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Minimize a scalar function using the COBYQA method. The Constrained Optimization BY Quadratic Approximations (COBYQA) method is a derivative-free optimization method designed to solve general nonlinear optimization problems. A complete description of COBYQA is given in [3]_. Parameters ---------- fun : {callable, None} Objective function to be minimized. ``fun(x, *args) -> float`` where ``x`` is an array with shape (n,) and `args` is a tuple. If `fun` is ``None``, the objective function is assumed to be the zero function, resulting in a feasibility problem. x0 : array_like, shape (n,) Initial guess. args : tuple, optional Extra arguments passed to the objective function. bounds : {`scipy.optimize.Bounds`, array_like, shape (n, 2)}, optional Bound constraints of the problem. It can be one of the cases below. #. An instance of `scipy.optimize.Bounds`. For the time being, the argument ``keep_feasible`` is disregarded, and all the constraints are considered unrelaxable and will be enforced. #. An array with shape (n, 2). The bound constraints for ``x[i]`` are ``bounds[i][0] <= x[i] <= bounds[i][1]``. Set ``bounds[i][0]`` to :math:`-\infty` if there is no lower bound, and set ``bounds[i][1]`` to :math:`\infty` if there is no upper bound. The COBYQA method always respect the bound constraints. constraints : {Constraint, list}, optional General constraints of the problem. It can be one of the cases below. #. An instance of `scipy.optimize.LinearConstraint`. The argument ``keep_feasible`` is disregarded. #. An instance of `scipy.optimize.NonlinearConstraint`. The arguments ``jac``, ``hess``, ``keep_feasible``, ``finite_diff_rel_step``, and ``finite_diff_jac_sparsity`` are disregarded. #. A list, each of whose elements are described in the cases above. callback : callable, optional A callback executed at each objective function evaluation. The method terminates if a ``StopIteration`` exception is raised by the callback function. Its signature can be one of the following: ``callback(intermediate_result)`` where ``intermediate_result`` is a keyword parameter that contains an instance of `scipy.optimize.OptimizeResult`, with attributes ``x`` and ``fun``, being the point at which the objective function is evaluated and the value of the objective function, respectively. The name of the parameter must be ``intermediate_result`` for the callback to be passed an instance of `scipy.optimize.OptimizeResult`. Alternatively, the callback function can have the signature: ``callback(xk)`` where ``xk`` is the point at which the objective function is evaluated. Introspection is used to determine which of the signatures to invoke. options : dict, optional Options passed to the solver. Accepted keys are: disp : bool, optional Whether to print information about the optimization procedure. Default is ``False``. maxfev : int, optional Maximum number of function evaluations. Default is ``500 * n``. maxiter : int, optional Maximum number of iterations. Default is ``1000 * n``. target : float, optional Target on the objective function value. The optimization procedure is terminated when the objective function value of a feasible point is less than or equal to this target. Default is ``-numpy.inf``. feasibility_tol : float, optional Tolerance on the constraint violation. If the maximum constraint violation at a point is less than or equal to this tolerance, the point is considered feasible. Default is ``numpy.sqrt(numpy.finfo(float).eps)``. radius_init : float, optional Initial trust-region radius. Typically, this value should be in the order of one tenth of the greatest expected change to `x0`. Default is ``1.0``. radius_final : float, optional Final