(phrSrSS/rSSKJr SSKrSSKJrJrJ r J r J r J r J r JrJrJrJrJr SSKJr SS KJr SS KJr SS KJr SS KJr SS KJr SSKJ r SSK!J"r" SSK#J$r$ SSK%J&r& SSK'J(r(J)r)J*r*J+r+J,r,J-r-J.r.J/r/ SSK0J1r1 /SQr2/SQr3/SQr4S Sjr5S!Sjr6Sr7S"Sjr8Sr9Sr:Sr;Sr<S#Sjr=g)$z Unified interfaces to minimization algorithms. Functions --------- - minimize : minimization of a function of several variables. - minimize_scalar : minimization of a function of one variable. minimizeminimize_scalar)warnN) _minimize_neldermead_minimize_powell _minimize_cg_minimize_bfgs_minimize_newtoncg_minimize_scalar_brent_minimize_scalar_bounded_minimize_scalar_golden MemoizeJacOptimizeResult_wrap_callback_recover_from_bracket_error)_minimize_dogleg)_minimize_trust_ncg)_minimize_trust_krylov)_minimize_trustregion_exact)_minimize_trustregion_constr)_minimize_lbfgsb) _minimize_tnc)_minimize_cobyla)_minimize_cobyqa)_minimize_slsqp)old_bound_to_newnew_bounds_to_oldold_constraint_to_newnew_constraint_to_oldNonlinearConstraintLinearConstraintBoundsPreparedConstraint) FD_METHODS) nelder-meadpowellcgbfgs newton-cgl-bfgs-btnccobylacobyqaslsqp trust-constrdogleg trust-ncg trust-exact trust-krylov) r&r'r(r)r*r+r0r1r2r3r4r.)brentboundedgoldenc  [R"[R"U55nURS:wa [ S5eUR R [RS;a[R"U[S9n[U[5(dU4nUcU(aSnOUbSnOSn[U5(aS n OUR5n U c0n U S ;a#[U5(a[S US 3[S S9 U S;aUb[S US3[S S9 U S;aUb[S US3[S S9 U S;a.[R "U5(a[S US3[S S9 U S;aUb[S US3[S S9 U S;a*U R#SS5(a[S US3[S S9 [U5(aOMUSLa[%U5nUR&nO0U[(;aU S;aOU S;aSnOUb[U5SLaSnOSnU b[+U 5n U S:Xa$U R-S U 5 U R-S!U 5 U S";aU R-S#U 5 U S$;aU R-S%U 5 U S&;aU R-S'U 5 U S(;aU R-S)U 5 U S*:XaU R-S+U 5 U S,:Xa6U R-S#U 5 U R-S'U 5 U R-S-U 5 U S :XaU"X4X$XVXxU S..U D6$[/XU 5nSn UGb[1XqS/5n[3XqU 5nU S0;Ga{UR4UR6:Hn[R8"U5(a [;XXUS19$[U5(+nUH'n[UR#S2S55(aM%SnM) UR!5=(a U=(d U S3:Hn U (aUR4UnX)n[=X~5n[?XU5n[U 5(a [?XU5n [U5(a [?XNUSS49nUVs/sHnURA5PM nnUHKn[?US5UUSS6S79US5'[UR#S2S55(dM8[?US2UUS SS79US2'MM [1XqU 5n[CX5n U S:Xa[EXX*4S8U0U D6nGOLU S9:Xa[GXX*U40U D6nGO6U S::Xa[IXX$U 40U D6nGO U S;:Xa[KXX$U 40U D6nGO U S<:Xa[MXX$XVU 40U D6nOU S=:Xa[OXX$U4S>U 0U D6nOU S3:Xa[QXX$U4S>U 0U D6nOU S?:Xa[SXX(4U US@.U D6nOU S*:Xa[UXX'X40U D6nOU SA:Xa[WXX$UU4S>U 0U D6nOU S,:Xa[YXX$XVXx4S>U 0U D6nOjU SB:Xa[[XX$U4S>U 0U D6nOSU SC:Xa[]XX$XV4S>U 0U D6nOU 0U D6nO%U SE:Xa[aXX$U4S>U 0U D6nO[ SFU35eU (aS[cURdWW5Ul2[cURfU[Rh5Ul3SGU;aSUl5[mU SHS5(aSUl7SIUl8SJUl9U$s snf)KaZMinimization of scalar function of one or more variables. Parameters ---------- fun : callable The objective function to be minimized:: fun(x, *args) -> float where ``x`` is a 1-D array with shape (n,) and ``args`` is a tuple of the fixed parameters needed to completely specify the function. Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where ``my_args`` and ``my_kwargs`` are required positional and keyword arguments. Rather than passing ``f0`` as the callable, wrap it to accept only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been gathered before invoking this function. x0 : ndarray, shape (n,) Initial guess. Array of real elements of size (n,), where ``n`` is the number of independent variables. args : tuple, optional Extra arguments passed to the objective function and its derivatives (`fun`, `jac` and `hess` functions). method : str or callable, optional Type of solver. Should be one of - 'Nelder-Mead' :ref:`(see here) ` - 'Powell' :ref:`(see here) ` - 'CG' :ref:`(see here) ` - 'BFGS' :ref:`(see here) ` - 'Newton-CG' :ref:`(see here) ` - 'L-BFGS-B' :ref:`(see here) ` - 'TNC' :ref:`(see here) ` - 'COBYLA' :ref:`(see here) ` - 'COBYQA' :ref:`(see here) ` - 'SLSQP' :ref:`(see here) ` - 'trust-constr':ref:`(see here) ` - 'dogleg' :ref:`(see here) ` - 'trust-ncg' :ref:`(see here) ` - 'trust-exact' :ref:`(see here) ` - 'trust-krylov' :ref:`(see here) ` - custom - a callable object, see below for description. If not given, chosen to be one of ``BFGS``, ``L-BFGS-B``, ``SLSQP``, depending on