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

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

658

659

660

661

662

663

664

665

666

667

668

669

670

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

698

699

700

701

702

703

704

705

706

707

708

709

710

711

712

713

714

715

716

717

718

719

720

721

722

723

724

725

726

727

728

729

730

731

732

733

734

735

736

737

738

739

740

741

742

743

744

745

746

747

748

749

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765

766

767

768

769

770

771

772

773

774

775

776

777

778

779

780

781

782

783

784

785

786

787

788

789

790

791

792

793

794

795

796

797

798

799

800

801

802

803

804

805

806

807

808

809

810

811

812

813

814

815

816

817

818

819

820

821

822

823

824

825

826

827

828

829

830

831

832

833

834

835

836

837

838

839

840

841

842

843

844

845

846

847

848

849

850

851

852

853

854

855

856

857

858

859

860

861

862

863

864

865

866

867

868

869

870

871

872

873

874

875

876

877

878

879

880

881

882

883

884

885

886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

901

902

903

904

905

906

907

908

909

910

911

912

913

914

915

916

917

918

919

920

921

922

923

924

925

926

927

928

929

930

931

932

933

934

935

936

937

938

939

940

941

942

943

944

945

946

947

948

949

950

951

952

953

954

955

956

957

958

959

960

961

962

963

964

965

966

967

968

969

970

971

972

973

974

975

976

977

978

979

980

981

982

983

984

985

986

987

988

989

990

991

992

993

994

995

996

997

998

999

1000

1001

1002

1003

1004

1005

1006

1007

1008

1009

1010

1011

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

1024

1025

1026

1027

1028

1029

1030

1031

1032

1033

1034

1035

1036

1037

1038

1039

1040

1041

1042

1043

1044

1045

1046

1047

1048

1049

1050

1051

1052

1053

1054

1055

1056

1057

1058

1059

1060

1061

1062

1063

1064

1065

1066

1067

1068

1069

1070

1071

1072

1073

1074

1075

1076

1077

1078

1079

1080

1081

1082

1083

1084

1085

1086

1087

1088

1089

1090

1091

1092

1093

1094

1095

1096

1097

1098

1099

1100

1101

1102

1103

1104

1105

1106

1107

1108

1109

1110

1111

1112

1113

1114

1115

1116

1117

1118

1119

1120

1121

1122

1123

1124

1125

1126

1127

1128

1129

1130

1131

1132

1133

1134

1135

1136

1137

1138

1139

1140

1141

1142

1143

1144

1145

1146

1147

"""Interior-point method for linear programming 

 

The *interior-point* method uses the primal-dual path following algorithm 

outlined in [1]_. This algorithm supports sparse constraint matrices and 

is typically faster than the simplex methods, especially for large, sparse 

problems. Note, however, that the solution returned may be slightly less 

accurate than those of the simplex methods and will not, in general, 

correspond with a vertex of the polytope defined by the constraints. 

 

.. versionadded:: 1.0.0 

 

References 

---------- 

.. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

""" 

# Author: Matt Haberland 

 

from __future__ import print_function, division, absolute_import 

import numpy as np 

import scipy as sp 

import scipy.sparse as sps 

import numbers 

from warnings import warn 

from scipy.linalg import LinAlgError 

from .optimize import OptimizeWarning, OptimizeResult, _check_unknown_options 

from ._linprog_util import _postsolve 

has_umfpack = True 

has_cholmod = True 

try: 

from sksparse.cholmod import cholesky as cholmod 

except ImportError: 

has_cholmod = False 

try: 

import scikits.umfpack # test whether to use factorized 

except ImportError: 

has_umfpack = False 

 

 

def _get_solver(M, sparse=False, lstsq=False, sym_pos=True, 

cholesky=True, permc_spec='MMD_AT_PLUS_A'): 

""" 

Given solver options, return a handle to the appropriate linear system 

solver. 

 

Parameters 

---------- 

M : 2D array 

As defined in [4] Equation 8.31 

sparse : bool (default = False) 

True if the system to be solved is sparse. This is typically set 

True when the original ``A_ub`` and ``A_eq`` arrays are sparse. 

lstsq : bool (default = False) 

True if the system is ill-conditioned and/or (nearly) singular and 

thus a more robust least-squares solver is desired. This is sometimes 

needed as the solution is approached. 

sym_pos : bool (default = True) 

True if the system matrix is symmetric positive definite 

Sometimes this needs to be set false as the solution is approached, 

even when the system should be symmetric positive definite, due to 

numerical difficulties. 

cholesky : bool (default = True) 

True if the system is to be solved by Cholesky, rather than LU, 

decomposition. This is typically faster unless the problem is very 

small or prone to numerical difficulties. 

permc_spec : str (default = 'MMD_AT_PLUS_A') 

Sparsity preservation strategy used by SuperLU. Acceptable values are: 

 

- ``NATURAL``: natural ordering. 

- ``MMD_ATA``: minimum degree ordering on the structure of A^T A. 

- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. 

- ``COLAMD``: approximate minimum degree column ordering. 

 

See SuperLU documentation. 

