Pathways to Solutions, Fixed Points, and Equilibria 0136535011
Numerical methods for the solution of systems of nonlinear equations have long been available. Until quite recently, how
229 61 6MB
English Pages 496 Year 1981
Table of contents :
Zangwill W.I.,Garcia C.B. Pathways to Solutions, Fixed Points, and Equilibria (series in computational mathematics)(PH,1981)(ISBN 0136535011)(T)(K)(600dpi)(496p) 2......Page 2
Copyright 3......Page 3
Contents iii 4......Page 4
FOREWORD xi 9......Page 9
PREFACE xiii 11......Page 11
I BASIC THEORY 1 17......Page 17
1.1 Solving Equations 3 19......Page 19
1.2 The Homotopy Principle 4 20......Page 20
1.3 Varieties of Homotopies 12 28......Page 28
1.4 Path Existence 14 30......Page 30
Summary 20 36......Page 36
Exercises 21 37......Page 37
Notes 23 39......Page 39
2.1 Movement along a Path 24 40......Page 40
2.2 Orientation 30 46......Page 46
2.3 The Piecewise Case 33 49......Page 49
Summary 34 50......Page 50
Appendix 35 51......Page 51
Exercises 40 56......Page 56
Notes 42 58......Page 58
3.1 Preliminaries 43 59......Page 59
3.2 Fixed-Point Theorems 46 62......Page 62
3.3 Nonlinear Equations 51 67......Page 67
3.4 Degree Theory 52 68......Page 68
Exercises 57 73......Page 73
Notes 60 76......Page 76
II APPLICATIONS 61 77......Page 77
4 NONLINEAR PROGRAMMING: DYNAMIC, PARAMETRIC, AND ALGORITHMIC 63 79......Page 79
4.1 Statement of NLP 64 80......Page 80
4.2 The Kuhn-Tucker Equations 65 81......Page 81
4.3 Dynamic and Parametric NLP 68 84......Page 84
4.4 Path Existence for the Dynamic NLP 71 87......Page 87
4.5 Solving the NLP by Path Following 72 88......Page 88
Exercises 78 94......Page 94
Notes 79 95......Page 95
5.1 The Equilibrium Programming Problem 81 97......Page 97
5.2 Nonlinear Programming versus Equilibrium Programming 84 100......Page 100
5.3 Examples of EP 86 102......Page 102
5.4 Proof of Equilibrium-Point Existence 96 112......Page 112
5.5 The Dynamic EP 99 5.6 The Algorithm 100 116......Page 116
Summary 105 121......Page 121
Appendix 106 122......Page 122
Exercises 107 123......Page 123
Notes 110 126......Page 126
6.1 The Economic Equilibrium Model 112 128......Page 128
6.2 Transforming EE into EP 116 132......Page 132
6.3 Competitive Equilibrium Extended 125 141......Page 141
Summary 128 144......Page 144
Appendix 128 Exercises 130 146......Page 146
Notes 131 147......Page 147
7.1 The Edgeworth Box 132 148......Page 148
7.2 Obtaining the Competitive Equilibrium 139 153......Page 153
7.3 Equilibrium for M Individuals 143 159......Page 159
Summary 144 160......Page 160
Notes 145 161......Page 161
8.1 Two-Person Games 147 163......Page 163
8.2 Mixed Strategies 151 167......Page 167
8.3 M-Person Games 155 171 ......Page 171
8.4 Some Dilemmas Concerning the Equilibrium Concept 160 176......Page 176
Exercises 164 180......Page 180
Notes 165 181......Page 181
9.1 Network Equilibrium Introduced 166 182......Page 182
9.2 The Paradox of Adding or Removing a Link 174 190......Page 190
9.3 Formulation of the Network Equilibrium Model 178 194......Page 194
9.4 The Equilibrium Programming Formulation 181 197......Page 197
9.5 Existence of a Network Equilibrium 184 200......Page 200
9.6 An Important Special Case 186 202......Page 202
9.7 Elasticity and Strength of Materials 189 205......Page 205
Exercises 195 211......Page 211
Notes 196 212......Page 212
10.1 Unconstrained Catastrophes 198 214......Page 214
10.2 Second-Order Conditions 203 219......Page 219
10.3 Other Forms of Catastrophe 209 225......Page 225
Exercises 214 230......Page 230
Notes 215 231......Page 231
III ALGORITHMS AND SOLUTION PROCEDURES 217 233......Page 233
Overview of Part III 219 235......Page 235
11.1 Getting Started 220 236......Page 236
11.2 Simplices 222 238......Page 238
11.3 Functions on Simplices 224 240......Page 240
11.4 Creating a Path 227 243......Page 243
11.5 Simplicial Algorithms 230 246......Page 246
