| """ |
| ===================================================== |
| Optimization and root finding (:mod:`scipy.optimize`) |
| ===================================================== |
| |
| .. currentmodule:: scipy.optimize |
| |
| .. toctree:: |
| :hidden: |
| |
| optimize.cython_optimize |
| |
| SciPy ``optimize`` provides functions for minimizing (or maximizing) |
| objective functions, possibly subject to constraints. It includes |
| solvers for nonlinear problems (with support for both local and global |
| optimization algorithms), linear programming, constrained |
| and nonlinear least-squares, root finding, and curve fitting. |
| |
| Common functions and objects, shared across different solvers, are: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| show_options - Show specific options optimization solvers. |
| OptimizeResult - The optimization result returned by some optimizers. |
| OptimizeWarning - The optimization encountered problems. |
| |
| |
| Optimization |
| ============ |
| |
| Scalar functions optimization |
| ----------------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| minimize_scalar - Interface for minimizers of univariate functions |
| |
| The `minimize_scalar` function supports the following methods: |
| |
| .. toctree:: |
| |
| optimize.minimize_scalar-brent |
| optimize.minimize_scalar-bounded |
| optimize.minimize_scalar-golden |
| |
| Local (multivariate) optimization |
| --------------------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| minimize - Interface for minimizers of multivariate functions. |
| |
| The `minimize` function supports the following methods: |
| |
| .. toctree:: |
| |
| optimize.minimize-neldermead |
| optimize.minimize-powell |
| optimize.minimize-cg |
| optimize.minimize-bfgs |
| optimize.minimize-newtoncg |
| optimize.minimize-lbfgsb |
| optimize.minimize-tnc |
| optimize.minimize-cobyla |
| optimize.minimize-cobyqa |
| optimize.minimize-slsqp |
| optimize.minimize-trustconstr |
| optimize.minimize-dogleg |
| optimize.minimize-trustncg |
| optimize.minimize-trustkrylov |
| optimize.minimize-trustexact |
| |
| Constraints are passed to `minimize` function as a single object or |
| as a list of objects from the following classes: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| NonlinearConstraint - Class defining general nonlinear constraints. |
| LinearConstraint - Class defining general linear constraints. |
| |
| Simple bound constraints are handled separately and there is a special class |
| for them: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| Bounds - Bound constraints. |
| |
| Quasi-Newton strategies implementing `HessianUpdateStrategy` |
| interface can be used to approximate the Hessian in `minimize` |
| function (available only for the 'trust-constr' method). Available |
| quasi-Newton methods implementing this interface are: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| BFGS - Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update strategy. |
| SR1 - Symmetric-rank-1 Hessian update strategy. |
| |
| .. _global_optimization: |
| |
| Global optimization |
| ------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| basinhopping - Basinhopping stochastic optimizer. |
| brute - Brute force searching optimizer. |
| differential_evolution - Stochastic optimizer using differential evolution. |
| |
| shgo - Simplicial homology global optimizer. |
| dual_annealing - Dual annealing stochastic optimizer. |
| direct - DIRECT (Dividing Rectangles) optimizer. |
| |
| Least-squares and curve fitting |
| =============================== |
| |
| Nonlinear least-squares |
| ----------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| least_squares - Solve a nonlinear least-squares problem with bounds on the variables. |
| |
| Linear least-squares |
| -------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| nnls - Linear least-squares problem with non-negativity constraint. |
| lsq_linear - Linear least-squares problem with bound constraints. |
| isotonic_regression - Least squares problem of isotonic regression via PAVA. |
| |
| Curve fitting |
| ------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| curve_fit -- Fit curve to a set of points. |
| |
| Root finding |
| ============ |
| |
| Scalar functions |
| ---------------- |
| .. autosummary:: |
| :toctree: generated/ |
| |
| root_scalar - Unified interface for nonlinear solvers of scalar functions. |
| brentq - quadratic interpolation Brent method. |
| brenth - Brent method, modified by Harris with hyperbolic extrapolation. |
| ridder - Ridder's method. |