 

Returns 

------- 

solve : function 

Handle to the appropriate solver function 

 

""" 

try: 

if sparse: 

if lstsq: 

def solve(r, sym_pos=False): 

return sps.linalg.lsqr(M, r)[0] 

elif cholesky: 

solve = cholmod(M) 

else: 

if has_umfpack and sym_pos: 

solve = sps.linalg.factorized(M) 

else: # factorized doesn't pass permc_spec 

solve = sps.linalg.splu(M, permc_spec=permc_spec).solve 

 

else: 

if lstsq: # sometimes necessary as solution is approached 

def solve(r): 

return sp.linalg.lstsq(M, r)[0] 

elif cholesky: 

L = sp.linalg.cho_factor(M) 

 

def solve(r): 

return sp.linalg.cho_solve(L, r) 

else: 

# this seems to cache the matrix factorization, so solving 

# with multiple right hand sides is much faster 

def solve(r, sym_pos=sym_pos): 

return sp.linalg.solve(M, r, sym_pos=sym_pos) 

# There are many things that can go wrong here, and it's hard to say 

# what all of them are. It doesn't really matter: if the matrix can't be 

# factorized, return None. get_solver will be called again with different 

# inputs, and a new routine will try to factorize the matrix. 

except KeyboardInterrupt: 

raise 

except Exception: 

return None 

return solve 

 

 

def _get_delta( 

A, 

b, 

c, 

x, 

y, 

z, 

tau, 

kappa, 

gamma, 

eta, 

sparse=False, 

lstsq=False, 

sym_pos=True, 

cholesky=True, 

pc=True, 

ip=False, 

permc_spec='MMD_AT_PLUS_A' 

): 

""" 

Given standard form problem defined by ``A``, ``b``, and ``c``; 

current variable estimates ``x``, ``y``, ``z``, ``tau``, and ``kappa``; 

algorithmic parameters ``gamma and ``eta; 

and options ``sparse``, ``lstsq``, ``sym_pos``, ``cholesky``, ``pc`` 

(predictor-corrector), and ``ip`` (initial point improvement), 

get the search direction for increments to the variable estimates. 

 

Parameters 

---------- 

As defined in [4], except: 

sparse : bool 

True if the system to be solved is sparse. This is typically set 

True when the original ``A_ub`` and ``A_eq`` arrays are sparse. 

lstsq : bool 

True if the system is ill-conditioned and/or (nearly) singular and 

thus a more robust least-squares solver is desired. This is sometimes 

needed as the solution is approached. 

sym_pos : bool 

True if the system matrix is symmetric positive definite 

Sometimes this needs to be set false as the solution is approached, 

even when the system should be symmetric positive definite, due to 

numerical difficulties. 

cholesky : bool 

True if the system is to be solved by Cholesky, rather than LU, 

decomposition. This is typically faster unless the problem is very 

small or prone to numerical difficulties. 

pc : bool 

True if the predictor-corrector method of Mehrota is to be used. This 

is almost always (if not always) beneficial. Even though it requires 

the solution of an additional linear system, the factorization 

is typically (implicitly) reused so solution is efficient, and the 

number of algorithm iterations is typically reduced. 

ip : bool 

True if the improved initial point suggestion due to [4] section 4.3 

is desired. It's unclear whether this is beneficial. 

permc_spec : str (default = 'MMD_AT_PLUS_A') 

(Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = 

True``.) A matrix is factorized in each iteration of the algorithm. 

This option specifies how to permute the columns of the matrix for 

sparsity preservation. Acceptable values are: 

 

- ``NATURAL``: natural ordering. 

- ``MMD_ATA``: minimum degree ordering on the structure of A^T A. 

- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. 

- ``COLAMD``: approximate minimum degree column ordering. 

 

This option can impact the convergence of the 

interior point algorithm; test different values to determine which 

performs best for your problem. For more information, refer to 

``scipy.sparse.linalg.splu``. 

 

Returns 

------- 

Search directions as defined in [4] 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

 

""" 

if A.shape[0] == 0: 

# If there are no constraints, some solvers fail (understandably) 

# rather than returning empty solution. This gets the job done. 

sparse, lstsq, sym_pos, cholesky = False, False, True, False 

n_x = len(x) 

 

# [4] Equation 8.8 

r_P = b * tau - A.dot(x) 

r_D = c * tau - A.T.dot(y) - z 

r_G = c.dot(x) - b.transpose().dot(y) + kappa 

mu = (x.dot(z) + tau * kappa) / (n_x + 1) 

 

# Assemble M from [4] Equation 8.31 

Dinv = x / z 

 

if sparse: 

M = A.dot(sps.diags(Dinv, 0, format="csc").dot(A.T)) 

else: 

M = A.dot(Dinv.reshape(-1, 1) * A.T) 

solve = _get_solver(M, sparse, lstsq, sym_pos, cholesky, permc_spec) 

 

# pc: "predictor-corrector" [4] Section 4.1 

# In development this option could be turned off 

# but it always seems to improve performance substantially 

n_corrections = 1 if pc else 0 

 

i = 0 

alpha, d_x, d_z, d_tau, d_kappa = 0, 0, 0, 0, 0 

while i <= n_corrections: 

# Reference [4] Eq. 8.6 

rhatp = eta(gamma) * r_P 

rhatd = eta(gamma) * r_D 

rhatg = np.array(eta(gamma) * r_G).reshape((1,)) 