Summary 000 Appendix 234 250......Page 250
Exercises 238 254......Page 254
Notes 239 255......Page 255
12.1 Notation 240 256......Page 256
12.2 Behavior of the Flex Simplicial 241 257......Page 257
12.3 Prevention of Cycling 244 260......Page 260
12.4 Large Simplices 246 262......Page 262
Appendix 252 268......Page 268
Exercises 254 270 ......Page 270
Notes 255 271 ......Page 271
13.1 The Triangulation 257 273......Page 273
13.2 The Triangulation Algorithm 259 275......Page 275
13.3 Complementarity 262 278......Page 278
Exercises 265 281......Page 281
Notes 267 283......Page 283
14 INTEGER LABELS 268 284......Page 284
14.1 A Trivial Piecewise-Linear Map 269 285......Page 285
14.2 The Integer Algorithm 272 288......Page 288
14.3 Sperner’s Lemma 275 291......Page 291
14.4 The Knaster-Kuratowski-Mazurkiewicz Lemma 283 299......Page 299
Exercises 286 302......Page 302
Notes 288 304......Page 304
15.1 Euler’s Method 289 305......Page 305
15.2 The Homotopy Differential Equations 291 307......Page 307
15.3 Trouble with Euler and Possible Alternatives 295 311......Page 311
15.4 A Restart Method 297 313......Page 313
15.5 Newton’s Solution Method 300 316......Page 316
Appendix 306 322......Page 322
Exercises 308 324......Page 324
Notes 309 325......Page 325
16.1 The Basic Idea of Predictor-Corrector Methods 310 326......Page 326
16.2 Horizontal Corrector 315 331......Page 331
16.3 Failure of the Horizontal Corrector 317 333......Page 333
16.4 The Euler Predictor-Corrector Algorithm 320 336......Page 336
16.5 General Discussion 322 338......Page 338
Summary 323 339......Page 339
Exercises 324 340......Page 340
Notes 325 341......Page 341
17.1 Contraction 326 342......Page 342
17.2 Separable Homotopies 329 345......Page 345
17.3 Example 335 351......Page 351
Summary 338 354......Page 354
Exercises 339 355......Page 355
Notes 340 356......Page 356
IV FUNDAMENTAL CONCEPTS AND EXTENSIONS 341 357......Page 357
18 ALL SOLUTIONS 343 359......Page 359
18.1 Complex Spaces 344 360......Page 360
18.2 Development of the Homotopy 350 366......Page 366
18.3 Conditions for Path Finiteness 354 370......Page 370
18.4 Further Considerations 359 375......Page 375
Summary 361 377......Page 377
Exercises 362 378......Page 378
Notes 363 379......Page 379
19.1 The Linear Complementarity Problem 364 380......Page 380
19.2 Solving the LC by Path Following 368 384......Page 384
19.3 Lemke’s Method 371 387......Page 387
Exercises 379 395......Page 395
Notes 380 396......Page
20.1 Existence of an LC Solution 381 397......Page 397
20.2 Quadratic Programs 384 400......Page 400
20.3 Bimatrix Games 388 404......Page 404
Exercises 394 410......Page 410
Notes 396 412......Page 412
21.1 Point-to-Set Maps 397 413......Page 413
21.2 The Kakutani Theorem 403 419......Page 419
21.3 Economic Equilibrium Existence Revisited 409 425......Page 425
Exercises 416 432......Page 432
Notes 418 434......Page 434
22 RELAXATION OF REGULARITY AND DIFFERENTIABILITY 419 435......Page 435
22.1 The Two Main Theorems 420 436......Page 436
22.2 The Weierstrass Theorem 425 441......Page 441
22.3 Applying the Sard and Weierstrass Theorems 427 443......Page 443
22.4 Assumption Relaxation for the Homotopy Invariance Theorem and the Fixed- Point Theorem 428 444......Page 444
22.5 Fixed-Point Theorems 434 450......Page 450
Summary 435 451......Page 451
Appendices 436 452......Page 452
Exercises 441 457......Page 457
Notes 442 458......Page 458
A.l Functions 443 459......Page 459
A.2 Mean Value Theorems 445 461......Page 461
A.3 The Implicit Function Theorem 446 462......Page 462
A.4 Existence and Uniqueness of Solutions for Ordinary Differential Equations 448 464......Page 464
B.l Convex Sets 450 466......Page 466
B.2 Convex and Concave Functions 451 467......Page 467
C.l The Nonlinear Programming Problem 455 471......Page 471
C.2 The Kuhn-Tucker Necessary Conditions 456 472......Page 472
BIBLIOGRAPHY 459 473......Page 473
