem. basis : array An array of the indices of the basic variables, such that basis[i] contains the column corresponding to the basic variable for row i. Basis is modified in place by _solve_simplex maxiter : int The maximum number of iterations to perform before aborting the optimization. phase : int The phase of the optimization being executed. In phase 1 a basic feasible solution is sought and the T has an additional row representing an alternate objective function. callback : callable, optional If a callback function is provided, it will be called within each iteration of the simplex algorithm. The callback must have the signature `callback(xk, **kwargs)` where xk is the current solution vector and kwargs is a dictionary containing the following:: "T" : The current Simplex algorithm T "nit" : The current iteration. "pivot" : The pivot (row, column) used for the next iteration. "phase" : Whether the algorithm is in Phase 1 or Phase 2. "basis" : The indices of the columns of the basic variables. tol : float The tolerance which determines when a solution is "close enough" to zero in Phase 1 to be considered a basic feasible solution or close enough to positive to to serve as an optimal solution. nit0 : int The initial iteration number used to keep an accurate iteration total in a two-phase problem. bland : bool If True, choose pivots using Bland's rule [3]. In problems which fail to converge due to cycling, using Bland's rule can provide convergence at the expense of a less optimal path about the simplex. 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. Possible values for the ``status`` attribute are: 0 : Optimization terminated successfully 1 : Iteration limit reached 2 : Problem appears to be infeasible 3 : Problem appears to be unbounded See `OptimizeResult` for a description of other attributes. Fr r r/ z1Argument 'phase' to _solve_simplex must be 1 or 2c s1 g | ]' } � | � j d d k r | � q S)r r/ )�shape)�.0�row)r+ r r r � <listcomp>E s z"_solve_simplex.<locals>.<listcomp>c s, g | ]" } � � | f d k r | � q S)r r )r4 �col)r+ r r r r6 G s N�dtypeT� r r r r r r r ) r3 � ValueError�range�size�lenr �zeros�float64�maxr. r' r2 )r+ �nr �maxiterr �callbackr, �nit0r- r r �mZ non_zero_rowr! ZpivvalZirow�solutionZ pivcol_found�statusZ pivrow_foundr )r+ r r r �_solve_simplex� sn ^ ,- * L &4 $ * LrH c 2 ( K sY t | � d } d d d d d d d d d d i } d } t j | � } d } t | � } | d k rv t j | � n t j d t | � g � } | d k r� t j | � n t j d t | � g � } | d k r� t j t j | � � n t j d g � } | d k r!t j t j | � � n t j d g � } t j | d t j �} t j | d t j �t j } | d k st | � d k r�n�t | � d k r9t | d d � r9| d d k r�| d n t j } | d d k r�| d n t j } t j | | g d t j �} t j | | g d t j �} n� t | � | k rZd. } d } n� y� x� t | � D]� } t | | � d k r�t � � | | d d k r�| | d n t j | | <| | d d k r�| | d n t j | | <qjWWn t k rd/ } d } Yn Xt j | t j k � r�| d } t j t j d g � | g � } t j t j t j g � | g � } t j t j d g � | g � } t j t j | j d d g � | g � } t j t j | j d d g � | g � } d } x�t | � D]�} | | | | k r6d0 } d | | f } t j | | � re| | d k red1 } d } t j | | � r�| | d k r�d2 } d } t j | | � r| | d k rt j | t j | � g � } d3 | d4 | f <t j | t j | | g � g � } d | | <t j | | � rt j | t j | � g � } d | d5 | f <t j | t j | | g � g � } t j | | <qWx`t d | � D]O} | | d k r�t j | | � r2| | d k r2| | d d � | f | | } | | d d � | f | | } | | | | | } n� | d d � d f | d d � | f | d d � d f <| d d � d f | d d � | f | d d � d f <| d | | | d <t j | | � r�| | d k r�d6 } d } q�Wt | � } t | � } | | } | } | t j | d k � } y | j \ }! }" Wn t k rct d � � Yn Xy | j \ }# }$ Wn t k r�t d � � Yn X|# | k r�d7 } d } |! | k r�d8 } d } |$ d k r�|$ | k r�d9 } d } |" d k r|" | k rd: } d } | d k r(t | � � t j | d | | | d g � }% | |% d; d | � f <| |% d> <|% d d? � d@ f } | d k r�| |% d | � d | � f <| | d | � <| d k r% | |% | | | � d | � f <| | | | | � <t j |% | | � | | | � f d � d }& d }' t j | d t �}( t j | d t �}) x� t | � D]� } | | k s� | | d k r | | |' |( | <| |) |' <|' d 7}' | | d k r� | | dA 9<|% | d dB � f dC 9<d |% | |( | f <d |% dD |( | f <qh | |& |( | <|& d 7}& qh WxH |) D]@ }* |% dE d d � f |% |* d d � f |% dF d d � f <q6 Wt |% | |( d d d | d | d | d | �\ }+ } t |% dI � | k r |% d dJ � d d � f }% t j |% t j | | | | | � d � }% n d } | d k rr | | } | r? t | � t d! t j d" |% dM d# |+ d$ | d% | d& d � St |% | |( d | |+ d d d | d | d' |+ d | �\ }, } t j | | | � }- |% d | � dN f |- |( d | � <|- d | � }. |- | | | � }/ t j j | d( t j | � d) d* �j! � }0 |. |0 }. | r� x< t d | � D]+ } t j | | � r^ |. | |. d 8<q^ W|. d d � }. |% dQ }1 | dR k r� | r t | | � t d+ j" |1 � � t d, j" |, � � n' | r t | | � t d, j" |, � � t d! |. d" |1 d# t |, � d$ | d- |/ d% | | d& | d k � S)Sa� Solve the following linear programming problem via a two-phase simplex algorithm. minimize: c^T * x subject to: A_ub * x <= b_ub A_eq * x == b_eq Parameters ---------- c : array_like Coefficients of the linear objective function to be minimized. A_ub : array_like 2-D array which, when matrix-multiplied by x, gives the values of the upper-bound inequality constraints at x. b_ub : array_like 1-D array of values representing the upper-bound of each inequality constraint (row) in A_ub. A_eq : array_like 2-D array which, when matrix-multiplied by x, gives the values of the equality constraints at x. b_eq : array_like 1-D array of values representing the RHS of each equality constraint (row) in A_eq. bounds : array_like The bounds for each independent variable in the solution, which can take one of three forms:: None : The default bounds, all variables are non-negative. (lb, ub) : If a 2-element sequence is provided, the same lower bound (lb) and upper bound (ub) will be applied to all variables. [(lb_0, ub_0), (lb_1, ub_1), ...] : If an n x 2 sequence is provided, each variable x_i will be bounded by lb[i] and ub[i]. Infinite bounds are specified using -np.inf (negative) or np.inf (positive). callback : callable If a callback function is provide, it will be called within each iteration of the simplex algorithm. The callback must have the signature `callback(xk, **kwargs)` where xk is the current solution vector and kwargs is a dictionary containing the following:: "tableau" : The current Simplex algorithm tableau "nit" : The current iteration. "pivot" : The pivot (row, column) used for the next iteration. "phase" : Whether the algorithm is in Phase 1 or Phase 2. "bv" : A structured array containing a string representation of each basic variable and its current value. Options ------- maxiter : int The maximum number of iterations to perform. disp : bool If True, print exit status message to sys.stdout tol : float The tolerance which determines when a solution is "close enough" to zero in Phase 1 to be considered a basic feasible solution or close enough to positive to to serve as an optimal solution. bland : bool If True, use Bland's anti-cycling rule [3] to choose pivots to prevent cycling. If False, choose pivots which should lead to a converged solution more quickly. The latter method is subject to cycling (non-convergence) in rare instances. Returns ------- A scipy.optimize.OptimizeResult consisting of the following fields:: x : ndarray The independent variable vector which optimizes the linear programming problem. fun : float Value of the objective function. slack : ndarray The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, then the corresponding constraint is active. success : bool Returns True if the algorithm succeeded in finding an optimal solution. status : int An integer representing the exit status of the optimization:: 0 : Optimization terminated successfully 1 : Iteration limit reached 2 : Problem appears to be infeasible 3 : Problem appears to be unbounded nit : int The number of iterations performed. message : str A string descriptor of the exit status of the optimization. Examples -------- Consider the following problem: Minimize: f = -1*x[0] + 4*x[1] Subject to: -3*x[0] + 1*x[1] <= 6 1*x[0] + 2*x[1] <= 4 x[1] >= -3 where: -inf <= x[0] <= inf This problem deviates from the standard linear programming problem. In standard form, linear programming problems assume the variables x are non-negative. Since the