 

# Reference [4] Eq. 8.7 

rhatxs = gamma * mu - x * z 

rhattk = np.array(gamma * mu - tau * kappa).reshape((1,)) 

 

if i == 1: 

if ip: # if the correction is to get "initial point" 

# Reference [4] Eq. 8.23 

rhatxs = ((1 - alpha) * gamma * mu - 

x * z - alpha**2 * d_x * d_z) 

rhattk = np.array( 

(1 - 

alpha) * 

gamma * 

mu - 

tau * 

kappa - 

alpha**2 * 

d_tau * 

d_kappa).reshape( 

(1, 

)) 

else: # if the correction is for "predictor-corrector" 

# Reference [4] Eq. 8.13 

rhatxs -= d_x * d_z 

rhattk -= d_tau * d_kappa 

 

# sometimes numerical difficulties arise as the solution is approached 

# this loop tries to solve the equations using a sequence of functions 

# for solve. For dense systems, the order is: 

# 1. scipy.linalg.cho_factor/scipy.linalg.cho_solve, 

# 2. scipy.linalg.solve w/ sym_pos = True, 

# 3. scipy.linalg.solve w/ sym_pos = False, and if all else fails 

# 4. scipy.linalg.lstsq 

# For sparse systems, the order is: 

# 1. sksparse.cholmod.cholesky (if available) 

# 2. scipy.sparse.linalg.factorized (if umfpack available) 

# 3. scipy.sparse.linalg.splu 

# 4. scipy.sparse.linalg.lsqr 

solved = False 

while(not solved): 

try: 

# [4] Equation 8.28 

p, q = _sym_solve(Dinv, A, c, b, solve) 

# [4] Equation 8.29 

u, v = _sym_solve(Dinv, A, rhatd - 

(1 / x) * rhatxs, rhatp, solve) 

if np.any(np.isnan(p)) or np.any(np.isnan(q)): 

raise LinAlgError 

solved = True 

except (LinAlgError, ValueError, TypeError) as e: 

# Usually this doesn't happen. If it does, it happens when 

# there are redundant constraints or when approaching the 

# solution. If so, change solver. 

if cholesky: 

cholesky = False 

warn( 

"Solving system with option 'cholesky':True " 

"failed. It is normal for this to happen " 

"occasionally, especially as the solution is " 

"approached. However, if you see this frequently, " 

"consider setting option 'cholesky' to False.", 

OptimizeWarning, stacklevel=5) 

elif sym_pos: 

sym_pos = False 

warn( 

"Solving system with option 'sym_pos':True " 

"failed. It is normal for this to happen " 

"occasionally, especially as the solution is " 

"approached. However, if you see this frequently, " 

"consider setting option 'sym_pos' to False.", 

OptimizeWarning, stacklevel=5) 

elif not lstsq: 

lstsq = True 

warn( 

"Solving system with option 'sym_pos':False " 

"failed. This may happen occasionally, " 

"especially as the solution is " 

"approached. However, if you see this frequently, " 

"your problem may be numerically challenging. " 

"If you cannot improve the formulation, consider " 

"setting 'lstsq' to True. Consider also setting " 

"`presolve` to True, if it is not already.", 

OptimizeWarning, stacklevel=5) 

else: 

raise e 

solve = _get_solver(M, sparse, lstsq, sym_pos, 

cholesky, permc_spec) 

# [4] Results after 8.29 

d_tau = ((rhatg + 1 / tau * rhattk - (-c.dot(u) + b.dot(v))) / 

(1 / tau * kappa + (-c.dot(p) + b.dot(q)))) 

d_x = u + p * d_tau 

d_y = v + q * d_tau 

 

# [4] Relations between after 8.25 and 8.26 

d_z = (1 / x) * (rhatxs - z * d_x) 

d_kappa = 1 / tau * (rhattk - kappa * d_tau) 

 

# [4] 8.12 and "Let alpha be the maximal possible step..." before 8.23 

alpha = _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, 1) 

if ip: # initial point - see [4] 4.4 

gamma = 10 

else: # predictor-corrector, [4] definition after 8.12 

beta1 = 0.1 # [4] pg. 220 (Table 8.1) 

gamma = (1 - alpha)**2 * min(beta1, (1 - alpha)) 

i += 1 

 

return d_x, d_y, d_z, d_tau, d_kappa 

 

 

def _sym_solve(Dinv, A, r1, r2, solve): 

""" 

An implementation of [4] equation 8.31 and 8.32 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

 

""" 

# [4] 8.31 

r = r2 + A.dot(Dinv * r1) 

v = solve(r) 

# [4] 8.32 

u = Dinv * (A.T.dot(v) - r1) 

return u, v 

 

 

def _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, alpha0): 

""" 

An implementation of [4] equation 8.21 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

 

""" 

# [4] 4.3 Equation 8.21, ignoring 8.20 requirement 

# same step is taken in primal and dual spaces 

# alpha0 is basically beta3 from [4] Table 8.1, but instead of beta3 

# the value 1 is used in Mehrota corrector and initial point correction 

i_x = d_x < 0 

i_z = d_z < 0 

alpha_x = alpha0 * np.min(x[i_x] / -d_x[i_x]) if np.any(i_x) else 1 

alpha_tau = alpha0 * tau / -d_tau if d_tau < 0 else 1 

alpha_z = alpha0 * np.min(z[i_z] / -d_z[i_z]) if np.any(i_z) else 1 

alpha_kappa = alpha0 * kappa / -d_kappa if d_kappa < 0 else 1 

alpha = np.min([1, alpha_x, alpha_tau, alpha_z, alpha_kappa]) 

return alpha 

 

 

def _get_message(status): 

""" 

Given problem status code, return a more detailed message. 

