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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. maxfev : int, optional Maximum number of function evaluations. maxiter : int, optional Maximum number of iterations. 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. 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. 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`. radius_final : float, optional Final trust-region radius. It should indicate the accuracy required in the final values of the variables. nb_points : int, optional Number of interpolation points used to build the quadratic models of the objective and constraint functions. scale : bool, optional Whether to scale the variables according to the bounds. 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. store_history : bool, optional Whether to store the history of the function evaluations. history_size : int, optional Maximum number of function evaluations to store in the history. 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. 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. increase_radius_factor : float, optional Factor by which the trust-region radius is increased when the reduction ratio is large. increase_radius_threshold : float, optional Threshold that controls the increase of the trust-region radius when the reduction ratio is large. decrease_radius_threshold : float, optional Threshold used to determine whether the trust-region radius should be reduced to the resolution. decrease_resolution_factor : float, optional Factor by which the resolution is reduced when the current value is far from its final value. large_resolution_threshold : float, optional Threshold used to determine whether the resolution is far from its final value. moderate_resolution_threshold : float, optional Threshold used to determine whether the resolution is close to its final value. low_ratio : float, optional Threshold used to determine whether the reduction ratio is low. high_ratio : float, optional Threshold used to determine whether the reduction ratio is high. 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. penalty_increase_threshold : float, optional Threshold used to determine whether the penalty parameter should be increased. penalty_increase_factor : float, optional Factor by which the penalty parameter is increased. short_step_threshold : float, optional Factor used to determine whether the trial step is too short. low_radius_factor : float, optional Factor used to determine which interpolation point should be removed from the interpolation set at each iteration. 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. threshold_ratio_constraints : float, optional Threshold used to determine which constraints should be taken into account when decreasing the penalty parameter. large_shift_factor : float, optional Factor used to determine whether the point around which the quadratic models are built should be updated. large_gradient_factor : float, optional Factor used to determine whether the models should be reset. resolution_factor : float, optional Factor by which the resolution is decreased. improve_tcg : bool, optional Whether to improve the steps computed by the truncated conjugate gradient method when the trust-region boundary is reached. 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 rr2r2_s~bgga"&&)2771bff+=>> FF # 99??qd "fiioo!