| bisect - Bisection method. |
| newton - Newton's method (also Secant and Halley's methods). |
| toms748 - Alefeld, Potra & Shi Algorithm 748. |
| RootResults - The root finding result returned by some root finders. |
| |
| The `root_scalar` function supports the following methods: |
| |
| .. toctree:: |
| |
| optimize.root_scalar-brentq |
| optimize.root_scalar-brenth |
| optimize.root_scalar-bisect |
| optimize.root_scalar-ridder |
| optimize.root_scalar-newton |
| optimize.root_scalar-toms748 |
| optimize.root_scalar-secant |
| optimize.root_scalar-halley |
| |
| |
| |
| The table below lists situations and appropriate methods, along with |
| *asymptotic* convergence rates per iteration (and per function evaluation) |
| for successful convergence to a simple root(*). |
| Bisection is the slowest of them all, adding one bit of accuracy for each |
| function evaluation, but is guaranteed to converge. |
| The other bracketing methods all (eventually) increase the number of accurate |
| bits by about 50% for every function evaluation. |
| The derivative-based methods, all built on `newton`, can converge quite quickly |
| if the initial value is close to the root. They can also be applied to |
| functions defined on (a subset of) the complex plane. |
| |
| +-------------+----------+----------+-----------+-------------+-------------+----------------+ |
| | Domain of f | Bracket? | Derivatives? | Solvers | Convergence | |
| + + +----------+-----------+ +-------------+----------------+ |
| | | | `fprime` | `fprime2` | | Guaranteed? | Rate(s)(*) | |
| +=============+==========+==========+===========+=============+=============+================+ |
| | `R` | Yes | N/A | N/A | - bisection | - Yes | - 1 "Linear" | |
| | | | | | - brentq | - Yes | - >=1, <= 1.62 | |
| | | | | | - brenth | - Yes | - >=1, <= 1.62 | |
| | | | | | - ridder | - Yes | - 2.0 (1.41) | |
| | | | | | - toms748 | - Yes | - 2.7 (1.65) | |
| +-------------+----------+----------+-----------+-------------+-------------+----------------+ |
| | `R` or `C` | No | No | No | secant | No | 1.62 (1.62) | |
| +-------------+----------+----------+-----------+-------------+-------------+----------------+ |
| | `R` or `C` | No | Yes | No | newton | No | 2.00 (1.41) | |
| +-------------+----------+----------+-----------+-------------+-------------+----------------+ |
| | `R` or `C` | No | Yes | Yes | halley | No | 3.00 (1.44) | |
| +-------------+----------+----------+-----------+-------------+-------------+----------------+ |
| |
| .. seealso:: |
| |
| `scipy.optimize.cython_optimize` -- Typed Cython versions of root finding functions |
| |
| Fixed point finding: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| fixed_point - Single-variable fixed-point solver. |
| |
| Multidimensional |
| ---------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| root - Unified interface for nonlinear solvers of multivariate functions. |
| |
| The `root` function supports the following methods: |
| |
| .. toctree:: |
| |
| optimize.root-hybr |
| optimize.root-lm |
| optimize.root-broyden1 |
| optimize.root-broyden2 |
| optimize.root-anderson |
| optimize.root-linearmixing |
| optimize.root-diagbroyden |
| optimize.root-excitingmixing |
| optimize.root-krylov |
| optimize.root-dfsane |
| |
| Elementwise Minimization and Root Finding |
| ========================================= |
| |
| .. toctree:: |
| :maxdepth: 3 |
| |
| optimize.elementwise |
| |
| Linear programming / MILP |
| ========================= |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| milp -- Mixed integer linear programming. |
| linprog -- Unified interface for minimizers of linear programming problems. |
| |
| The `linprog` function supports the following methods: |
| |
| .. toctree:: |
| |
| optimize.linprog-simplex |
| optimize.linprog-interior-point |
| optimize.linprog-revised_simplex |
| optimize.linprog-highs-ipm |
| optimize.linprog-highs-ds |
| optimize.linprog-highs |
| |
| The simplex, interior-point, and revised simplex methods support callback |
| functions, such as: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| linprog_verbose_callback -- Sample callback function for linprog (simplex). |
| |
| Assignment problems |
| =================== |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| linear_sum_assignment -- Solves the linear-sum assignment problem. |
| quadratic_assignment -- Solves the quadratic assignment problem. |
| |
| The `quadratic_assignment` function supports the following methods: |
| |
| .. toctree:: |
| |
| optimize.qap-faq |
| optimize.qap-2opt |
| |