 

Parameters 

---------- 

status : int 

An integer representing the exit status of the optimization:: 

 

0 : Optimization terminated successfully 

1 : Iteration limit reached 

2 : Problem appears to be infeasible 

3 : Problem appears to be unbounded 

4 : Serious numerical difficulties encountered 

 

Returns 

------- 

message : str 

A string descriptor of the exit status of the optimization. 

 

""" 

messages = ( 

["Optimization terminated successfully.", 

"The iteration limit was reached before the algorithm converged.", 

"The algorithm terminated successfully and determined that the " 

"problem is infeasible.", 

"The algorithm terminated successfully and determined that the " 

"problem is unbounded.", 

"Numerical difficulties were encountered before the problem " 

"converged. Please check your problem formulation for errors, " 

"independence of linear equality constraints, and reasonable " 

"scaling and matrix condition numbers. If you continue to " 

"encounter this error, please submit a bug report." 

]) 

return messages[status] 

 

 

def _do_step(x, y, z, tau, kappa, d_x, d_y, d_z, d_tau, d_kappa, alpha): 

""" 

An implementation of [4] Equation 8.9 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

 

""" 

x = x + alpha * d_x 

tau = tau + alpha * d_tau 

z = z + alpha * d_z 

kappa = kappa + alpha * d_kappa 

y = y + alpha * d_y 

return x, y, z, tau, kappa 

 

 

def _get_blind_start(shape): 

""" 

Return the starting point from [4] 4.4 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

 

""" 

m, n = shape 

x0 = np.ones(n) 

y0 = np.zeros(m) 

z0 = np.ones(n) 

tau0 = 1 

kappa0 = 1 

return x0, y0, z0, tau0, kappa0 

 

 

def _indicators(A, b, c, c0, x, y, z, tau, kappa): 

""" 

Implementation of several equations from [4] used as indicators of 

the status of optimization. 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

 

""" 

 

# residuals for termination are relative to initial values 

x0, y0, z0, tau0, kappa0 = _get_blind_start(A.shape) 

 

# See [4], Section 4 - The Homogeneous Algorithm, Equation 8.8 

def r_p(x, tau): 

return b * tau - A.dot(x) 

 

def r_d(y, z, tau): 

return c * tau - A.T.dot(y) - z 

 

def r_g(x, y, kappa): 

return kappa + c.dot(x) - b.dot(y) 

 

# np.dot unpacks if they are arrays of size one 

def mu(x, tau, z, kappa): 

return (x.dot(z) + np.dot(tau, kappa)) / (len(x) + 1) 

 

obj = c.dot(x / tau) + c0 

 

def norm(a): 

return np.linalg.norm(a) 

 

# See [4], Section 4.5 - The Stopping Criteria 

r_p0 = r_p(x0, tau0) 

r_d0 = r_d(y0, z0, tau0) 

r_g0 = r_g(x0, y0, kappa0) 

mu_0 = mu(x0, tau0, z0, kappa0) 

rho_A = norm(c.T.dot(x) - b.T.dot(y)) / (tau + norm(b.T.dot(y))) 

rho_p = norm(r_p(x, tau)) / max(1, norm(r_p0)) 

rho_d = norm(r_d(y, z, tau)) / max(1, norm(r_d0)) 

rho_g = norm(r_g(x, y, kappa)) / max(1, norm(r_g0)) 

rho_mu = mu(x, tau, z, kappa) / mu_0 

return rho_p, rho_d, rho_A, rho_g, rho_mu, obj 

 

 

def _display_iter(rho_p, rho_d, rho_g, alpha, rho_mu, obj, header=False): 

""" 

Print indicators of optimization status to the console. 

 

Parameters 

---------- 

rho_p : float 

The (normalized) primal feasibility, see [4] 4.5 

rho_d : float 

The (normalized) dual feasibility, see [4] 4.5 

rho_g : float 

The (normalized) duality gap, see [4] 4.5 

alpha : float 

The step size, see [4] 4.3 

rho_mu : float 

The (normalized) path parameter, see [4] 4.5 

obj : float 

The objective function value of the current iterate 

header : bool 

True if a header is to be printed 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

 

""" 

if header: 

print("Primal Feasibility ", 

"Dual Feasibility ", 

"Duality Gap ", 

"Step ", 

"Path Parameter ", 

"Objective ") 

 

# no clue why this works 

fmt = '{0:<20.13}{1:<20.13}{2:<20.13}{3:<17.13}{4:<20.13}{5:<20.13}' 

print(fmt.format( 

float(rho_p), 

float(rho_d), 

float(rho_g), 

float(alpha) if isinstance(alpha, numbers.Number) else alpha, 

float(rho_mu), 

float(obj))) 

 

 

def _ip_hsd(A, b, c, c0, alpha0, beta, maxiter, disp, tol, sparse, lstsq, 

sym_pos, cholesky, pc, ip, permc_spec, callback, _T_o): 

r""" 

Solve a linear programming problem in standard form: 

 

Minimize:: 

 

c @ x 

 

Subject to:: 

 

A @ x == b 

x >= 0 

 

using the interior point method of [4]. 