whether or not the problem has constraints or bounds. jac : {callable, '2-point', '3-point', 'cs', bool}, optional Method for computing the gradient vector. Only for CG, BFGS, Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg, trust-krylov, trust-exact and trust-constr. If it is a callable, it should be a function that returns the gradient vector:: jac(x, *args) -> array_like, shape (n,) where ``x`` is an array with shape (n,) and ``args`` is a tuple with the fixed parameters. If `jac` is a Boolean and is True, `fun` is assumed to return a tuple ``(f, g)`` containing the objective function and the gradient. Methods 'Newton-CG', 'trust-ncg', 'dogleg', 'trust-exact', and 'trust-krylov' require that either a callable be supplied, or that `fun` return the objective and gradient. If None or False, the gradient will be estimated using 2-point finite difference estimation with an absolute step size. Alternatively, the keywords {'2-point', '3-point', 'cs'} can be used to select a finite difference scheme for numerical estimation of the gradient with a relative step size. These finite difference schemes obey any specified `bounds`. hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy}, optional Method for computing the Hessian matrix. Only for Newton-CG, dogleg, trust-ncg, trust-krylov, trust-exact and trust-constr. If it is callable, it should return the Hessian matrix:: hess(x, *args) -> {LinearOperator, spmatrix, array}, (n, n) where ``x`` is a (n,) ndarray and ``args`` is a tuple with the fixed parameters. The keywords {'2-point', '3-point', 'cs'} can also be used to select a finite difference scheme for numerical estimation of the hessian. Alternatively, objects implementing the `HessianUpdateStrategy` interface can be used to approximate the Hessian. Available quasi-Newton methods implementing this interface are: - `BFGS` - `SR1` Not all of the options are available for each of the methods; for availability refer to the notes. hessp : callable, optional Hessian of objective function times an arbitrary vector p. Only for Newton-CG, trust-ncg, trust-krylov, trust-constr. Only one of `hessp` or `hess` needs to be given. If `hess` is provided, then `hessp` will be ignored. `hessp` must compute the Hessian times an arbitrary vector:: hessp(x, p, *args) -> ndarray shape (n,) where ``x`` is a (n,) ndarray, ``p`` is an arbitrary vector with dimension (n,) and ``args`` is a tuple with the fixed parameters. bounds : sequence or `Bounds`, optional Bounds on variables for Nelder-Mead, L-BFGS-B, TNC, SLSQP, Powell, trust-constr, COBYLA, and COBYQA methods. There are two ways to specify the bounds: 1. Instance of `Bounds` class. 2. Sequence of ``(min, max)`` pairs for each element in `x`. None is used to specify no bound. constraints : {Constraint, dict} or List of {Constraint, dict}, optional Constraints definition. Only for COBYLA, COBYQA, SLSQP and trust-constr. Constraints for 'trust-constr' and 'cobyqa' are defined as a single object or a list of objects specifying constraints to the optimization problem. Available constraints are: - `LinearConstraint` - `NonlinearConstraint` Constraints for COBYLA, SLSQP are defined as a list of dictionaries. Each dictionary with fields: type : str Constraint type: 'eq' for equality, 'ineq' for inequality. fun : callable The function defining the constraint. jac : callable, optional The Jacobian of `fun` (only for SLSQP). args : sequence, optional Extra arguments to be passed to the function and Jacobian. Equality constraint means that the constraint function result is to be zero whereas inequality means that it is to be non-negative. Note that COBYLA only supports inequality constraints. tol : float, optional Tolerance for termination. When `tol` is specified, the selected minimization algorithm sets some relevant solver-specific tolerance(s) equal to `tol`. For