scipy optimize minimize исходный код

scipy.optimize.minimize_scalar¶

Minimization of scalar function of one variable.

Parameters fun callable

Objective function. Scalar function, must return a scalar.

bracket sequence, optional

bounds sequence, optional

For method ‘bounded’, bounds is mandatory and must have two 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:

Tolerance for termination. For detailed control, use solver-specific options.

options dict, optional

A dictionary of solver options.

Maximum number of iterations to perform.

Set to True to print convergence messages.

See 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.

Interface to minimization algorithms for scalar multivariate functions

Additional options accepted by the solvers

This section describes the available solvers that can be selected by the ‘method’ parameter. The default method is Brent.

Method Brent uses Brent’s algorithm to find a local minimum. The algorithm uses inverse parabolic interpolation when possible to speed up convergence of the golden section method.

Method Golden uses the golden section search technique. It uses analog of the bisection method to decrease the bracketed interval. It is usually preferable to use the Brent method.

Method Bounded can perform bounded minimization. It uses the Brent method to find a local minimum in the interval x1 method parameter.

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.

New in version 0.11.0.

Consider the problem of minimizing the following function.

Using the Brent method, we find the local minimum as:

Using the Bounded method, we find a local minimum with specified bounds as:

© Copyright 2008-2021, The SciPy community.

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scipy.optimize.minimize¶

Minimization of scalar function of one or more variables.

New in version 0.11.0.

fun : callable

x0 : ndarray

args : tuple, optional

Extra arguments passed to the objective function and its derivatives (Jacobian, Hessian).

method : str or callable, optional

Type of solver. Should be one of

jac : bool or callable, optional

Jacobian (gradient) of objective function. Only for CG, BFGS, Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If jac is a Boolean and is True, fun is assumed to return the gradient along with the objective function. If False, the gradient will be estimated numerically. jac can also be a callable returning the gradient of the objective. In this case, it must accept the same arguments as fun.

hess, hessp : callable, optional

Hessian (matrix of second-order derivatives) of objective function or Hessian of objective function times an arbitrary vector p. Only for Newton-CG, dogleg, trust-ncg. Only one of hessp or hess needs to be given. If hess is provided, then hessp will be ignored. If neither hess nor hessp is provided, then the Hessian product will be approximated using finite differences on jac. hessp must compute the Hessian times an arbitrary vector.

bounds : sequence, optional

constraints : dict or sequence of dict, optional

Constraints definition (only for COBYLA and SLSQP). Each constraint is defined in a dictionary with fields:

Constraint type: ‘eq’ for equality, ‘ineq’ for inequality.

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. For detailed control, use solver-specific options.

options : dict, optional

A dictionary of solver options. All methods accept the following generic options:

Maximum number of iterations to perform.

Set to True to print convergence messages.

callback : callable, optional

res : OptimizeResult

This section describes the available solvers that can be selected by the ‘method’ parameter. The default method is BFGS.

Unconstrained minimization

Method Nelder-Mead uses the Simplex algorithm [R101], [R102]. This algorithm has been successful in many applications but other algorithms using the first and/or second derivatives information might be preferred for their better performances and robustness in general.

Method Powell is a modification of Powell’s method [R103], [R104] 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.

Method CG uses a nonlinear conjugate gradient algorithm by Polak and Ribiere, a variant of the Fletcher-Reeves method described in [R105] pp. 120-122. Only the first derivatives are used.

Method BFGS uses the quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [R105] 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 Newton-CG uses a Newton-CG algorithm [R105] 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.

Method Anneal uses simulated annealing, which is a probabilistic metaheuristic algorithm for global optimization. It uses no derivative information from the function being optimized.

Method dogleg uses the dog-leg trust-region algorithm [R105] for unconstrained minimization. This algorithm requires the gradient and Hessian; furthermore the Hessian is required to be positive definite.

Method trust-ncg uses the Newton conjugate gradient trust-region algorithm [R105] 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.

Constrained minimization

Method L-BFGS-B uses the L-BFGS-B algorithm [R106], [R107] for bound constrained minimization.

Method TNC uses a truncated Newton algorithm [R105], [R108] 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.

Method COBYLA uses the Constrained Optimization BY Linear Approximation (COBYLA) method [R109], [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.

Method 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.

Custom minimizers

A simple application of the Nelder-Mead method is:

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scipy.optimize.minimize¶

Minimization of scalar function of one or more variables.

New in version 0.11.0.

fun : callable

x0 : ndarray

args : tuple, optional

Extra arguments passed to the objective function and its derivatives (Jacobian, Hessian).

method : str, optional

Type of solver. Should be one of

jac : bool or callable, optional

Jacobian of objective function. Only for CG, BFGS, Newton-CG, dogleg, trust-ncg. If jac is a Boolean and is True, fun is assumed to return the value of Jacobian along with the objective function. If False, the Jacobian will be estimated numerically. jac can also be a callable returning the Jacobian of the objective. In this case, it must accept the same arguments as fun.

hess, hessp : callable, optional

Hessian of objective function or Hessian of objective function times an arbitrary vector p. Only for Newton-CG, dogleg, trust-ncg. Only one of hessp or hess needs to be given. If hess is provided, then hessp will be ignored. If neither hess nor hessp is provided, then the hessian product will be approximated using finite differences on jac. hessp must compute the Hessian times an arbitrary vector.

bounds : sequence, optional

constraints : dict or sequence of dict, optional

Constraints definition (only for COBYLA and SLSQP). Each constraint is defined in a dictionary with fields:

Constraint type: ‘eq’ for equality, ‘ineq’ for inequality.