 

Parameters 

---------- 

A : 2D array 

2D array such that ``A @ x``, gives the values of the equality 

constraints at ``x``. 

b : 1D array 

1D array of values representing the RHS of each equality constraint 

(row) in ``A`` (for standard form problem). 

c : 1D array 

Coefficients of the linear objective function to be minimized (for 

standard form problem). 

c0 : float 

Constant term in objective function due to fixed (and eliminated) 

variables. (Purely for display.) 

alpha0 : float 

The maximal step size for Mehrota's predictor-corrector search 

direction; see :math:`\beta_3`of [4] Table 8.1 

beta : float 

The desired reduction of the path parameter :math:`\mu` (see [6]_) 

maxiter : int 

The maximum number of iterations of the algorithm. 

disp : bool 

Set to ``True`` if indicators of optimization status are to be printed 

to the console each iteration. 

tol : float 

Termination tolerance; see [4]_ Section 4.5. 

sparse : bool 

Set to ``True`` if the problem is to be treated as sparse. However, 

the inputs ``A_eq`` and ``A_ub`` should nonetheless be provided as 

(dense) arrays rather than sparse matrices. 

lstsq : bool 

Set to ``True`` if the problem is expected to be very poorly 

conditioned. This should always be left as ``False`` unless severe 

numerical difficulties are frequently encountered, and a better option 

would be to improve the formulation of the problem. 

sym_pos : bool 

Leave ``True`` if the problem is expected to yield a well conditioned 

symmetric positive definite normal equation matrix (almost always). 

cholesky : bool 

Set to ``True`` if the normal equations are to be solved by explicit 

Cholesky decomposition followed by explicit forward/backward 

substitution. This is typically faster for moderate, dense problems 

that are numerically well-behaved. 

pc : bool 

Leave ``True`` if the predictor-corrector method of Mehrota is to be 

used. This is almost always (if not always) beneficial. 

ip : bool 

Set to ``True`` if the improved initial point suggestion due to [4]_ 

Section 4.3 is desired. It's unclear whether this is beneficial. 

permc_spec : str (default = 'MMD_AT_PLUS_A') 

(Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = 

True``.) A matrix is factorized in each iteration of the algorithm. 

This option specifies how to permute the columns of the matrix for 

sparsity preservation. Acceptable values are: 

 

- ``NATURAL``: natural ordering. 

- ``MMD_ATA``: minimum degree ordering on the structure of A^T A. 

- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. 

- ``COLAMD``: approximate minimum degree column ordering. 

 

This option can impact the convergence of the 

interior point algorithm; test different values to determine which 

performs best for your problem. For more information, refer to 

``scipy.sparse.linalg.splu``. 

callback : callable, optional 

If a callback function is provided, it will be called within each 

iteration of the algorithm. The callback function must accept a single 

`scipy.optimize.OptimizeResult` consisting of the following fields: 

 

x : 1D array 

Current solution vector 

fun : float 

Current value of the objective function 

success : bool 

True only when an algorithm has completed successfully, 

so this is always False as the callback function is called 

only while the algorithm is still iterating. 

slack : 1D array 

The values of the slack variables. Each slack variable 

corresponds to an inequality constraint. If the slack is zero, 

the corresponding constraint is active. 

con : 1D array 

The (nominally zero) residuals of the equality constraints, 

that is, ``b - A_eq @ x`` 

phase : int 

The phase of the algorithm being executed. This is always 

1 for the interior-point method because it has only one phase. 

status : int 

For revised simplex, this is always 0 because if a different 

status is detected, the algorithm terminates. 

nit : int 

The number of iterations performed. 

message : str 

A string descriptor of the exit status of the optimization. 

 

Returns 

------- 

x_hat : float 

Solution vector (for standard form problem). 

status : int 

An integer representing the exit status of the optimization:: 

 

0 : Optimization terminated successfully 

1 : Iteration limit reached 

2 : Problem appears to be infeasible 

3 : Problem appears to be unbounded 

4 : Serious numerical difficulties encountered 

 

message : str 

A string descriptor of the exit status of the optimization. 

iteration : int 

The number of iterations taken to solve the problem 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

.. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear 

Programming based on Newton's Method." Unpublished Course Notes, 

March 2004. Available 2/25/2017 at: 

https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf 

 

""" 

 

iteration = 0 

 

# default initial point 

x, y, z, tau, kappa = _get_blind_start(A.shape) 

 

# first iteration is special improvement of initial point 

ip = ip if pc else False 

 

# [4] 4.5 

rho_p, rho_d, rho_A, rho_g, rho_mu, obj = _indicators( 

A, b, c, c0, x, y, z, tau, kappa) 

go = rho_p > tol or rho_d > tol or rho_A > tol # we might get lucky : ) 

 

if disp: 

_display_iter(rho_p, rho_d, rho_g, "-", rho_mu, obj, header=True) 

if callback is not None: 

x_o, fun, slack, con, _, _ = _postsolve(x/tau, *_T_o, 

tol=tol) 

res = OptimizeResult({'x': x_o, 'fun': fun, 'slack': slack, 

'con': con, 'nit': iteration, 'phase': 1, 

'complete': False, 'status': 0, 

'message': "", 'success': False}) 

callback(res) 

 

status = 0 

message = "Optimization terminated successfully." 