detailed control, use solver-specific options. options : dict, optional A dictionary of solver options. All methods except `TNC` accept the following generic options: maxiter : int Maximum number of iterations to perform. Depending on the method each iteration may use several function evaluations. For `TNC` use `maxfun` instead of `maxiter`. disp : bool Set to True to print convergence messages. For method-specific options, see :func:`show_options()`. callback : callable, optional A callable called after each iteration. All methods except TNC, SLSQP, and COBYLA support a callable with the signature:: callback(intermediate_result: OptimizeResult) where ``intermediate_result`` is a keyword parameter containing an `OptimizeResult` with attributes ``x`` and ``fun``, the present values of the parameter vector and objective function. Note that the name of the parameter must be ``intermediate_result`` for the callback to be passed an `OptimizeResult`. These methods will also terminate if the callback raises ``StopIteration``. All methods except trust-constr (also) support a signature like:: callback(xk) where ``xk`` is the current parameter vector. Introspection is used to determine which of the signatures above to invoke. 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. See `OptimizeResult` for a description of other attributes. See also -------- minimize_scalar : Interface to minimization algorithms for scalar univariate functions 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 *BFGS*. **Unconstrained minimization** Method :ref:`CG ` uses a nonlinear conjugate gradient algorithm by Polak and Ribiere, a variant of the Fletcher-Reeves method described in [5]_ pp.120-122. Only the first derivatives are used. Method :ref:`BFGS ` uses the quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [5]_ pp. 136. It uses the first derivatives only. BFGS has proven good performance even for non-smooth optimizations. This method also returns an approximation of the Hessian inverse, stored as `hess_inv` in the OptimizeResult object. Method :ref:`Newton-CG ` uses a Newton-CG algorithm [5]_ pp. 168 (also known as the truncated Newton method). It uses a CG method to the compute the search direction. See also *TNC* method for a box-constrained minimization with a similar algorithm. Suitable for large-scale problems. Method :ref:`dogleg ` uses the dog-leg trust-region algorithm [5]_ for unconstrained minimization. This algorithm requires the gradient and Hessian; furthermore the Hessian is required to be positive definite. Method :ref:`trust-ncg ` uses the Newton conjugate gradient trust-region algorithm [5]_ for unconstrained minimization. This algorithm requires the gradient and either the Hessian or a function that computes the product of the Hessian with a given vector. Suitable for large-scale problems. Method :ref:`trust-krylov ` uses the Newton GLTR trust-region algorithm [14]_, [15]_ for unconstrained minimization. This algorithm requires the gradient and either the Hessian or a function that computes the product of the Hessian with a given vector. Suitable for large-scale problems. On indefinite problems it requires usually less iterations than the `trust-ncg` method and is recommended for medium and large-scale problems. Method :ref:`trust-exact ` is a trust-region method for unconstrained minimization in which quadratic subproblems are solved almost exactly [13]_. This algorithm requires the gradient and the Hessian (which is *not* required to be positive definite). It is, in many situations, the Newton method to converge in fewer iterations and the most recommended for small and medium-size problems. **Bound-Constrained minimization** Method :ref:`Nelder-Mead ` uses the Simplex algorithm [1]_, [2]_. This algorithm is robust in many applications. However, if numerical computation of