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. For detailed control, use solver-specific options.

options : dict, optional

A dictionary of solver options. All methods accept the following generic options:

Maximum number of iterations to perform.

Set to True to print convergence messages.

For method-specific options, see show_options(‘minimize’, method).

callback : callable, optional

res : Result

The optimization result represented as a Result 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 Result for a description of other attributes.

This section describes the available solvers that can be selected by the ‘method’ parameter. The default method is BFGS.

Unconstrained minimization

Method Nelder-Mead uses the Simplex algorithm [R93], [R94]. This algorithm has been successful in many applications but other algorithms using the first and/or second derivatives information might be preferred for their better performances and robustness in general.

Method Powell is a modification of Powell’s method [R95], [R96] 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.

Method CG uses a nonlinear conjugate gradient algorithm by Polak and Ribiere, a variant of the Fletcher-Reeves method described in [R97] pp. 120-122. Only the first derivatives are used.

Method BFGS uses the quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [R97] 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 Result object.

Method Newton-CG uses a Newton-CG algorithm [R97] 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.

Method Anneal uses simulated annealing, which is a probabilistic metaheuristic algorithm for global optimization. It uses no derivative information from the function being optimized.

Method dogleg uses the dog-leg trust-region algorithm [R97] for unconstrained minimization. This algorithm requires the gradient and Hessian; furthermore the Hessian is required to be positive definite.

Method trust-ncg uses the Newton conjugate gradient trust-region algorithm [R97] 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.

Constrained minimization

Method L-BFGS-B uses the L-BFGS-B algorithm [R98], [R99] for bound constrained minimization.

Method TNC uses a truncated Newton algorithm [R97], [R100] 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.

Method COBYLA uses the Constrained Optimization BY Linear Approximation (COBYLA) method [R101], [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.

Method 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].

A simple application of the Nelder-Mead method is:

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scipy.optimize.minimize¶

Minimization of scalar function of one or more variables.

In general, the optimization problems are of the form:

where x is a vector of one or more variables. g_i(x) are the inequality constraints. h_j(x) are the equality constrains.

Optionally, the lower and upper bounds for each element in x can also be specified using the bounds argument.

fun : callable

x0 : ndarray

args : tuple, optional

Extra arguments passed to the objective function and its derivatives (Jacobian, Hessian).

method : str or callable, optional

Type of solver. Should be one of

jac : bool or callable, optional

Jacobian (gradient) of objective function. Only for CG, BFGS, Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If jac is a Boolean and is True, fun is assumed to return the gradient along with the objective function. If False, the gradient will be estimated numerically. jac can also be a callable returning the gradient of the objective. In this case, it must accept the same arguments as fun.

hess, hessp : callable, optional

Hessian (matrix of second-order derivatives) of objective function or Hessian of objective function times an arbitrary vector p. Only for Newton-CG, dogleg, trust-ncg. Only one of hessp or hess needs to be given. If hess is provided, then hessp will be ignored. If neither hess nor hessp is provided, then the Hessian product will be approximated using finite differences on jac. hessp must compute the Hessian times an arbitrary vector.

bounds : sequence, optional

constraints : dict or sequence of dict, optional

Constraints definition (only for COBYLA and SLSQP). Each constraint is defined in a dictionary with fields:

Constraint type: ‘eq’ for equality, ‘ineq’ for inequality.

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. For detailed control, use solver-specific options.

options : dict, optional

A dictionary of solver options. All methods accept the following generic options:

Maximum number of iterations to perform.

Set to True to print convergence messages.

callback : callable, optional

res : OptimizeResult

This section describes the available solvers that can be selected by the ‘method’ parameter. The default method is BFGS.

Unconstrained minimization

Method Nelder-Mead uses the Simplex algorithm [R174], [R175]. 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 Powell is a modification of Powell’s method [R176], [R177] 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.

Method CG uses a nonlinear conjugate gradient algorithm by Polak and Ribiere, a variant of the Fletcher-Reeves method described in [R178] pp. 120-122. Only the first derivatives are used.

Method BFGS uses the quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [R178] 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 Newton-CG uses a Newton-CG algorithm [R178] 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.

Method dogleg uses the dog-leg trust-region algorithm [R178] for unconstrained minimization. This algorithm requires the gradient and Hessian; furthermore the Hessian is required to be positive definite.

Method trust-ncg uses the Newton conjugate gradient trust-region algorithm [R178] 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.

Constrained minimization

Method L-BFGS-B uses the L-BFGS-B algorithm [R179], [R180] for bound constrained minimization.

Method TNC uses a truncated Newton algorithm [R178], [R181] 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.

Method COBYLA uses the Constrained Optimization BY Linear Approximation (COBYLA) method [R182], [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 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.

Custom minimizers

New in version 0.11.0.

A simple application of the Nelder-Mead method is:

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