Numerical optimization: theoretical and practical aspects
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| Format: | Book |
|---|---|
| Language: | English French |
| Published: |
Berlin [u.a.]
Springer
2006
|
| Edition: | 2. ed. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 490 S. graph. Darst. |
| ISBN: | 354035445X 9783540354451 |
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| 650 | 4 | |a Optimisation mathématique | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Computer algorithms | |
| 650 | 4 | |a Mathematical optimization | |
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Record in the Search Index
| _version_ | 1804135972872388608 |
|---|---|
| adam_text | J. FREDERIC BONNANS * J. CHARLES GILBERT CLAUDE LEMARECHAL * CLAUDIA A.
SAGASTIZABAL NUMERICAL OPTIMIZATION THEORETICAL AND PRACTICAL ASPECTS
SECOND EDITION WITH 52 FIGURES A LJ SPRINGER TABLE OF CONTENTS
PRELIMINARIES GENERAL INTRODUCTION 3 1.1 GENERALITIES ON OPTIMIZATION 3
1.1.1 THE PROBLEM 3 1.1.2 CLASSIFICATION 4 1.2 MOTIVATION AND EXAMPLES 5
1.2.1 MOLECULAR BIOLOGY 5 1.2.2 METEOROLOGY 6 1.2.3 TRAJECTORY OF A
DEEPWATER VEHICLE 8 1.2.4 OPTIMIZATION OF POWER MANAGEMENT 9 1.3 GENERAL
PRINCIPLES OF RESOLUTION 10 1.4 CONVERGENCE: GLOBAL ASPECTS 12 1.5
CONVERGENCE: LOCAL ASPECTS 14 1.6 COMPUTING THE GRADIENT 16
BIBLIOGRAPHICAL COMMENTS 19 PART I UNCONSTRAINED PROBLEMS 2 BASIC
METHODS 25 2.1 EXISTENCE QUESTIONS 25 2.2 OPTIMALITY CONDITIONS 26 2.3
FIRST-ORDER METHODS 27 2.3.1 GAUSS-SEIDEL 27 2.3.2 METHOD OF SUCCESSIVE
APPROXIMATIONS, OR GRADIENT METHOD 28 2.4 LINK WITH THE GENERAL DESCENT
SCHEME 28 2.4.1 CHOOSING THE 4-NONN 29 2.4.2 CHOOSING THE 2 -NORM 30
2.5 STEEPEST-DESCENT METHOD 30 2.6 IMPLEMENTATION 34 BIBLIOGRAPHICAL
COMMENTS 35 VIII TABLE OF CONTENTS 3 LINE-SEARCHES 37 3.1 GENERAL SCHEME
37 3.2 COMPUTING THE NEW T 40 3.3 OPTIMAL STEPSIZE (FOR THE RECORD ONLY)
42 3.4 MODERN LINE-SEARCH: WOLFE S RULE 43 3.5 OTHER LINE-SEARCHES:
GOLDSTEIN AND PRICE, ARMIJO 47 3.5.1 GOLDSTEIN AND PRICE 47 3.5.2 ARMIJO
47 3.5.3 REMARK ON THE CHOICE OF CONSTANTS 48 3.6 IMPLEMENTATION
CONSIDERATIONS 49 BIBLIOGRAPHICAL COMMENTS 50 4 NEWTONIAN METHODS 51 4.1
PRELIMINARIES 51 4.2 FORCING GLOBAL CONVERGENCE 52 4.3 ALLEVIATING THE
METHOD 53 4.4 QUASI-NEWTON METHODS 54 4.5 GLOBAL CONVERGENCE 57 4.6
LOCAL CONVERGENCE: GENERALITIES 59 4.7 LOCAL CONVERGENCE: BFGS 61
BIBLIOGRAPHICAL COMMENTS 65 5 CONJUGATE GRADIENT 67 5.1 OUTLINE OF
CONJUGATE GRADIENT 67 5.2 DEVELOPING THE METHOD 69 5.3 COMPUTING THE
DIRECTION 70 5.4 THE ALGORITHM SEEN AS AN ORTHOGONALIZATION PROCESS 70
5.5 APPLICATION TO NON-QUADRATIC FUNCTIONS 72 5.6 RELATION WITH
QUASI-NEWTON 74 BIBLIOGRAPHICAL COMMENTS 75 6 SPECIAL METHODS 77 6.1
TRUST-REGIONS 77 6.1.1 THE ELEMENTARY PROBLEM 78 6.1.2 THE ELEMENTARY
MECHANISM: CURVILINEAR SEARCH 79 6.1.3 INCIDENCE ON THE SEQUENCE XK 81
6.2 LEAST-SQUARES PROBLEMS: GAUSS-NEWTON 82 6.3 LARGE-SCALE PROBLEMS:
LIMITED-MEMORY QUASI-NEWTON 84 6.4 TRUNCATED NEWTON 86 6.5 QUADRATIC
PROGRAMMING 88 6.5.1 THE BASIC MECHANISM 89 6.5.2 THE SOLUTION ALGORITHM
90 6.5.3 CONVERGENCE 92 BIBLIOGRAPHICAL COMMENTS 95 TABLE OF CONTENTS IX
A CASE STUDY: SEISMIC REFLECTION TOMOGRAPHY 97 7.1 MODELLING 97 7.2
COMPUTATION OF THE REFLECTION POINTS 99 7.3 GRADIENT OF THE TRAVELTIME
100 7.4 THE LEAST-SQUARES PROBLEM TO SOLVE 101 7.5 SOLVING THE SEISMIC
REFLECTION TOMOGRAPHY PROBLEM 102 GENERAL CONCLUSION 103 PART II
NONSMOOTH OPTIMIZATION 8 INTRODUCTION TO NONSMOOTH OPTIMIZATION 109 8.1
FIRST ELEMENTS OF CONVEX ANALYSIS 109 8.2 LAGRANGIAN RELAXATION AND
DUALITY ILL 8.2.1 PRIMAL-DUAL RELATIONS ILL 8.2.2 BACK TO THE PRIMAL.