AUTHOR INDEX 473 489......Page 489
INDEX 476 492......Page 492
cover......Page 1
Zangwill W.I.,Garcia C.B. Pathways to Solutions, Fixed Points, and Equilibria (series in computational mathematics)(PH,1981)(ISBN 0136535011)(T)(K)(600dpi)(496p) 2......Page 2
Copyright 3......Page 3
Contents iii 4......Page 4
FOREWORD xi 9......Page 9
PREFACE xiii 11......Page 11
I BASIC THEORY 1 17......Page 17
1.1 Solving Equations 3 19......Page 19
1.2 The Homotopy Principle 4 20......Page 20
1.3 Varieties of Homotopies 12 28......Page 28
1.4 Path Existence 14 30......Page 30
Summary 20 36......Page 36
Exercises 21 37......Page 37
Notes 23 39......Page 39
2.1 Movement along a Path 24 40......Page 40
2.2 Orientation 30 46......Page 46
2.3 The Piecewise Case 33 49......Page 49
Summary 34 50......Page 50
Appendix 35 51......Page 51
Exercises 40 56......Page 56
Notes 42 58......Page 58
3.1 Preliminaries 43 59......Page 59
3.2 Fixed-Point Theorems 46 62......Page 62
3.3 Nonlinear Equations 51 67......Page 67
3.4 Degree Theory 52 68......Page 68
Exercises 57 73......Page 73
Notes 60 76......Page 76
II APPLICATIONS 61 77......Page 77
4 NONLINEAR PROGRAMMING: DYNAMIC, PARAMETRIC, AND ALGORITHMIC 63 79......Page 79
4.1 Statement of NLP 64 80......Page 80
4.2 The Kuhn-Tucker Equations 65 81......Page 81
4.3 Dynamic and Parametric NLP 68 84......Page 84
4.4 Path Existence for the Dynamic NLP 71 87......Page 87
4.5 Solving the NLP by Path Following 72 88......Page 88
Exercises 78 94......Page 94
Notes 79 95......Page 95
5.1 The Equilibrium Programming Problem 81 97......Page 97
5.2 Nonlinear Programming versus Equilibrium Programming 84 100......Page 100
5.3 Examples of EP 86 102......Page 102
5.4 Proof of Equilibrium-Point Existence 96 112......Page 112
5.5 The Dynamic EP 99 5.6 The Algorithm 100 116......Page 116
Summary 105 121......Page 121
Appendix 106 122......Page 122
Exercises 107 123......Page 123
Notes 110 126......Page 126
6.1 The Economic Equilibrium Model 112 128......Page 128
6.2 Transforming EE into EP 116 132......Page 132
6.3 Competitive Equilibrium Extended 125 141......Page 141
Summary 128 144......Page 144
Appendix 128 Exercises 130 146......Page 146
Notes 131 147......Page 147
7.1 The Edgeworth Box 132 148......Page 148
7.2 Obtaining the Competitive Equilibrium 139 153......Page 153
7.3 Equilibrium for M Individuals 143 159......Page 159
Summary 144 160......Page 160
Notes 145 161......Page 161
8.1 Two-Person Games 147 163......Page 163
8.2 Mixed Strategies 151 167......Page 167
8.3 M-Person Games 155 171 ......Page 171
8.4 Some Dilemmas Concerning the Equilibrium Concept 160 176......Page 176
Exercises 164 180......Page 180
Notes 165 181......Page 181
9.1 Network Equilibrium Introduced 166 182......Page 182
9.2 The Paradox of Adding or Removing a Link 174 190......Page 190
9.3 Formulation of the Network Equilibrium Model 178 194......Page 194
9.4 The Equilibrium Programming Formulation 181 197......Page 197
9.5 Existence of a Network Equilibrium 184 200......Page 200
9.6 An Important Special Case 186 202......Page 202
9.7 Elasticity and Strength of Materials 189 205......Page 205
Exercises 195 211......Page 211
Notes 196 212......Page 212
10.1 Unconstrained Catastrophes 198 214......Page 214
10.2 Second-Order Conditions 203 219......Page 219
10.3 Other Forms of Catastrophe 209 225......Page 225
Exercises 214 230......Page 230
Notes 215 231......Page 231
III ALGORITHMS AND SOLUTION PROCEDURES 217 233......Page 233
Overview of Part III 219 235......Page 235
11.1 Getting Started 220 236......Page 236
11.2 Simplices 222 238......Page 238
11.3 Functions on Simplices 224 240......Page 240
11.4 Creating a Path 227 243......Page 243
11.5 Simplicial Algorithms 230 246......Page 246
Summary 000 Appendix 234 250......Page 250
Exercises 238 254......Page 254
Notes 239 255......Page 255
12.1 Notation 240 256......Page 256
12.2 Behavior of the Flex Simplicial 241 257......Page 257