 

if sparse: 

A = sps.csc_matrix(A) 

A.T = A.transpose() # A.T is defined for sparse matrices but is slow 

# Redefine it to avoid calculating again 

# This is fine as long as A doesn't change 

 

while go: 

 

iteration += 1 

 

if ip: # initial point 

# [4] Section 4.4 

gamma = 1 

 

def eta(g): 

return 1 

else: 

# gamma = 0 in predictor step according to [4] 4.1 

# if predictor/corrector is off, use mean of complementarity [6] 

# 5.1 / [4] Below Figure 10-4 

gamma = 0 if pc else beta * np.mean(z * x) 

# [4] Section 4.1 

 

def eta(g=gamma): 

return 1 - g 

 

try: 

# Solve [4] 8.6 and 8.7/8.13/8.23 

d_x, d_y, d_z, d_tau, d_kappa = _get_delta( 

A, b, c, x, y, z, tau, kappa, gamma, eta, 

sparse, lstsq, sym_pos, cholesky, pc, ip, permc_spec) 

 

if ip: # initial point 

# [4] 4.4 

# Formula after 8.23 takes a full step regardless if this will 

# take it negative 

alpha = 1.0 

x, y, z, tau, kappa = _do_step( 

x, y, z, tau, kappa, d_x, d_y, 

d_z, d_tau, d_kappa, alpha) 

x[x < 1] = 1 

z[z < 1] = 1 

tau = max(1, tau) 

kappa = max(1, kappa) 

ip = False # done with initial point 

else: 

# [4] Section 4.3 

alpha = _get_step(x, d_x, z, d_z, tau, 

d_tau, kappa, d_kappa, alpha0) 

# [4] Equation 8.9 

x, y, z, tau, kappa = _do_step( 

x, y, z, tau, kappa, d_x, d_y, d_z, d_tau, d_kappa, alpha) 

 

except (LinAlgError, FloatingPointError, 

ValueError, ZeroDivisionError): 

# this can happen when sparse solver is used and presolve 

# is turned off. Also observed ValueError in AppVeyor Python 3.6 

# Win32 build (PR #8676). I've never seen it otherwise. 

status = 4 

message = _get_message(status) 

break 

 

# [4] 4.5 

rho_p, rho_d, rho_A, rho_g, rho_mu, obj = _indicators( 

A, b, c, c0, x, y, z, tau, kappa) 

go = rho_p > tol or rho_d > tol or rho_A > tol 

 

if disp: 

_display_iter(rho_p, rho_d, rho_g, alpha, rho_mu, obj) 

if callback is not None: 

x_o, fun, slack, con, _, _ = _postsolve(x/tau, *_T_o, 

tol=tol) 

res = OptimizeResult({'x': x_o, 'fun': fun, 'slack': slack, 

'con': con, 'nit': iteration, 'phase': 1, 

'complete': False, 'status': 0, 

'message': "", 'success': False}) 

callback(res) 

 

# [4] 4.5 

inf1 = (rho_p < tol and rho_d < tol and rho_g < tol and tau < tol * 

max(1, kappa)) 

inf2 = rho_mu < tol and tau < tol * min(1, kappa) 

if inf1 or inf2: 

# [4] Lemma 8.4 / Theorem 8.3 

if b.transpose().dot(y) > tol: 

status = 2 

else: # elif c.T.dot(x) < tol: ? Probably not necessary. 

status = 3 

message = _get_message(status) 

break 

elif iteration >= maxiter: 

status = 1 

message = _get_message(status) 

break 

 

x_hat = x / tau 

# [4] Statement after Theorem 8.2 

return x_hat, status, message, iteration 

 

 

def _linprog_ip( 

c, 

c0=0, 

A=None, 

b=None, 

callback=None, 

_T_o=[], 

alpha0=.99995, 

beta=0.1, 

maxiter=1000, 

disp=False, 

tol=1e-8, 

sparse=False, 

lstsq=False, 

sym_pos=True, 

cholesky=None, 

pc=True, 

ip=False, 

permc_spec='MMD_AT_PLUS_A', 

**unknown_options): 

r""" 

Minimize a linear objective function subject to linear 

equality and non-negativity constraints using the interior point method 

of [4]_. Linear programming is intended to solve problems 

of the following form: 

 

Minimize:: 

 

c @ x 

 

Subject to:: 

 

A @ x == b 

x >= 0 

 

Parameters 

---------- 

c : 1D array 

Coefficients of the linear objective function to be minimized. 

c0 : float 

Constant term in objective function due to fixed (and eliminated) 

variables. (Purely for display.) 

A : 2D array 

2D array such that ``A @ x``, gives the values of the equality 

constraints at ``x``. 

b : 1D array 

1D array of values representing the right hand side of each equality 

constraint (row) in ``A``. 