derivative can be trusted, other algorithms using the first and/or second derivatives information might be preferred for their better performance in general. Method :ref:`L-BFGS-B ` uses the L-BFGS-B algorithm [6]_, [7]_ for bound constrained minimization. Method :ref:`Powell ` is a modification of Powell's method [3]_, [4]_ which is a conjugate direction method. It performs sequential one-dimensional minimizations along each vector of the directions set (`direc` field in `options` and `info`), which is updated at each iteration of the main minimization loop. The function need not be differentiable, and no derivatives are taken. If bounds are not provided, then an unbounded line search will be used. If bounds are provided and the initial guess is within the bounds, then every function evaluation throughout the minimization procedure will be within the bounds. If bounds are provided, the initial guess is outside the bounds, and `direc` is full rank (default has full rank), then some function evaluations during the first iteration may be outside the bounds, but every function evaluation after the first iteration will be within the bounds. If `direc` is not full rank, then some parameters may not be optimized and the solution is not guaranteed to be within the bounds. Method :ref:`TNC ` uses a truncated Newton algorithm [5]_, [8]_ to minimize a function with variables subject to bounds. This algorithm uses gradient information; it is also called Newton Conjugate-Gradient. It differs from the *Newton-CG* method described above as it wraps a C implementation and allows each variable to be given upper and lower bounds. **Constrained Minimization** Method :ref:`COBYLA ` uses the Constrained Optimization BY Linear Approximation (COBYLA) method [9]_, [10]_, [11]_. The algorithm is based on linear approximations to the objective function and each constraint. The method wraps a FORTRAN implementation of the algorithm. The constraints functions 'fun' may return either a single number or an array or list of numbers. Method :ref:`COBYQA ` uses the Constrained Optimization BY Quadratic Approximations (COBYQA) method [18]_. The algorithm is a derivative-free trust-region SQP method based on quadratic approximations to the objective function and each nonlinear constraint. The bounds are treated as unrelaxable constraints, in the sense that the algorithm always respects them throughout the optimization process. Method :ref:`SLSQP ` uses Sequential Least SQuares Programming to minimize a function of several variables with any combination of bounds, equality and inequality constraints. The method wraps the SLSQP Optimization subroutine originally implemented by Dieter Kraft [12]_. Note that the wrapper handles infinite values in bounds by converting them into large floating values. Method :ref:`trust-constr ` is a trust-region algorithm for constrained optimization. It switches between two implementations depending on the problem definition. It is the most versatile constrained minimization algorithm implemented in SciPy and the most appropriate for large-scale problems. For equality constrained problems it is an implementation of Byrd-Omojokun Trust-Region SQP method described in [17]_ and in [5]_, p. 549. When inequality constraints are imposed as well, it switches to the trust-region interior point method described in [16]_. This interior point algorithm, in turn, solves inequality constraints by introducing slack variables and solving a sequence of equality-constrained barrier problems for progressively smaller values of the barrier parameter. The previously described equality constrained SQP method is used to solve the subproblems with increasing levels of accuracy as the iterate gets closer to a solution. **Finite-Difference Options** For Method :ref:`trust-constr ` the gradient and the Hessian may be approximated