&=4QCzBC C  #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 rr3r3xs+t$GK,K"n ! j"2 3 MB  MB  % % LL((R0   $7 8 B   B " ( (#NN((R0   D )Z':f+=F,!!NOOJ&hz%7H.I !LMM ! 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' '( "",,-+.gg6J6J.K*LGG & &' w}}**OGMM,JK#' (>#?GGMM    g))002 2 MM,SE3^Q Grc t|}|jtjjt tjt |tj|tjj<|tjdks|tjdk\r td|jtjjt tjt |tj|tjj<|tjdkr tdtj|vr!|tjdkr tdtj|vr!|tjdkr tdtj|vrEtj|vr3|tj|tjk\rNtdtj|vr_tjt tjdd|tjzzg|tjj<ntj|vr\tjt tjd |tjzg|tjj<ndt tj|tjj<t tj|tjj<|jtjjt tjt |tj|tjj<|tjdks|tjdk\r td tj|vr!|tjdkr td tj |vr!|tj dkr td tj|vrEtj |vr3|tj |tjkDrEtd tj|vrYtjt tj |tjg|tj j<ntj |vrYtjt tj|tj g|tjj<ndt tj|tjj<t tj |tj j<tj"|vr7|tj"dks|tj"dk\r tdtj$|vr7|tj$dks|tj$dk\r tdtj"|vrEtj$|vr3|tj"|tj$kDrEtdtj"|vrYtjt tj$|tj"g|tj$j<ntj$|vrYtjt tj"|tj$g|tj"j<ndt tj"|tj"j<t tj$|tj$j<|jtj&jt tj&t |tj&|tj&j<|tj&dks|tj&dk\r tdtj(|vr!|tj(dkr tdtj*|vr!|tj*dkr tdtj(|vrEtj*|vr3|tj*|tj(krEtdtj(|vrYtjt tj*|tj(g|tj*j<ntj*|vrYtjt tj(|tj*g|tj(j<ndt tj(|tj(j<t tj*|tj*j<|jtj,jt tj,t |tj,|tj,j<|tj,dks|tj,dk\r td|jtj.jt tj.t |tj.|tj.j<|tj.dks|tj.dk\r td|jtj0jt tj0t |tj0|tj0j<|tj0dks|tj0dk\r td|jtj2jt tj2t |tj2|tj2j<|tj2dkr td|jtj4jt tj4t |tj4|tj4j<|tj4dkr td|jtj6jt tj6t |tj6|tj6j<|tj6dkr td|jtj8jt tj8t |tj8|tj8j<|tj8dkr td|jtj:jt tj:t=|tj:|tj:j<|D]B}|tj>jAvs$tCjDd|dtFdD|S)z$ 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?rzGThe 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 rr6r6%sQ V I ((..)::;9>)2239Ii..445 )223s: Y55 6# =   ++11)==> I   ++y8 i99 :c A L   ((I5  / /9 < i99 :99: ;4   ) )Y 6?Avv!)"E"EFsYy'G'GHHI @ )55;;<  , , 9<>FF!)"B"BCi C CDD = )22889=N  , ,= )22889 iAA B )55;;< ,,22)>>?=B)667=Ii22889 )6673> Y99 :c A   ,, 9 i:: ;s B M   //9< i== ># E   ,, 9  3 3y @ i== > <<= >>   - - :CE66!)"I"IJ)>>? D )99??@  0 0I =@B!)"F"FG)AAB A )66<<= iBB C )66<<= iEE F )99??@i')%%&#- Y(( )S 0 D  y()&&'3. Y)) *c 1 E  i'I,@,@I,M Y(( )Ii6J6J,K KD     )02!)"6"67)--. 1 )&&,,-    */1vv!)"5"56)../ 0 )%%++,0A   0 )%%++,1B  1 )&&,,-  &&)22316)**+1Ii&&,,- )**+s2 Y-- .# 5 I   ,, 9 i:: ;c A *  ))Y6 i77 8C ? J   ,, 9  - - : i77 8 <<= >.   - - :=?VV!)"C"CD)>>? > )3399:  * *i 7@B!)"F"FG);;< A )66<<= iBB C )66<<=>O  - -> )3399:&&,,)8897<)0017Ii,,223 )001S8 Y33 4 ; O  ##)))55649)--.4Ii))//0 )--.#5 Y00 1S 8 L  &&,,)8897<)0017Ii,,223 )001S8 Y33 4 ; O  --33)??@>C)778>Ii3399:6673> N  $$**)6675:)../5Ii**001--.4() ) ''--)99:8=)1128Ii--334001S8 H  ##)))55649)--.4Ii))//0,,-4 D  ##)//0.2)''(.Ii##))*  i++224 4 MM.se15~q I rch|j|tjk\rt|j|z}||\}}}|j |||}||tj kr||tjkrt|jr||tjkrt|||fS)z: Evaluate the objective and constraint functions. ) rsrr?rrJrprr%ris_feasibilityr) rrrr{x_evalrrrr_vals rrYrY|s yyGG,,--    $F "6 GWg HHVWg .E77>>** WW445 5 Ugg.E.E&FF GW $$rc<|j|\}}}|xr,tj|xrtj|}|tjtj fvr|xr||t jk}t} tjdtjdtjdtjdtj dtjdtjdtjdtjd i j!|d | _|| _|j&| _|j+|| _|| _|| _|j2| _|| _|t j8r"|j:| _|j<| _|t j>rMtA| j"|| j,| j.| j0| j4| j6| S) z7 Build the result of the optimization process. z##&@##&E##&5##%K!B!" c&'(# N$FNLLFMzz!}FHFJFL))FKFJw$$%^^!//w NN  HH JJ LL KK JJ  MrcXtt|dtd|dtd|d|jstd|jd|dtd|dtjd it 5td|ddddy#1swYyxYw) 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 rrqrqs G WIQ- ,VHA 67 "6(! ,-    }Bwiq9: *5' 34  )= ) %aS*+ * ) )s B  B))r Nr NN)%rnumpyrEscipy.optimizerrrrrrproblemr r r r r utilsrrrrrsettingsrrrrrrrr2r3r4r6rYr9rqr rrrs{#   x v 2B5JeHPTn %&2j ,r