variables don't have standard bounds where 0 <= x <= inf, the bounds of the variables must be explicitly set. There are two upper-bound constraints, which can be expressed as dot(A_ub, x) <= b_ub The input for this problem is as follows: >>> from scipy.optimize import linprog >>> c = [-1, 4] >>> A = [[-3, 1], [1, 2]] >>> b = [6, 4] >>> x0_bnds = (None, None) >>> x1_bnds = (-3, None) >>> res = linprog(c, A, b, bounds=(x0_bnds, x1_bnds)) >>> print(res) fun: -22.0 message: 'Optimization terminated successfully.' nit: 1 slack: array([ 39., 0.]) status: 0 success: True x: array([ 10., -3.]) References ---------- .. [1] Dantzig, George B., Linear programming and extensions. Rand Corporation Research Study Princeton Univ. Press, Princeton, NJ, 1963 .. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to Mathematical Programming", McGraw-Hill, Chapter 4. .. [3] Bland, Robert G. New finite pivoting rules for the simplex method. Mathematics of Operations Research (2), 1977: pp. 103-107. r z%Optimization terminated successfully.r zIteration limit reached.r/ z>Optimization failed. Unable to find a feasible starting point.r9 z9Optimization failed. The problem appears to be unbounded.� z1Optimization failed. Singular matrix encountered.FNr8 �__len__ziInvalid input for linprog with method = 'simplex'. Length of bounds is inconsistent with the length of czkInvalid input for linprog with method = 'simplex'. bounds must be a n x 2 sequence/array where n = len(c).TzaInvalid input for linprog with method = 'simplex'. Lower bound %d is greater than upper bound %dzTInvalid input for linprog with method = 'simplex'. Lower bound may not be +infinityzTInvalid input for linprog with method = 'simplex'. Upper bound may not be -infinityzPInvalid input for linprog with method = 'simplex'. Upper bound may not be -inf.z,Invalid input. A_ub must be two-dimensionalz,Invalid input. A_eq must be two-dimensionalz|Invalid input for linprog with method = 'simplex'. The number of rows in A_eq must be equal to the number of values in b_eqz|Invalid input for linprog with method = 'simplex'. The number of rows in A_ub must be equal to the number of values in b_ubzlInvalid input for linprog with method = 'simplex'. Number of columns in A_eq must be equal to the size of czlInvalid input for linprog with method = 'simplex'. Number of columns in A_ub must be equal to the size of cr rC rB r, r- r Zfunr rG �message�successrD r) Z fill_valueg z, Current function value: {0: <12.6f}z Iterations: {0:d}�slackr r r r r r r r r r r r r ���rN r )rN r rN r r r r r r r r r )r r r r r )r r r r r )r r )r r )#r r Zasarrayr= �emptyZravelr> r? Zones�inf�hasattrr; � IndexError�anyZ concatenate�arrayZhstackr3 �isinf�isfiniteZvstackZ count_nonzeror: Z fill_diagonal�intrH �abs�deleteZs_r r r' r$ Zfilledr )2�c�A_ub�b_ub�A_eq�b_eq�boundsrB ZdisprC r, r- Zunknown_optionsrG ZmessagesZhave_floor_variableZccZf0rA ZAeqZAubZbeqZbub�L�U�a�brK �iZmubZmeqrE Zn_slackZ n_artificialZAub_rowsZAub_colsZAeq_rowsZAeq_colsr+ ZslcountZavcountr Z r_artificial�rZnit1Znit2rF r rM Zmasked_L�objr r r �_linprog_simplex� sV � 6666 &$# " 07 !$!++ ### & %#"":: % * > 1 & $- *rg �simplexc C so | j � } | d k r i } | d k r[ t | d | d | d | d | d | d | | �St d | � � d S) a� Minimize a linear objective function subject to linear equality and inequality constraints. Linear Programming is intended to solve the following problem form: Minimize: c^T * x Subject to: A_ub * x <= b_ub A_eq * x == b_eq Parameters ---------- c : array_like Coefficients of the linear objective function to be minimized. A_ub : array_like, optional 2-D array which, when matrix-multiplied by x, gives the values of the upper-bound inequality constraints at x. b_ub : array_like, optional 1-D array of values representing the upper-bound of each inequality constraint (row) in A_ub. A_eq : array_like, optional 2-D array which, when matrix-multiplied by x, gives the values of the equality constraints at x. b_eq : array_like, optional 1-D array of values representing the RHS of each equality constraint (row) in A_eq. bounds : sequence, optional ``(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. By default bounds are ``(0, None)`` (non-negative) If a sequence containing a single tuple is provided, then ``min`` and ``max`` will be applied to all variables in the problem. method : str, optional Type of solver. At this time only 'simplex' is supported :ref:`(see here) <optimize.linprog-simplex>`. callback : callable, optional If a callback function is provide, it will be called within each iteration of the simplex algorithm. The callback must have the signature `callback(xk, **kwargs)` where xk is the current solution vector and kwargs is a dictionary containing the following:: "tableau" : The current Simplex algorithm tableau "nit" : The current iteration. "pivot" : The pivot (row, column) used for the next iteration. "phase" : Whether the algorithm is in Phase 1 or Phase 2. "basis" : The indices of the columns of the basic variables. 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('linprog')`. Returns ------- A `scipy.optimize.OptimizeResult` consisting of the following fields: x : ndarray The independent variable vector which optimizes the linear programming problem. fun : float Value of the objective function. slack : ndarray The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, then the corresponding constraint is active. success : bool Returns True if the algorithm succeeded in finding an optimal solution. status : int An integer representing the exit status of the optimization:: 0 : Optimization terminated successfully 1 : Iteration limit reached 2 : Problem appears to be infeasible 3 : Problem appears to be unbounded nit : int The number of iterations performed. message : str A string descriptor of the exit status of the optimization. See Also -------- 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 :ref:`Simplex <optimize.linprog-simplex>`. Method *Simplex* uses the Simplex algorithm (as it relates to Linear Programming, NOT the Nelder-Mead Simplex) [1]_, [2]_. This algorithm should be reasonably reliable and fast. .. versionadded:: 0.15.0 References ---------- .. [1] Dantzig, George B., Linear programming and extensions. Rand Corporation Research Study Princeton Univ. Press, Princeton, NJ, 1963 .. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to Mathematical Programming", McGraw-Hill, Chapter 4. .. [3] Bland, Robert G. New finite pivoting rules for the simplex method. Mathematics of Operations Research (2), 1977: pp. 103-107. Examples -------- Consider the following problem: Minimize: f = -1*x[0] + 4*x[1] Subject to: -3*x[0] + 1*x[1] <= 6 1*x[0] + 2*x[1] <= 4 x[1] >= -3 where: -inf <= x[0] <= inf This problem deviates from the standard linear programming problem. In standard form, linear programming problems assume the variables x are non-negative. Since the variables don't have standard bounds where 0 <= x <= inf, the bounds of the variables must be explicitly set. There are two upper-bound constraints, which can be expressed as dot(A_ub, x) <= b_ub The input for this problem is as follows: >>> c = [-1, 4] >>> A = [[-3, 1], [1, 2]] >>> b = [6, 4] >>> x0_bounds = (None, None) >>> x1_bounds = (-3, None) >>> from scipy.optimize import linprog >>> res = linprog(c, A_ub=A, b_ub=b, bounds=(x0_bounds, x1_bounds), ... options={"disp": True}) Optimization terminated successfully. Current function value: -22.000000 Iterations: 1 >>> print(res) fun: -22.0 message: 'Optimization terminated successfully.' nit: 1 slack: array([ 39., 0.]) status: 0 success: True x: array([ 10., -3.]) Note the actual objective value is 11.428571. In this case we minimized the negative of the objective function. Nrh r[ r\ r] r^ r_ rC zUnknown solver %s)�lowerrg r: ) rZ r[ r\ r] r^ r_ �methodrC Zoptions�methr r r r 8 s � !)�__doc__Z __future__r r r Znumpyr �optimizer r �__all__Z __docformat__r r r. r2 rH rg r r r r r �<module> s$ A ,$' � � �
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cobyla.cpython-35.pyc
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lbfgsb.cpython-35.pyc
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N4ST4R_ID | Naxtarrr