using three finite-difference schemes: {'2-point', '3-point', 'cs'}. The scheme 'cs' is, potentially, the most accurate but it requires the function to correctly handle complex inputs and to be differentiable in the complex plane. The scheme '3-point' is more accurate than '2-point' but requires twice as many operations. If the gradient is estimated via finite-differences the Hessian must be estimated using one of the quasi-Newton strategies. **Method specific options for the** `hess` **keyword** +--------------+------+----------+-------------------------+-----+ | method/Hess | None | callable | '2-point/'3-point'/'cs' | HUS | +==============+======+==========+=========================+=====+ | Newton-CG | x | (n, n) | x | x | | | | LO | | | +--------------+------+----------+-------------------------+-----+ | dogleg | | (n, n) | | | +--------------+------+----------+-------------------------+-----+ | trust-ncg | | (n, n) | x | x | +--------------+------+----------+-------------------------+-----+ | trust-krylov | | (n, n) | x | x | +--------------+------+----------+-------------------------+-----+ | trust-exact | | (n, n) | | | +--------------+------+----------+-------------------------+-----+ | trust-constr | x | (n, n) | x | x | | | | LO | | | | | | sp | | | +--------------+------+----------+-------------------------+-----+ where LO=LinearOperator, sp=Sparse matrix, HUS=HessianUpdateStrategy **Custom minimizers** It may be useful to pass a custom minimization method, for example when using a frontend to this method such as `scipy.optimize.basinhopping` or a different library. You can simply pass a callable as the ``method`` parameter. The callable is called as ``method(fun, x0, args, **kwargs, **options)`` where ``kwargs`` corresponds to any other parameters passed to `minimize` (such as `callback`, `hess`, etc.), except the `options` dict, which has its contents also passed as `method` parameters pair by pair. Also, if `jac` has been passed as a bool type, `jac` and `fun` are mangled so that `fun` returns just the function values and `jac` is converted to a function returning the Jacobian. The method shall return an `OptimizeResult` object. The provided `method` callable must be able to accept (and possibly ignore) arbitrary parameters; the set of parameters accepted by `minimize` may expand in future versions and then these parameters will be passed to the method. You can find an example in the scipy.optimize tutorial. References ---------- .. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function Minimization. The Computer Journal 7: 308-13. .. [2] Wright M H. 1996. Direct search methods: Once scorned, now respectable, in Numerical Analysis 1995: Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis (Eds. D F Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK. 191-208. .. [3] Powell, M J D. 1964. An efficient method for finding the minimum of a function of several variables without calculating derivatives. The Computer Journal 7: 155-162. .. [4] Press W, S A Teukolsky, W T Vetterling and B P Flannery. Numerical Recipes (any edition), Cambridge University Press. .. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization. Springer New York. .. [6] Byrd, R H and P Lu and J. Nocedal. 1995. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM Journal on Scientific and Statistical Computing 16 (5): 1190-1208. .. [7] Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization. ACM Transactions on Mathematical Software 23 (4): 550-560. .. [8] Nash, S G. Newton-Type Minimization Via the Lanczos Method. 1984. SIAM Journal of Numerical Analysis 21: 770-778. .. [9] Powell, M J D. A direct search optimization method that models the objective and constraint functions by linear interpolation. 