RECOVERING PRIMAL SOLUTIONS .... 113 8.3 TWO CONVEX NONDIFFERENTIABLE
FUNCTIONS 116 8.3.1 FINITE MINIMAX PROBLEMS 116 8.3.2 DUAL FUNCTIONS IN
LAGRANGIAN DUALITY 117 9 SOME METHODS IN NONSMOOTH OPTIMIZATION 119 9.1
WHY SPECIAL METHODS? 119 9.2 DESCENT METHODS 120 9.2.1 STEEPEST-DESCENT
METHOD 121 9.2.2 STABILIZATION. A DUAL APPROACH. THE E-SUBDIFFERENTIAL
124 9.3 TWO BLACK-BOX METHODS 126 9.3.1 SUBGRADIENT METHODS , 127 9.3.2
CUTTING-PLANES METHOD 130 10 BUNDLE METHODS. THE QUEST FOR DESCENT 137
10.1 STABILIZATION. A PRIMAL APPROACH 137 10.2 SOME EXAMPLES OF
STABILIZED PROBLEMS 140 10.3 PENALIZED BUNDLE METHODS 141 10.3.1 A TRIP
TO THE DUAL SPACE 144 10.3.2 MANAGING THE BUNDLE. AGGREGATION 147 10.3.3
UPDATING THE PENALIZATION PARAMETER. REVERSAL FORMS 150 10.3.4
CONVERGENCE ANALYSIS 154 11 APPLICATIONS OF NONSMOOTH OPTIMIZATION 161
11.1 DIVIDE TO CONQUER. DECOMPOSITION METHODS 161 11.1.1 PRICE
DECOMPOSITION 163 11.1.2 RESOURCE DECOMPOSITION 167 11.1.3 VARIABLE
PARTITIONING OR BENDERS DECOMPOSITION 169 11.1.4 OTHER DECOMPOSITION
METHODS 171 X TABLE OF CONTENTS 11.2 TRANSPASSING FRONTIERS 172 11.2.1
DYNAMIC BUNDLE METHODS 173 11.2.2 CONSTRAINED BUNDLE METHODS 177 11.2.3
BUNDLE METHODS FOR GENERALIZED EQUATIONS 180 12 COMPUTATIONAL EXERCISES
183 12.1 BUILDING PROTOTYPICAL NSO BLACK BOXES 183 12.1.1 THE FUNCTION
MAXQUAD 183 12.1.2 THE FUNCTION MAXANAL 184 12.2 IMPLEMENTATION OF SOME
NSO METHODS 185 12.3 RUNNING THE CODES 186 12.4 IMPROVING THE BUNDLE
IMPLEMENTATION 187 12.5 DECOMPOSITION APPLICATION 187 PART III NEWTON S
METHODS IN CONSTRAINED OPTIMIZATION 13 BACKGROUND 197 13.1 DIFFERENTIAL
CALCULUS 197 13.2 EXISTENCE AND UNIQUENESS OF SOLUTIONS 199 13.3
FIRST-ORDER OPTIMALLY CONDITIONS 200 13.4 SECOND-ORDER OPTIMALITY
CONDITIONS 202 13.5 SPEED OF CONVERGENCE 203 13.6 PROJECTION ONTO A
CLOSED CONVEX SET 205 13.7 THE NEWTON METHOD 205 13.8 THE HANGING CHAIN
PROJECT I 208 NOTES 213 EXERCISES 214 14 LOCAL METHODS FOR PROBLEMS WITH
EQUALITY CONSTRAINTS .. 215 14.1 NEWTON S METHOD 216 14.2 ADAPTED
DECOMPOSITIONS OF R N 222 14.3 LOCAL ANALYSIS OF NEWTON S METHOD 227