| Utilities |
| ========= |
| |
| Finite-difference approximation |
| ------------------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| approx_fprime - Approximate the gradient of a scalar function. |
| check_grad - Check the supplied derivative using finite differences. |
| |
| |
| Line search |
| ----------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| bracket - Bracket a minimum, given two starting points. |
| line_search - Return a step that satisfies the strong Wolfe conditions. |
| |
| Hessian approximation |
| --------------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| LbfgsInvHessProduct - Linear operator for L-BFGS approximate inverse Hessian. |
| HessianUpdateStrategy - Interface for implementing Hessian update strategies |
| |
| Benchmark problems |
| ------------------ |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| rosen - The Rosenbrock function. |
| rosen_der - The derivative of the Rosenbrock function. |
| rosen_hess - The Hessian matrix of the Rosenbrock function. |
| rosen_hess_prod - Product of the Rosenbrock Hessian with a vector. |
| |
| Legacy functions |
| ================ |
| |
| The functions below are not recommended for use in new scripts; |
| all of these methods are accessible via a newer, more consistent |
| interfaces, provided by the interfaces above. |
| |
| Optimization |
| ------------ |
| |
| General-purpose multivariate methods: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| fmin - Nelder-Mead Simplex algorithm. |
| fmin_powell - Powell's (modified) conjugate direction method. |
| fmin_cg - Non-linear (Polak-Ribiere) conjugate gradient algorithm. |
| fmin_bfgs - Quasi-Newton method (Broydon-Fletcher-Goldfarb-Shanno). |
| fmin_ncg - Line-search Newton Conjugate Gradient. |
| |
| Constrained multivariate methods: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| fmin_l_bfgs_b - Zhu, Byrd, and Nocedal's constrained optimizer. |
| fmin_tnc - Truncated Newton code. |
| fmin_cobyla - Constrained optimization by linear approximation. |
| fmin_slsqp - Minimization using sequential least-squares programming. |
| |
| Univariate (scalar) minimization methods: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| fminbound - Bounded minimization of a scalar function. |
| brent - 1-D function minimization using Brent method. |
| golden - 1-D function minimization using Golden Section method. |
| |
| Least-squares |
| ------------- |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| leastsq - Minimize the sum of squares of M equations in N unknowns. |
| |
| Root finding |
| ------------ |
| |
| General nonlinear solvers: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| fsolve - Non-linear multivariable equation solver. |
| broyden1 - Broyden's first method. |
| broyden2 - Broyden's second method. |
| NoConvergence - Exception raised when nonlinear solver does not converge. |
| |
| Large-scale nonlinear solvers: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| newton_krylov |
| anderson |
| |
| BroydenFirst |
| InverseJacobian |
| KrylovJacobian |
| |
| Simple iteration solvers: |
| |
| .. autosummary:: |
| :toctree: generated/ |
| |
| excitingmixing |
| linearmixing |
| diagbroyden |
| |
| """ |
|
|
| from ._optimize import * |
| from ._minimize import * |
| from ._root import * |
| from ._root_scalar import * |
| from ._minpack_py import * |
| from ._zeros_py import * |
| from ._lbfgsb_py import fmin_l_bfgs_b, LbfgsInvHessProduct |
| from ._tnc import fmin_tnc |
| from ._cobyla_py import fmin_cobyla |
| from ._nonlin import * |
| from ._slsqp_py import fmin_slsqp |
| from ._nnls import nnls |
| from ._basinhopping import basinhopping |
| from ._linprog import linprog, linprog_verbose_callback |
| from ._lsap import linear_sum_assignment |
| from ._differentialevolution import differential_evolution |
| from ._lsq import least_squares, lsq_linear |
| from ._isotonic import isotonic_regression |
| from ._constraints import (NonlinearConstraint, |
| LinearConstraint, |
| Bounds) |
| from ._hessian_update_strategy import HessianUpdateStrategy, BFGS, SR1 |
| from ._shgo import shgo |
| from ._dual_annealing import dual_annealing |
| from ._qap import quadratic_assignment |
| from ._direct_py import direct |
| from ._milp import milp |
|
|
| |
| from . import ( |
| cobyla, lbfgsb, linesearch, minpack, minpack2, moduleTNC, nonlin, optimize, |
| slsqp, tnc, zeros |
| ) |
|
|
| __all__ = [s for s in dir() if not s.startswith('_')] |
|
|
| from scipy._lib._testutils import PytestTester |
| test = PytestTester(__name__) |
| del PytestTester |
|
|