1994. Advances in Optimization and Numerical Analysis, eds. S. Gomez and J-P Hennart, Kluwer Academic (Dordrecht), 51-67. .. [10] Powell M J D. Direct search algorithms for optimization calculations. 1998. Acta Numerica 7: 287-336. .. [11] Powell M J D. A view of algorithms for optimization without derivatives. 2007.Cambridge University Technical Report DAMTP 2007/NA03 .. [12] Kraft, D. A software package for sequential quadratic programming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace Center -- Institute for Flight Mechanics, Koln, Germany. .. [13] Conn, A. R., Gould, N. I., and Toint, P. L. Trust region methods. 2000. Siam. pp. 169-200. .. [14] F. Lenders, C. Kirches, A. Potschka: "trlib: A vector-free implementation of the GLTR method for iterative solution of the trust region problem", :arxiv:`1611.04718` .. [15] N. Gould, S. Lucidi, M. Roma, P. Toint: "Solving the Trust-Region Subproblem using the Lanczos Method", SIAM J. Optim., 9(2), 504--525, (1999). .. [16] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. 1999. An interior point algorithm for large-scale nonlinear programming. SIAM Journal on Optimization 9.4: 877-900. .. [17] Lalee, Marucha, Jorge Nocedal, and Todd Plantenga. 1998. On the implementation of an algorithm for large-scale equality constrained optimization. SIAM Journal on Optimization 8.3: 682-706. .. [18] Ragonneau, T. M. *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 -------- Let us consider the problem of minimizing the Rosenbrock function. This function (and its respective derivatives) is implemented in `rosen` (resp. `rosen_der`, `rosen_hess`) in the `scipy.optimize`. >>> from scipy.optimize import minimize, rosen, rosen_der A simple application of the *Nelder-Mead* method is: >>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2] >>> res = minimize(rosen, x0, method='Nelder-Mead', tol=1e-6) >>> res.x array([ 1., 1., 1., 1., 1.]) Now using the *BFGS* algorithm, using the first derivative and a few options: >>> res = minimize(rosen, x0, method='BFGS', jac=rosen_der, ... options={'gtol': 1e-6, 'disp': True}) Optimization terminated successfully. Current function value: 0.000000 Iterations: 26 Function evaluations: 31 Gradient evaluations: 31 >>> res.x array([ 1., 1., 1., 1., 1.]) >>> print(res.message) Optimization terminated successfully. >>> res.hess_inv array([ [ 0.00749589, 0.01255155, 0.02396251, 0.04750988, 0.09495377], # may vary [ 0.01255155, 0.02510441, 0.04794055, 0.09502834, 0.18996269], [ 0.02396251, 0.04794055, 0.09631614, 0.19092151, 0.38165151], [ 0.04750988, 0.09502834, 0.19092151, 0.38341252, 0.7664427 ], [ 0.09495377, 0.18996269, 0.38165151, 0.7664427, 1.53713523] ]) Next, consider a minimization problem with several constraints (namely Example 16.4 from [5]_). The objective function is: >>> fun = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2 There are three constraints defined as: >>> cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2}, ... {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6}, ... {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2}) And variables must be positive, hence the following bounds: >>> bnds = ((0, None), (0, None)) The optimization problem is solved using the SLSQP method as: >>> res = minimize(fun, (2, 0), method='SLSQP', bounds=bnds, ... constraints=cons) It should converge to the theoretical solution (1.4 ,1.7). rz"'x0' must only have one dimension. 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S U* w- =   w ,   w , 1 1   vs + 9 9   vs + @ @   vs + ( (   uc * 8    0# 6 > !   vs +   vs +   }c 2 yc4D#'4+24 4*+4@KK #F6!&d3 / /yyFII-Gvvg9k &c]*I"t 455 $I# "++-HY-G$%-K!99g.\,V='g>H%%0GLHC==+C'!LC6AA[csxxz[ A&C!23u:w3:A:;"=CJ t 455%6s5z77>>?&AE '$F5h-H }"3D.6.%,.  sJ'J 3DxC7C SdEWE   $T(,#*,  s6=(0=4;= CT''%'  s9H&,9079  sk*!(* ct&)I4<I@GI  *3Dt+1I4<I@GI  s4=(0=4;=  !