14.4 COMPUTATION OF THE NEWTON STEP 230 14.5 REDUCED HESSIAN ALGORITHM
235 14.6 A COMPARISON OF THE ALGORITHMS 243 14.7 THE HANGING CHAIN
PROJECT II 245 NOTES 250 EXERCISES 251 15 LOCAL METHODS FOR PROBLEMS
WITH EQUALITY AND INEQUALITY CONSTRAINTS 255 15.1 THE SQP ALGORITHM 256
15.2 PRIMAL-DUAL QUADRATIC CONVERGENCE 259 15.3 PRIMAL SUPERLINEAR
CONVERGENCE 264 TABLE OF CONTENTS XI 15.4 THE HANGING CHAIN PROJECT III
267 NOTES 270 EXERCISE 270 16 EXACT PENALIZATION 271 16.1 OVERVIEW 271
16.2 THE LAGRANGIAN . 274 16.3 THE AUGMENTED LAGRANGIAN 275 16.4
NONDIFFERENTIABLE AUGMENTED FUNCTION 279 NOTES 284 EXERCISES 285 17
GLOBALIZATION BY LINE-SEARCH 289 17.1 LINE-SEARCH SQP ALGORITHMS 291
17.2 TRUNCATED SQP 298 17.3 FROM GLOBAL TO LOCAL 307 17.4 THE HANGING
CHAIN PROJECT IV 316 NOTES 320 EXERCISES 321 18 QUASI-NEWTON VERSIONS
323 18.1 PRINCIPLES 323 18.2 QUASI-NEWTON SQP 327 18.3 REDUCED
QUASI-NEWTON ALGORITHM 331 18.4 THE HANGING CHAIN PROJECT V 340 PART IV
INTERIOR-POINT ALGORITHMS FOR LINEAR AND QUADRATIC OPTIMIZATION 19
LINEARLY CONSTRAINED OPTIMIZATION AND SIMPLEX ALGORITHM 353 19.1
EXISTENCE OF SOLUTIONS 353 19.1.1 EXISTENCE RESULT 353 19.1.2 BASIC
POINTS AND EXTENSIONS 355 19.2 DUALITY 356 19.2.1 INTRODUCING THE DUAL
PROBLEM 357 19.2.2 CONCEPT OF SADDLE-POINT 358 19.2.3 OTHER FORMULATIONS
362 19.2.4 STRICT COMPLEMENTARITY 363 19.3 THE SIMPLEX ALGORITHM 364
19.3.1 COMPUTING THE DESCENT DIRECTION 364 19.3.2 STATING THE ALGORITHM
365 19.3.3 DUAL SIMPLEX 367 19.4 COMMENTS 368 XII TABLE OF CONTENTS 20
LINEAR MONOTONE COMPLEMENTARITY AND ASSOCIATED VECTOR FIELDS 371 20.1
LOGARITHMIC PENALTY AND CENTRAL PATH 371 20.1.1 LOGARITHMIC PENALTY 371
20.1.2 CENTRAL PATH 372 20.2 LINEAR MONOTONE COMPLEMENTARITY 373 20.2.1
GENERAL FRAMEWORK 374 20.2.2 A GROUP OF TRANSFORMATIONS 377 20.2.3
STANDARD FORM 378 20.2.4 PARTITION OF VARIABLES AND CANONICAL FORM 379
20.2.5 MAGNITUDES IN A NEIGHBORHOOD OF THE CENTRAL PATH . . 380 20.3
VECTOR FIELDS ASSOCIATED WITH THE CENTRAL PATH 382 20.3.1 GENERAL
FRAMEWORK 383 20.3.2 SCALING THE PROBLEM 383 20.3.3 ANALYSIS OF THE
DIRECTIONS 384 20.3.4 MODIFIED FIELD 387 20.4 CONTINUOUS TRAJECTORIES
