Hide keyboard shortcuts

Hot-keys on this page

r m x p   toggle line displays

j k   next/prev highlighted chunk

0   (zero) top of page

1   (one) first highlighted chunk

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

""" 

===================================================== 

Optimization and Root Finding (:mod:`scipy.optimize`) 

===================================================== 

 

.. currentmodule:: scipy.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 programing, 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-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 

------------------- 

 

.. autosummary:: 

:toctree: generated/ 

 

basinhopping - Basinhopping stochastic optimizer. 

brute - Brute force searching optimizer. 

differential_evolution - stochastic minimization using differential evolution. 

 

shgo - simplicial homology global optimisation 

dual_annealing - Dual annealing stochastic 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. 

 

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 zeros 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 

 

Linear Programming 

================== 

 

.. autosummary:: 

:toctree: generated/ 

 

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 

 

The simplex method supports 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. 

 

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) level set 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. 

differential_evolution - stochastic minimization using differential evolution. 

 

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 multi-variable equation solver. 

broyden1 - Broyden's first method. 

broyden2 - Broyden's second method. 

 

Large-scale nonlinear solvers: 

 

.. autosummary:: 

:toctree: generated/ 

 

newton_krylov 

anderson 

 

Simple iteration solvers: 

 

.. autosummary:: 

:toctree: generated/ 

 

excitingmixing 

linearmixing 

diagbroyden 

 

:mod:`Additional information on the nonlinear solvers <scipy.optimize.nonlin>` 

""" 

 

from __future__ import division, print_function, absolute_import 

 

from .optimize import * 

from ._minimize import * 

from ._root import * 

from ._root_scalar import * 

from .minpack import * 

from .zeros import * 

from .lbfgsb import fmin_l_bfgs_b, LbfgsInvHessProduct 

from .tnc import fmin_tnc 

from .cobyla import fmin_cobyla 

from .nonlin import * 

from .slsqp import fmin_slsqp 

from .nnls import nnls 

from ._basinhopping import basinhopping 

from ._linprog import linprog, linprog_verbose_callback 

from ._hungarian import linear_sum_assignment 

from ._differentialevolution import differential_evolution 

from ._lsq import least_squares, lsq_linear 

from ._constraints import (NonlinearConstraint, 

LinearConstraint, 

Bounds) 

from ._hessian_update_strategy import HessianUpdateStrategy, BFGS, SR1 

from ._shgo import shgo 

from ._dual_annealing import dual_annealing 

 

__all__ = [s for s in dir() if not s.startswith('_')] 

 

from scipy._lib._testutils import PytestTester 

test = PytestTester(__name__) 

del PytestTester