#4d@+3@7>@  $SdC.6C:AC  )#4dH3;H?FH?6(344ceeWg6"&&9  CLx)511  : JMBs[c[U[5(dU4n[U5(aSnOUcUcSOSnOUR5nUc0nUbUS;aSUS3n[ U5eUbV[ U5nUS:XaSU;a[ S[S S 9 XVS'O+US:XaURS U5 OURS U5 URS 5n [U [5(aS [U 5-US 'US:Xa U"U4X1US.UD6n ObUS:Xa[[XU40UD6n OIUS:XaUc [ S5e[XU40UD6n O'US:Xa[[XU40UD6n O[ SU35e[ R""U R$5SU l[ R&"U R(U R$R*5SU lU $)a$Local minimization of scalar function of one variable. Parameters ---------- fun : callable Objective function. Scalar function, must return a scalar. Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where ``my_args`` and ``my_kwargs`` are required positional and keyword arguments. Rather than passing ``f0`` as the callable, wrap it to accept only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been gathered before invoking this function. bracket : sequence, optional For methods 'brent' and 'golden', `bracket` defines the bracketing interval and is required. Either a triple ``(xa, xb, xc)`` satisfying ``xa < xb < xc`` and ``func(xb) < func(xa) and func(xb) < func(xc)``, or a pair ``(xa, xb)`` to be used as initial points for a downhill bracket search (see `scipy.optimize.bracket`). The minimizer ``res.x`` will not necessarily satisfy ``xa <= res.x <= xb``. bounds : sequence, optional For method 'bounded', `bounds` is mandatory and must have two finite items corresponding to the optimization bounds. args : tuple, optional Extra arguments passed to the objective function. method : str or callable, optional Type of solver. Should be one of: - :ref:`Brent ` - :ref:`Bounded ` - :ref:`Golden ` - custom - a callable object (added in version 0.14.0), see below Default is "Bounded" if bounds are provided and "Brent" otherwise. See the 'Notes' section for details of each solver. tol : float, optional Tolerance for termination. For detailed control, use solver-specific options. options : dict, optional A dictionary of solver options. maxiter : int Maximum number of iterations to perform. disp : bool Set to True to print convergence messages. See :func:`show_options()` for solver-specific options. 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. See `OptimizeResult` for a description of other attributes. See also -------- minimize : Interface to minimization algorithms for scalar multivariate functions 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 the ``"Bounded"`` Brent method if `bounds` are passed and unbounded ``"Brent"`` otherwise. Method :ref:`Brent ` uses Brent's algorithm [1]_ to find a local minimum. The algorithm uses inverse parabolic interpolation when possible to speed up convergence of the golden section method. Method :ref:`Golden ` uses the golden section search technique [1]_. It uses analog of the bisection method to decrease the bracketed interval. It is usually preferable to use the *Brent* method. Method :ref:`Bounded ` can perform bounded minimization [2]_ [3]_. It uses the Brent method to find a local minimum in the interval x1 < xopt < x2. Note that the Brent and Golden methods do not guarantee success unless a valid ``bracket`` triple is provided. If a three-point bracket cannot be found, consider `scipy.optimize.minimize`. Also, all methods are intended only for local minimization. When the function of interest has more than one local minimum, consider :ref:`global_optimization`. **Custom minimizers** It may be useful to pass a custom minimization method, for example when using some library frontend to minimize_scalar. You can simply pass a callable as the ``method`` parameter. The callable is called as ``method(fun, args, **kwargs, **options)`` where ``kwargs`` corresponds to any other parameters passed to `minimize` (such as `bracket`, `tol`, etc.), except the `options` dict, which has its contents also passed as `method` parameters pair by pair. The method shall return an `OptimizeResult` object. The provided `method` callable must be able to accept (and possibly ignore) arbitrary parameters; the set of parameters