 

Options 

------- 

maxiter : int (default = 1000) 

The maximum number of iterations of the algorithm. 

disp : bool (default = False) 

Set to ``True`` if indicators of optimization status are to be printed 

to the console each iteration. 

tol : float (default = 1e-8) 

Termination tolerance to be used for all termination criteria; 

see [4]_ Section 4.5. 

alpha0 : float (default = 0.99995) 

The maximal step size for Mehrota's predictor-corrector search 

direction; see :math:`\beta_{3}` of [4]_ Table 8.1. 

beta : float (default = 0.1) 

The desired reduction of the path parameter :math:`\mu` (see [6]_) 

when Mehrota's predictor-corrector is not in use (uncommon). 

sparse : bool (default = False) 

Set to ``True`` if the problem is to be treated as sparse after 

presolve. If either ``A_eq`` or ``A_ub`` is a sparse matrix, 

this option will automatically be set ``True``, and the problem 

will be treated as sparse even during presolve. If your constraint 

matrices contain mostly zeros and the problem is not very small (less 

than about 100 constraints or variables), consider setting ``True`` 

or providing ``A_eq`` and ``A_ub`` as sparse matrices. 

lstsq : bool (default = False) 

Set to ``True`` if the problem is expected to be very poorly 

conditioned. This should always be left ``False`` unless severe 

numerical difficulties are encountered. Leave this at the default 

unless you receive a warning message suggesting otherwise. 

sym_pos : bool (default = True) 

Leave ``True`` if the problem is expected to yield a well conditioned 

symmetric positive definite normal equation matrix 

(almost always). Leave this at the default unless you receive 

a warning message suggesting otherwise. 

cholesky : bool (default = True) 

Set to ``True`` if the normal equations are to be solved by explicit 

Cholesky decomposition followed by explicit forward/backward 

substitution. This is typically faster for problems 

that are numerically well-behaved. 

pc : bool (default = True) 

Leave ``True`` if the predictor-corrector method of Mehrota is to be 

used. This is almost always (if not always) beneficial. 

ip : bool (default = False) 

Set to ``True`` if the improved initial point suggestion due to [4]_ 

Section 4.3 is desired. Whether this is beneficial or not 

depends on the problem. 

permc_spec : str (default = 'MMD_AT_PLUS_A') 

(Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = 

True``, and no SuiteSparse.) 

A matrix is factorized in each iteration of the algorithm. 

This option specifies how to permute the columns of the matrix for 

sparsity preservation. Acceptable values are: 

 

- ``NATURAL``: natural ordering. 

- ``MMD_ATA``: minimum degree ordering on the structure of A^T A. 

- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. 

- ``COLAMD``: approximate minimum degree column ordering. 

 

This option can impact the convergence of the 

interior point algorithm; test different values to determine which 

performs best for your problem. For more information, refer to 

``scipy.sparse.linalg.splu``. 

 

Returns 

------- 

x : 1D array 

Solution vector. 

status : int 

An integer representing the exit status of the optimization:: 

 

0 : Optimization terminated successfully 

1 : Iteration limit reached 

2 : Problem appears to be infeasible 

3 : Problem appears to be unbounded 

4 : Serious numerical difficulties encountered 

 

message : str 

A string descriptor of the exit status of the optimization. 

iteration : int 

The number of iterations taken to solve the problem. 

 

Notes 

----- 

This method implements the algorithm outlined in [4]_ with ideas from [8]_ 

and a structure inspired by the simpler methods of [6]_. 

 

The primal-dual path following method begins with initial 'guesses' of 

the primal and dual variables of the standard form problem and iteratively 

attempts to solve the (nonlinear) Karush-Kuhn-Tucker conditions for the 

problem with a gradually reduced logarithmic barrier term added to the 

objective. This particular implementation uses a homogeneous self-dual 

formulation, which provides certificates of infeasibility or unboundedness 

where applicable. 

 

The default initial point for the primal and dual variables is that 

defined in [4]_ Section 4.4 Equation 8.22. Optionally (by setting initial 

point option ``ip=True``), an alternate (potentially improved) starting 

point can be calculated according to the additional recommendations of 

[4]_ Section 4.4. 

 

A search direction is calculated using the predictor-corrector method 

(single correction) proposed by Mehrota and detailed in [4]_ Section 4.1. 

(A potential improvement would be to implement the method of multiple 

corrections described in [4]_ Section 4.2.) In practice, this is 

accomplished by solving the normal equations, [4]_ Section 5.1 Equations 

8.31 and 8.32, derived from the Newton equations [4]_ Section 5 Equations 

8.25 (compare to [4]_ Section 4 Equations 8.6-8.8). The advantage of 

solving the normal equations rather than 8.25 directly is that the 

matrices involved are symmetric positive definite, so Cholesky 

decomposition can be used rather than the more expensive LU factorization. 

 

With default options, the solver used to perform the factorization depends 

on third-party software availability and the conditioning of the problem. 