389 20.4.1 LIMIT POINTS OF CONTINUOUS TRAJECTORIES 389 20.4.2 DEVELOPING
AFFINE TRAJECTORIES AND DIRECTIONS 391 20.4.3 MIZUNO S LEMMA 393 20.5
COMMENTS 393 21 PREDICTOR-CORRECTOR ALGORITHMS 395 21.1 OVERVIEW 395
21.2 STATEMENT OF THE METHODS 396 21.2.1 GENERAL FRAMEWORK FOR
PRIMAL-DUAL ALGORITHMS 396 21.2.2 WEIGHTING AFTER DISPLACEMENT 397
21.2.3 THE PREDICTOR-CORRECTOR METHOD 397 21.3 A SMALL-NEIGHBORHOOD
ALGORITHM 398 21.3.1 STATEMENT OF THE ALGORITHM. MAIN RESULT 398 21.3.2
ANALYSIS OF THE CENTRALIZATION MOVE 398 21.3.3 ANALYSIS OF THE AFFINE
STEP AND GLOBAL CONVERGENCE . 399 21.3.4 ASYMPTOTIC SPEED OF CONVERGENCE
401 21.4 A PREDICTOR-CORRECTOR ALGORITHM WITH MODIFIED FIELD 402 21.4.1
PRINCIPLE 402 21.4.2 STATEMENT OF THE ALGORITHM. MAIN RESULT 404 21.4.3
COMPLEXITY ANALYSIS 404 21.4.4 ASYMPTOTIC ANALYSIS 405 21.5 A
LARGE-NEIGHBORHOOD ALGORITHM 406 21.5.1 STATEMENT OF THE ALGORITHM. MAIN
RESULT 406 21.5.2 ANALYSIS OF THE CENTERING STEP 407 21.5.3 ANALYSIS OF
THE AFFINE STEP 408 21.5.4 ASYMPTOTIC CONVERGENCE 408 21.6 PRACTICAL
ASPECTS 408 21.7 COMMENTS 409 TABLE OF CONTENTS XIII 22 NON-FEASIBLE
ALGORITHMS 411 22.1 OVERVIEW 411 22.2 PRINCIPLE OF THE NON-FEASIBLE PATH
FOLLOWING 411 22.2.1 NON-FEASIBLE CENTRAL PATH 411 22.2.2 DIRECTIONS OF
MOVE 412 22.2.3 ORDERS OF MAGNITUDE OF APPROXIMATELY CENTERED POINTS 413
22.2.4 ANALYSIS OF DIRECTIONS 415 22.2.5 MODIFIED FIELD 418 22.3
NON-FEASIBLE PREDICTOR-CORRECTOR ALGORITHM 419 22.3.1 COMPLEXITY
ANALYSIS 420 22.3.2 ASYMPTOTIC ANALYSIS 422 22.4 COMMENTS 422 23
SELF-DUALITY 425 23.1 OVERVIEW 425 23.2 LINEAR PROBLEMS WITH INEQUALITY
CONSTRAINTS 425 23.2.1 A FAMILY OF SELF-DUAL LINEAR PROBLEMS 425 23.2.2
EMBEDDING IN A SELF-DUAL PROBLEM 427 23.3 LINEAR PROBLEMS IN STANDARD
FORM 429 23.3.1 THE ASSOCIATED SELF-DUAL HOMOGENEOUS SYSTEM 429 23.3.2
EMBEDDING IN A FEASIBLE SELF-DUAL PROBLEM 430 23.4 PRACTICAL ASPECTS 431
23.5 EXTENSION TO LINEAR MONOTONE COMPLEMENTARITY PROBLEMS. . 433 23.6
COMMENTS 434 24 ONE-STEP METHODS 435 24.1 OVERVIEW 435 24.2 THE
LARGEST-STEP SETHOD 436 24.2.1 LARGEST-STEP ALGORITHM 436 24.2.2