accepted by `minimize` may expand in future versions and then these parameters will be passed to the method. You can find an example in the scipy.optimize tutorial. .. versionadded:: 0.11.0 References ---------- .. [1] Press, W., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes in C. Cambridge University Press. .. [2] Forsythe, G.E., M. A. Malcolm, and C. B. Moler. "Computer Methods for Mathematical Computations." Prentice-Hall Series in Automatic Computation 259 (1977). .. [3] Brent, Richard P. Algorithms for Minimization Without Derivatives. Courier Corporation, 2013. Examples -------- Consider the problem of minimizing the following function. >>> def f(x): ... return (x - 2) * x * (x + 2)**2 Using the *Brent* method, we find the local minimum as: >>> from scipy.optimize import minimize_scalar >>> res = minimize_scalar(f) >>> res.fun -9.9149495908 The minimizer is: >>> res.x 1.28077640403 Using the *Bounded* method, we find a local minimum with specified bounds as: >>> res = minimize_scalar(f, bounds=(-3, -1), method='bounded') >>> res.fun # minimum 3.28365179850e-13 >>> res.x # minimizer -2.0000002026 r>r5r6>r5r7z-Use of `bounds` is incompatible with 'method=z'.rCz\Method 'bounded' does not support relative tolerance in x; defaulting to absolute tolerance.r?r@rHrEdisp)rKbracketrOz9The `bounds` parameter is mandatory for method `bounded`.r7rV)rbrcrdrer^rkrrgrlrirfintrr r rrZr\rTreshaperxshape) rTrrOrKrrHrrr}rrs rrrst dE " "w  .wi||~ d&99A&L!! w- 9 !7 5A / #G  Y    uc *   vs + ;;v D$c$i- ySOtVOwO )*@*-I@GI   >12 2&sDDGD  )*A*-I@GI?6(344 jj!"%CG JJsuucggmm ,R 0CE JcXURU)nURU)n[X#5$)z0Removes fixed variables from a `Bounds` instance)rprqr#)rOrrprqs rrtrts, G8 B G8 B ">rc$^^^^^UUUUU4SjnU$)z@Wraps a function such that fixed variables need not be passed incN>[R"TURS9nT UT'XT)'T"U/UQ70UD6n[R"U5nTS:Xa[R"U5nOTS:Xa[R "U5nTS:Xa UST)4nU$TS:Xa UT)T)4nU$)Nr:rr?.)rZ zeros_liker;arrayr[ atleast_2d) x_inrKkwargsx_outy_outfun_inrrUrSrs rfun_out"_remove_from_func..fun_outs gTZZ8 gwhu.t.v. a<MM%(E \MM%(E Q;#x-(E q[7(WH,-E rr)rrrrUrSrs````` rrurus$ NrcU)nURS:XaUSS2S4USSS24-n[R"X0RS9nX$U)'UR 5XC'U$)z%Adds fixed variables back to an arrayr?Nr:)r]rZrr;ravel)rrri_freers rrwrws[XF yyA~46$'?2 MM& 3E6'NJJLEM LrcSn[R"URUR:5(a [ U5eSn[R "URUR 5Ul[R "URUR 5UlU$![an[ U5UeSnAff=f)zCheck that bounds are valid.z:An upper bound is less than the corresponding lower bound.z?The number of bounds is not compatible with the length of `x0`.N)rZrhrqrpr^ broadcast_tor Exception)rOr~rmsges rroro s GC vvfii&))#$$o KC%OOFIIrxx8 OOFIIrxx8  M %o1$%sA B%% C/ B;;CcUS;a/[U[5(d[U5up4[X45nU$US;aC[U[5(a.[URUR UR S5nU$)z3Converts bounds to the form required by the solver.>rRr-r.r'r&r0)r+r,r/oldr)rbr#rrrprqr)rOr~rrprqs rrnrnss &&))%f-FBB^F M 4 4 ff % %&vyy&))RXXa[IF Mrc[[[4nUSSnUc/nO[X5(aU/nO [ U5nUS;a5[ U5H$upV[Xd5(aM[ XV5X'M& U$[ [ U55H=upV[Xd5(dM[Xa5nUSX'URUSS5 M? U$)z8Converts constraints to the form required by the solver.N)r0r.rRrr) r!r"rkrblist enumeraterr extend)rPr~rall_constraint_typesnew_constraint_typesirold_constraintss rrmrm*s/1A4H/4 K 6 6"m ;'  00 ,FAc88!6q!> -   [ 12FA#44"7"@!0!3 ""?12#67 3 rc LSnSnURnU(aSn[XGS5nSnUHdn [X5n U RU5n [R "U 5(dM<[ U[R "U 55nSnSUS3nMf [Xp"U/UQ76XVS SSS 9$) z Provides a default OptimizeResult for when a bounded minimization method has (lb == ub).all(). Parameters ---------- fun: callable bounds: Bounds method: str constraints: Constraint Tz/All independent variables were fixed by bounds.zVAll independent variables were fixed by bounds at values that satisfy the constraints.rRrFzAll independent variables were fixed by bounds, but the independent variables do not satisfy the constraints exactly. (Maximum violation: z).r)rxrTr{r}nfevnjevnhev)rprmr$ violationrZsummaxr) rTrOrrKrPr{r}r~maxcvcpcrs rrsrsDsG?G B4-kuE E   &LL$ 66)  rvvi01EGCCH'MG   #b.4.' 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