trust-region radius. It should indicate the accuracy required in the final values of the variables. Default is ``1e-6``. nb_points : int, optional Number of interpolation points used to build the quadratic models of the objective and constraint functions. Default is ``2 * n + 1``. scale : bool, optional Whether to scale the variables according to the bounds. Default is ``False``. filter_size : int, optional Maximum number of points in the filter. The filter is used to select the best point returned by the optimization procedure. Default is ``sys.maxsize``. store_history : bool, optional Whether to store the history of the function evaluations. Default is ``False``. history_size : int, optional Maximum number of function evaluations to store in the history. Default is ``sys.maxsize``. debug : bool, optional Whether to perform additional checks during the optimization procedure. This option should be used only for debugging purposes and is highly discouraged to general users. Default is ``False``. Other constants (from the keyword arguments) are described below. They are not intended to be changed by general users. They should only be changed by users with a deep understanding of the algorithm, who want to experiment with different settings. Returns ------- `scipy.optimize.OptimizeResult` Result of the optimization procedure, with the following fields: message : str Description of the cause of the termination. success : bool Whether the optimization procedure terminated successfully. status : int Termination status of the optimization procedure. x : `numpy.ndarray`, shape (n,) Solution point. fun : float Objective function value at the solution point. maxcv : float Maximum constraint violation at the solution point. nfev : int Number of function evaluations. nit : int Number of iterations. If ``store_history`` is True, the result also has the following fields: fun_history : `numpy.ndarray`, shape (nfev,) History of the objective function values. maxcv_history : `numpy.ndarray`, shape (nfev,) History of the maximum constraint violations. A description of the termination statuses is given below. .. list-table:: :widths: 25 75 :header-rows: 1 * - Exit status - Description * - 0 - The lower bound for the trust-region radius has been reached. * - 1 - The target objective function value has been reached. * - 2 - All variables are fixed by the bound constraints. * - 3 - The callback requested to stop the optimization procedure. * - 4 - The feasibility problem received has been solved successfully. * - 5 - The maximum number of function evaluations has been exceeded. * - 6 - The maximum number of iterations has been exceeded. * - -1 - The bound constraints are infeasible. * - -2 - A linear algebra error occurred. Other Parameters ---------------- decrease_radius_factor : float, optional Factor by which the trust-region radius is reduced when the reduction ratio is low or negative. Default is ``0.5``. increase_radius_factor : float, optional Factor by which the trust-region radius is increased when the reduction ratio is large. Default is ``numpy.sqrt(2.0)``. increase_radius_threshold : float, optional Threshold that controls the increase of the trust-region radius when the reduction ratio is large. Default is ``2.0``. decrease_radius_threshold : float, optional Threshold used to determine whether the trust-region radius should be reduced to the resolution. Default is ``1.4``. decrease_resolution_factor : float, optional Factor by which the resolution is reduced when the current value is far from its final value. Default is ``0.1``. large_resolution_threshold : float, optional Threshold used to determine whether the resolution is far from its final value. Default is ``250.0``. moderate_resolution_threshold : float, optional Threshold used to determine whether the resolution is close to its final value. Default is ``16.0``. low_ratio : float, optional Threshold used to determine whether the reduction ratio is low. Default is ``0.1``. high_ratio : float, optional Threshold used to determine whether the reduction ratio is high. Default is ``0.7``. very_low_ratio : float, optional Threshold used to determine whether the reduction ratio is very low. This is used to determine whether the models should be reset. Default is ``0.01``. penalty_increase_threshold : float, optional Threshold used to determine whether the penalty parameter should be increased. Default is ``1.5``. penalty_increase_factor : float, optional Factor by which the penalty parameter is increased. Default is ``2.0``. short_step_threshold : float, optional Factor used to determine whether the trial step is too short. Default is ``0.5``. low_radius_factor : float, optional Factor used to determine which interpolation point should be removed from the interpolation set at each iteration. Default is ``0.1``. byrd_omojokun_factor : float, optional Factor by which the trust-region radius is reduced for the computations of the normal step in the Byrd-Omojokun composite-step approach. Default is ``0.8``. threshold_ratio_constraints : float, optional Threshold used to determine which constraints should be taken into account when decreasing the penalty parameter. Default is ``2.0``. large_shift_factor : float, optional Factor used to determine whether the point around which the quadratic models are built should be updated. Default is ``10.0``. large_gradient_factor : float, optional Factor used to determine whether the models should be reset. Default is ``10.0``. resolution_factor : float, optional Factor by which the resolution is decreased. Default is ``2.0``. improve_tcg : bool, optional Whether to improve the steps computed by the truncated conjugate gradient method when the trust-region boundary is reached. Default is ``True``. References ---------- .. [1] J. Nocedal and S. J. Wright. *Numerical Optimization*. Springer Ser. Oper. Res. Financ. Eng. Springer, New York, NY, USA, second edition, 2006. `doi:10.1007/978-0-387-40065-5 `_. .. [2] M. J. D. Powell. A direct search optimization method that models the objective and constraint functions by linear interpolation. In S. Gomez and J.-P. Hennart, editors, *Advances in Optimization and Numerical Analysis*, volume 275 of Math. Appl., pages 51--67. Springer, Dordrecht, Netherlands, 1994. `doi:10.1007/978-94-015-8330-5_4 `_. .. [3] T. M. Ragonneau. *Model-Based Derivative-Free Optimization Methods and Software*. PhD thesis, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, 2022. URL: https://theses.lib.polyu.edu.hk/handle/200/12294. Examples -------- To demonstrate how to use `minimize`, we first minimize the Rosenbrock function implemented in `scipy.optimize` in an unconstrained setting. .. testsetup:: import numpy as np np.set_printoptions(precision=3, suppress=True) >>> from cobyqa import minimize >>> from scipy.optimize import rosen To solve the problem using COBYQA, run: >>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2] >>> res = minimize(rosen, x0) >>> res.x array([1., 1., 1., 1., 1.]) To see how bound and constraints are handled using `minimize`, we solve Example 16.4 of [1]_, defined as .. math:: \begin{aligned} \min_{x \in \mathbb{R}^2} & \quad (x_1 - 1)^2 + (x_2 - 2.5)^2\\ \text{s.t.