LARGEST-STEP ALGORITHM WITH SAFEGUARD 436 24.3 CENTRALIZATION IN THE
SPACE OF LARGE VARIABLES 437 24.3.1 ONE-SIDED DISTANCE 437 24.3.2
CONVERGENCE WITH STRICT COMPLEMENTARITY 441 24.3.3 CONVERGENCE WITHOUT
STRICT COMPLEMENTARITY 443 24.3.4 RELATIVE DISTANCE IN THE SPACE OF
LARGE VARIABLES.... 444 24.4 CONVERGENCE ANALYSIS 445 24.4.1 GLOBAL
CONVERGENCE OF THE LARGEST-STEP ALGORITHM . . 445 24.4.2 LOCAL
CONVERGENCE OF THE LARGEST-STEP ALGORITHM . . . 446 24.4.3 CONVERGENCE
OF THE LARGEST-STEP ALGORITHM WITH SAFEGUARD 447 24.5 COMMENTS 450 XIV
TABLE OF CONTENTS 25 COMPLEXITY OF LINEAR OPTIMIZATION PROBLEMS WITH
INTEGER DATA 451 25.1 OVERVIEW 451 25.2 MAIN RESULTS 452 25.2.1 GENERAL
HYPOTHESES 452 25.2.2 STATEMENT OF THE RESULTS 452 25.2.3 APPLICATION
453 25.3 SOLVING A SYSTEM OF LINEAR EQUATIONS 453 25.4 PROOFS OF THE
MAIN RESULTS 455 25.4.1 PROOF OF THEOREM25.1 455 25.4.2 PROOF OF THEOREM
25.2 455 25.5 COMMENTS 456 26 KARMARKAR S ALGORITHM 457 26.1 OVERVIEW
457 26.2 LINEAR PROBLEM IN PROJECTIVE FORM 457 26.2.1 PROJECTIVE FORM
AND KARMARKAR POTENTIAL 457 26.2.2 MINIMIZING THE POTENTIAL AND SOLVING
(PLP) 458 26.3 STATEMENT OF KARMARKAR S ALGORITHM 459 26.4 ANALYSIS OF
THE ALGORITHM 460 26.4.1 COMPLEXITY ANALYSIS 460 26.4.2 ANALYSIS OF THE
POTENTIAL DECREASE 460 26.4.3 ESTIMATING THE OPTIMAL COST 461 26.4.4
PRACTICAL ASPECTS 462 26.5 COMMENTS 463 REFERENCES 465 INDEX 485
|
| adam_txt |
J. FREDERIC BONNANS * J. CHARLES GILBERT CLAUDE LEMARECHAL * CLAUDIA A.
SAGASTIZABAL NUMERICAL OPTIMIZATION THEORETICAL AND PRACTICAL ASPECTS
SECOND EDITION WITH 52 FIGURES A \ LJ SPRINGER TABLE OF CONTENTS
PRELIMINARIES GENERAL INTRODUCTION 3 1.1 GENERALITIES ON OPTIMIZATION 3
1.1.1 THE PROBLEM 3 1.1.2 CLASSIFICATION 4 1.2 MOTIVATION AND EXAMPLES 5
1.2.1 MOLECULAR BIOLOGY 5 1.2.2 METEOROLOGY 6 1.2.3 TRAJECTORY OF A
DEEPWATER VEHICLE 8 1.2.4 OPTIMIZATION OF POWER MANAGEMENT 9 1.3 GENERAL
PRINCIPLES OF RESOLUTION 10 1.4 CONVERGENCE: GLOBAL ASPECTS 12 1.5
CONVERGENCE: LOCAL ASPECTS 14 1.6 COMPUTING THE GRADIENT 16
BIBLIOGRAPHICAL COMMENTS 19 PART I UNCONSTRAINED PROBLEMS 2 BASIC