 

For dense problems, solvers are tried in the following order: 

 

1. ``scipy.linalg.cho_factor`` 

 

2. ``scipy.linalg.solve`` with option ``sym_pos=True`` 

 

3. ``scipy.linalg.solve`` with option ``sym_pos=False`` 

 

4. ``scipy.linalg.lstsq`` 

 

For sparse problems: 

 

1. ``sksparse.cholmod.cholesky`` (if scikit-sparse and SuiteSparse are installed) 

 

2. ``scipy.sparse.linalg.factorized`` (if scikit-umfpack and SuiteSparse are installed) 

 

3. ``scipy.sparse.linalg.splu`` (which uses SuperLU distributed with SciPy) 

 

4. ``scipy.sparse.linalg.lsqr`` 

 

If the solver fails for any reason, successively more robust (but slower) 

solvers are attempted in the order indicated. Attempting, failing, and 

re-starting factorization can be time consuming, so if the problem is 

numerically challenging, options can be set to bypass solvers that are 

failing. Setting ``cholesky=False`` skips to solver 2, 

``sym_pos=False`` skips to solver 3, and ``lstsq=True`` skips 

to solver 4 for both sparse and dense problems. 

 

Potential improvements for combatting issues associated with dense 

columns in otherwise sparse problems are outlined in [4]_ Section 5.3 and 

[10]_ Section 4.1-4.2; the latter also discusses the alleviation of 

accuracy issues associated with the substitution approach to free 

variables. 

 

After calculating the search direction, the maximum possible step size 

that does not activate the non-negativity constraints is calculated, and 

the smaller of this step size and unity is applied (as in [4]_ Section 

4.1.) [4]_ Section 4.3 suggests improvements for choosing the step size. 

 

The new point is tested according to the termination conditions of [4]_ 

Section 4.5. The same tolerance, which can be set using the ``tol`` option, 

is used for all checks. (A potential improvement would be to expose 

the different tolerances to be set independently.) If optimality, 

unboundedness, or infeasibility is detected, the solve procedure 

terminates; otherwise it repeats. 

 

The expected problem formulation differs between the top level ``linprog`` 

module and the method specific solvers. The method specific solvers expect a 

problem in standard form: 

 

Minimize:: 

 

c @ x 

 

Subject to:: 

 

A @ x == b 

x >= 0 

 

Whereas the top level ``linprog`` module expects a problem of form: 

 

Minimize:: 

 

c @ x 

 

Subject to:: 

 

A_ub @ x <= b_ub 

A_eq @ x == b_eq 

lb <= x <= ub 

 

where ``lb = 0`` and ``ub = None`` unless set in ``bounds``. 

 

The original problem contains equality, upper-bound and variable constraints 

whereas the method specific solver requires equality constraints and 

variable non-negativity. 

 

``linprog`` module converts the original problem to standard form by 

converting the simple bounds to upper bound constraints, introducing 

non-negative slack variables for inequality constraints, and expressing 

unbounded variables as the difference between two non-negative variables. 

 

 

References 

---------- 

.. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point 

optimizer for linear programming: an implementation of the 

homogeneous algorithm." High performance optimization. Springer US, 

2000. 197-232. 

.. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear 

Programming based on Newton's Method." Unpublished Course Notes, 

March 2004. Available 2/25/2017 at 

https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf 

.. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear 

programming." Mathematical Programming 71.2 (1995): 221-245. 

.. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear 

programming." Athena Scientific 1 (1997): 997. 

.. [10] Andersen, Erling D., et al. Implementation of interior point methods 

for large scale linear programming. HEC/Universite de Geneve, 1996. 

 

""" 

 

_check_unknown_options(unknown_options) 

 

# These should be warnings, not errors 

if (cholesky or cholesky is None) and sparse and not has_cholmod: 

if cholesky: 

warn("Sparse cholesky is only available with scikit-sparse. " 

"Setting `cholesky = False`", 

OptimizeWarning, stacklevel=3) 

cholesky = False 

 

if sparse and lstsq: 

warn("Option combination 'sparse':True and 'lstsq':True " 

"is not recommended.", 

OptimizeWarning, stacklevel=3) 

 

if lstsq and cholesky: 

warn("Invalid option combination 'lstsq':True " 

"and 'cholesky':True; option 'cholesky' has no effect when " 

"'lstsq' is set True.", 

OptimizeWarning, stacklevel=3) 

 

valid_permc_spec = ('NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A', 'COLAMD') 

if permc_spec.upper() not in valid_permc_spec: 

warn("Invalid permc_spec option: '" + str(permc_spec) + "'. " 

"Acceptable values are 'NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A', " 

"and 'COLAMD'. Reverting to default.", 

OptimizeWarning, stacklevel=3) 

permc_spec = 'MMD_AT_PLUS_A' 

 

# This can be an error 

if not sym_pos and cholesky: 

raise ValueError( 

"Invalid option combination 'sym_pos':False " 

"and 'cholesky':True: Cholesky decomposition is only possible " 

"for symmetric positive definite matrices.") 

 

cholesky = cholesky or (cholesky is None and sym_pos and not lstsq) 

 

x, status, message, iteration = _ip_hsd(A, b, c, c0, alpha0, beta, 

maxiter, disp, tol, sparse, 

lstsq, sym_pos, cholesky, 

pc, ip, permc_spec, callback, 

_T_o) 

 

return x, status, message, iteration