} & \quad -x_1 + 2x_2 \le 2,\\ & \quad x_1 + 2x_2 \le 6,\\ & \quad x_1 - 2x_2 \le 2,\\ & \quad x_1 \ge 0,\\ & \quad x_2 \ge 0. \end{aligned} >>> import numpy as np >>> from scipy.optimize import Bounds, LinearConstraint Its objective function can be implemented as: >>> def fun(x): ... return (x[0] - 1.0)**2 + (x[1] - 2.5)**2 This problem can be solved using `minimize` as: >>> x0 = [2.0, 0.0] >>> bounds = Bounds([0.0, 0.0], np.inf) >>> constraints = LinearConstraint([ ... [-1.0, 2.0], ... [1.0, 2.0], ... [1.0, -2.0], ... ], -np.inf, [2.0, 6.0, 2.0]) >>> res = minimize(fun, x0, bounds=bounds, constraints=constraints) >>> res.x array([1.4, 1.7]) To see how nonlinear constraints are handled, we solve Problem (F) of [2]_, defined as .. math:: \begin{aligned} \min_{x \in \mathbb{R}^2} & \quad -x_1 - x_2\\ \text{s.t.} & \quad x_1^2 - x_2 \le 0,\\ & \quad x_1^2 + x_2^2 \le 1. \end{aligned} >>> from scipy.optimize import NonlinearConstraint Its objective and constraint functions can be implemented as: >>> def fun(x): ... return -x[0] - x[1] >>> >>> def cub(x): ... return [x[0]**2 - x[1], x[0]**2 + x[1]**2] This problem can be solved using `minimize` as: >>> x0 = [1.0, 1.0] >>> constraints = NonlinearConstraint(cub, -np.inf, [0.0, 1.0]) >>> res = minimize(fun, x0, constraints=constraints) >>> res.x array([0.707, 0.707]) Finally, to see how to supply linear and nonlinear constraints simultaneously, we solve Problem (G) of [2]_, defined as .. math:: \begin{aligned} \min_{x \in \mathbb{R}^3} & \quad x_3\\ \text{s.t.} & \quad 5x_1 - x_2 + x_3 \ge 0,\\ & \quad -5x_1 - x_2 + x_3 \ge 0,\\ & \quad x_1^2 + x_2^2 + 4x_2 \le x_3. \end{aligned} Its objective and nonlinear constraint functions can be implemented as: >>> def fun(x): ... return x[2] >>> >>> def cub(x): ... return x[0]**2 + x[1]**2 + 4.0*x[1] - x[2] This problem can be solved using `minimize` as: >>> x0 = [1.0, 1.0, 1.0] >>> constraints = [ ... LinearConstraint( ... [[5.0, -1.0, 1.0], [-5.0, -1.0, 1.0]], ... [0.0, 0.0], ... np.inf, ... ), ... 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NzThe bounds must have z elements.rzGThe shape of the bounds is not compatible with the number of variables.rrzPThe bounds must be an instance of scipy.optimize.Bounds or an array-like object.) rrEfullinfr.lbshapeubr*r0asarray TypeError)r7r5s rr2r2qs"~bga"&))271bf+=+=>>> FF # # 9?qd " "fio!&=&=BQBBBCC Cfi+++  # #  F## The lower bound of the nonlinear constraints must be a vector.z>The upper bound of the nonlinear constraints must be a vector.r_)eqineqz+The constraint type must be "eq" or "ineq".rvz)The constraint function must be callable.rxr)rvr_rxzrThe constraints must be instances of scipy.optimize.LinearConstraint, scipy.optimize.NonlinearConstraint, or dict.)r.r!r0rrrrappendArEbroadcast_arraysrrvr*callabler"r)ryrr constraintrrs rr3r3sO+t$$%GK,K,K%"n !77 j"2 3 36  MB  MB  % % L(R00      $7 8 8'  B   B " ( (#N(R00      D ) ) Z'':f+=F,,!!NOOOJ&&hz%7H.I.I& !LMMM ! 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'( ",-+.gg6J.K*L*LGG &' w}*OGM,JKKK#' (>#?#?GGM  HH g)0022 2 2 M3S333^Q G G GHHrc lt|}|tjjt tjt |tj|tjj<|tjdks|tjdkrtd|tjjt tjt |tj|tjj<|tjdkrtdtj |vr%|tj dkrtdtj |vr%|tj dkrtdtj |vrEtj |vr7|tj |tj krtdntj |vrTtj t tj dd|tj zzg|tj j<ntj |vrQtj t tj d |tj zg|tj j<nRt tj |tj j<t tj |tj j<|tjjt tjt |tj|tjj<|tjdks|tjdkrtd tj|vr%|tjdkrtd tj|vr%|tjdkrtd tj|vrEtj|vr7|tj|tjkrtd n tj|vrNtj t tj|tjg|tjj<ntj|vrNtj t tj|tjg|tjj<nRt tj|tjj<t tj|tjj<tj|vr;|tjdks|tjdkrtdtj|vr;|tjdks|tjdkrtdtj|vrEtj|vr7|tj|tjkrtdn tj|vrNtj t tj|tjg|tjj<ntj|vrNtj t tj|tjg|tjj<nRt tj|tjj<t tj|tjj<|tjjt tjt |tj|tjj<|tjdks|tjdkrtdtj|vr%|tjdkrtdtj|vr%|tjdkrtdtj|vrEtj|vr7|tj|tjkrtdn tj|vrNtj t tj|tjg|tjj<ntj|vrNtj t tj|tjg|tjj<nRt tj|tjj<t tj|tjj<|tjjt tjt |tj|tjj<|tjdks|tjdkrtd|tjjt tjt |tj|tjj<|tjdks|tjdkrtd|tjjt tjt |tj|tjj<|tjdks|tjdkrtd|tjjt tjt |tj|tjj<|tjdkrtd|tjjt tjt |tj|tjj<|tjdkrtd|tjjt tjt |tj|tjj<|tjdkrtd|tjjt tjt |tj|tjj<|tjdkrtd|tjjt tjt=|tj|tjj<|D]A}|tj vrtCj"d|dtFdB|S)z$ Set the