12.3 Prevention of Cycling 244 260......Page 260
12.4 Large Simplices 246 262......Page 262
Appendix 252 268......Page 268
Exercises 254 270 ......Page 270
Notes 255 271 ......Page 271
13.1 The Triangulation 257 273......Page 273
13.2 The Triangulation Algorithm 259 275......Page 275
13.3 Complementarity 262 278......Page 278
Exercises 265 281......Page 281
Notes 267 283......Page 283
14 INTEGER LABELS 268 284......Page 284
14.1 A Trivial Piecewise-Linear Map 269 285......Page 285
14.2 The Integer Algorithm 272 288......Page 288
14.3 Sperner’s Lemma 275 291......Page 291
14.4 The Knaster-Kuratowski-Mazurkiewicz Lemma 283 299......Page 299
Exercises 286 302......Page 302
Notes 288 304......Page 304
15.1 Euler’s Method 289 305......Page 305
15.2 The Homotopy Differential Equations 291 307......Page 307
15.3 Trouble with Euler and Possible Alternatives 295 311......Page 311
15.4 A Restart Method 297 313......Page 313
15.5 Newton’s Solution Method 300 316......Page 316
Appendix 306 322......Page 322
Exercises 308 324......Page 324
Notes 309 325......Page 325
16.1 The Basic Idea of Predictor-Corrector Methods 310 326......Page 326
16.2 Horizontal Corrector 315 331......Page 331
16.3 Failure of the Horizontal Corrector 317 333......Page 333
16.4 The Euler Predictor-Corrector Algorithm 320 336......Page 336
16.5 General Discussion 322 338......Page 338
Summary 323 339......Page 339
Exercises 324 340......Page 340
Notes 325 341......Page 341
17.1 Contraction 326 342......Page 342
17.2 Separable Homotopies 329 345......Page 345
17.3 Example 335 351......Page 351
Summary 338 354......Page 354
Exercises 339 355......Page 355
Notes 340 356......Page 356
IV FUNDAMENTAL CONCEPTS AND EXTENSIONS 341 357......Page 357
18 ALL SOLUTIONS 343 359......Page 359
18.1 Complex Spaces 344 360......Page 360
18.2 Development of the Homotopy 350 366......Page 366
18.3 Conditions for Path Finiteness 354 370......Page 370
18.4 Further Considerations 359 375......Page 375
Summary 361 377......Page 377
Exercises 362 378......Page 378
Notes 363 379......Page 379
19.1 The Linear Complementarity Problem 364 380......Page 380
19.2 Solving the LC by Path Following 368 384......Page 384
19.3 Lemke’s Method 371 387......Page 387
Exercises 379 395......Page 395
Notes 380 396......Page
20.1 Existence of an LC Solution 381 397......Page 397
20.2 Quadratic Programs 384 400......Page 400
20.3 Bimatrix Games 388 404......Page 404
Exercises 394 410......Page 410
Notes 396 412......Page 412
21.1 Point-to-Set Maps 397 413......Page 413
21.2 The Kakutani Theorem 403 419......Page 419
21.3 Economic Equilibrium Existence Revisited 409 425......Page 425
Exercises 416 432......Page 432
Notes 418 434......Page 434
22 RELAXATION OF REGULARITY AND DIFFERENTIABILITY 419 435......Page 435
22.1 The Two Main Theorems 420 436......Page 436
22.2 The Weierstrass Theorem 425 441......Page 441
22.3 Applying the Sard and Weierstrass Theorems 427 443......Page 443
22.4 Assumption Relaxation for the Homotopy Invariance Theorem and the Fixed- Point Theorem 428 444......Page 444
22.5 Fixed-Point Theorems 434 450......Page 450
Summary 435 451......Page 451
Appendices 436 452......Page 452
Exercises 441 457......Page 457
Notes 442 458......Page 458
A.l Functions 443 459......Page 459
A.2 Mean Value Theorems 445 461......Page 461
A.3 The Implicit Function Theorem 446 462......Page 462
A.4 Existence and Uniqueness of Solutions for Ordinary Differential Equations 448 464......Page 464
B.l Convex Sets 450 466......Page 466
B.2 Convex and Concave Functions 451 467......Page 467
C.l The Nonlinear Programming Problem 455 471......Page 471
C.2 The Kuhn-Tucker Necessary Conditions 456 472......Page 472
BIBLIOGRAPHY 459 473......Page 473
AUTHOR INDEX 473 489......Page 489
INDEX 476 492......Page 492
cover......Page 1
- Author / Uploaded
- Zangwill W.I.
- Garcia C.B.
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