default constants. rg?zCThe constant decrease_radius_factor must be in the interval (0, 1).z>The constant increase_radius_threshold must be greater than 1.z;The constant increase_radius_factor must be greater than 1.z>The constant decrease_radius_threshold must be greater than 1.zPThe constant decrease_radius_threshold must be less than increase_radius_factor.g?r zGThe constant decrease_resolution_factor must be in the interval (0, 1).z?The constant large_resolution_threshold must be greater than 1.zBThe constant moderate_resolution_threshold must be greater than 1.zVThe constant moderate_resolution_threshold must be at most large_resolution_threshold.z6The constant low_ratio must be in the interval (0, 1).z7The constant high_ratio must be in the interval (0, 1).z2The constant low_ratio must be at most high_ratio.z;The constant very_low_ratio must be in the interval (0, 1).zKThe constant penalty_increase_threshold must be greater than or equal to 1.zThe constant low_radius_factor must be in the interval (0, 1).zAThe constant byrd_omojokun_factor must be in the interval (0, 1).z@The constant threshold_ratio_constraints must be greater than 1.z4The constant large_shift_factor must be nonnegative.z:The constant large_gradient_factor must be greater than 1.z6The constant resolution_factor must be greater than 1.zUnknown constant: rr)$r!rrDECREASE_RADIUS_FACTORrrr&r*INCREASE_RADIUS_THRESHOLDINCREASE_RADIUS_FACTORDECREASE_RADIUS_THRESHOLDrErrVrTLARGE_RESOLUTION_THRESHOLDMODERATE_RESOLUTION_THRESHOLDrl HIGH_RATIOrfPENALTY_INCREASE_THRESHOLDPENALTY_INCREASE_FACTORrRLOW_RADIUS_FACTORr`THRESHOLD_RATIO_CONSTRAINTSrNrirW IMPROVE_TCGr$rrrrr)r|rrs rr6r67sL V I (.):;9>)2399Ii.45 )23s:: Y5 6# = =    +1)=> > I    +y88 i9 :c A A L    (I55  /9 < < i9 :9: ; ;4  ;  )Y 6 6?Av!)"EFsYy'GHHI @ @ )5;<<  , 9 9<>F!)"BCi CDD = = )2899=N  ,= )289 iA B )5;< ,2)>?=B)67==Ii289 )673>> Y9 :c A A    , 99 i: ;s B B M    /9<< i= ># E E    , 99  3y @ @ i= > <= > >>  >  - : :CE6!)"IJ)>? D D )9?@@  0I = =@B!)"FG)AB A A )6<== iB C )6<= iE F )9?@i'')%&#-- Y( )S 0 0 D   y(()&'3.. Y) *c 1 1 E   i''I,@I,M,M Y( )Ii6J,K K KD  L   ) )02!)"67)-. 1 1 )&,--   * */1v!)"56)./ 0 0 )%+,,0A  0 )%+,1B  1 )&,- &)2316)*+11Ii&,- )*+s22 Y- .# 5 5 I    , 99 i: ;c A A *   )Y66 i7 8C ? ? J    , 99  - : : i7 8 <= > >.  >  - : :=?V!)"CD)>? > > )39::  *i 7 7@B!)"FG);< A A )6<== iB C )6<=>O  -> )39:&,)897<)0177Ii,23 )01S88 Y3 4 ; ; O   #))5649)-.44Ii)/0 )-.#55 Y0 1S 8 8 L   &,)897<)0177Ii,23 )01S88 Y3 4 ; ; O   -3)?@>C)78>>Ii39:673>> N   $*)675:)./55Ii*01-.44()) ) '-)9:8=)1288Ii-3401S88 H   #))5649)-.44Ii)/0,-44 D   #)/0.2)'(..Ii#)* JJ i+2244 4 4 M5s555~q I I I rcj|j|tjkrt|j|z}|||j\}}}||||}||tjkr||tjkrt|j r||tjkrt|||fS)z: Evaluate the objective and constraint functions. ) rsrr?rrJrurprr%ris_feasibilityr) rrrr{x_evalrrrr_vals rrYrYs yGG,---   $F "69+< = =GWg HHVWg . .E77>*** WW45 5 5 Ugg.E&FFF GW $$rc||\}}}|o'tj|otj|}|tjtjfvr|o||t jk}t} tj dtjdtj dtj dtjdtj dtj dtjdtjd i |d | _|| _|j| _||| _|| _|| _|j| _|| _|t jr|j| _|j| _|t jr3tA| j|| j| j| j| j| j| S) z7 Build the result of the optimization process. z#&@#&E#&5#%K!B!" c&'((# N$FNLFMzz!}}FHFJFL)FKFJw$%0^!/w   N  H J L K J    Mrctt|dtd|dtd|d|jstd|jd|dtd|dtjd it 5td|dddddS#1swxYwYdS) zP Print information about the current state of the optimization process. rz Number of function evaluations: zNumber of iterations: zLeast value of z: zMaximum constraint violation: zCorresponding point: Nr)r<rfun_namerE printoptionsr)rrrrrrsrs rrqrqs2 GGG W--- 6V 6 6 6777 ,6 , , ,--- ; 9 99w999::: 35 3 3 3444  ) )= ) ),, *a***+++,,,,,,,,,,,,,,,,,,sB33B7:B7)rNrNN)%rnumpyrEscipy.optimizerrrrrrproblemr r r r r utilsrrrrrsettingsrrrrrrrr2r3r4r6rYr9rqrrrrs#"""""   J J J J Z   2B5B5B5JeHeHeHPTTTn %%%&222j , , , , ,r