METHODS 25 2.1 EXISTENCE QUESTIONS 25 2.2 OPTIMALITY CONDITIONS 26 2.3
FIRST-ORDER METHODS 27 2.3.1 GAUSS-SEIDEL 27 2.3.2 METHOD OF SUCCESSIVE
APPROXIMATIONS, OR GRADIENT METHOD 28 2.4 LINK WITH THE GENERAL DESCENT
SCHEME 28 2.4.1 CHOOSING THE 4-NONN 29 2.4.2 CHOOSING THE 2 -NORM 30
2.5 STEEPEST-DESCENT METHOD 30 2.6 IMPLEMENTATION 34 BIBLIOGRAPHICAL
COMMENTS 35 VIII TABLE OF CONTENTS 3 LINE-SEARCHES 37 3.1 GENERAL SCHEME
37 3.2 COMPUTING THE NEW T 40 3.3 OPTIMAL STEPSIZE (FOR THE RECORD ONLY)
42 3.4 MODERN LINE-SEARCH: WOLFE'S RULE 43 3.5 OTHER LINE-SEARCHES:
GOLDSTEIN AND PRICE, ARMIJO 47 3.5.1 GOLDSTEIN AND PRICE 47 3.5.2 ARMIJO
47 3.5.3 REMARK ON THE CHOICE OF CONSTANTS 48 3.6 IMPLEMENTATION
CONSIDERATIONS 49 BIBLIOGRAPHICAL COMMENTS 50 4 NEWTONIAN METHODS 51 4.1
PRELIMINARIES 51 4.2 FORCING GLOBAL CONVERGENCE 52 4.3 ALLEVIATING THE
METHOD 53 4.4 QUASI-NEWTON METHODS 54 4.5 GLOBAL CONVERGENCE 57 4.6
LOCAL CONVERGENCE: GENERALITIES 59 4.7 LOCAL CONVERGENCE: BFGS 61
BIBLIOGRAPHICAL COMMENTS 65 5 CONJUGATE GRADIENT 67 5.1 OUTLINE OF
CONJUGATE GRADIENT 67 5.2 DEVELOPING THE METHOD 69 5.3 COMPUTING THE
DIRECTION 70 5.4 THE ALGORITHM SEEN AS AN ORTHOGONALIZATION PROCESS 70
5.5 APPLICATION TO NON-QUADRATIC FUNCTIONS 72 5.6 RELATION WITH
QUASI-NEWTON 74 BIBLIOGRAPHICAL COMMENTS 75 6 SPECIAL METHODS 77 6.1
TRUST-REGIONS 77 6.1.1 THE ELEMENTARY PROBLEM 78 6.1.2 THE ELEMENTARY
MECHANISM: CURVILINEAR SEARCH 79 6.1.3 INCIDENCE ON THE SEQUENCE XK 81
6.2 LEAST-SQUARES PROBLEMS: GAUSS-NEWTON 82 6.3 LARGE-SCALE PROBLEMS:
LIMITED-MEMORY QUASI-NEWTON 84 6.4 TRUNCATED NEWTON 86 6.5 QUADRATIC
PROGRAMMING 88 6.5.1 THE BASIC MECHANISM 89 6.5.2 THE SOLUTION ALGORITHM
90 6.5.3 CONVERGENCE 92 BIBLIOGRAPHICAL COMMENTS 95 TABLE OF CONTENTS IX
A CASE STUDY: SEISMIC REFLECTION TOMOGRAPHY 97 7.1 MODELLING 97 7.2
COMPUTATION OF THE REFLECTION POINTS 99 7.3 GRADIENT OF THE TRAVELTIME
100 7.4 THE LEAST-SQUARES PROBLEM TO SOLVE 101 7.5 SOLVING THE SEISMIC
REFLECTION TOMOGRAPHY PROBLEM 102 GENERAL CONCLUSION 103 PART II
NONSMOOTH OPTIMIZATION 8 INTRODUCTION TO NONSMOOTH OPTIMIZATION 109 8.1
FIRST ELEMENTS OF CONVEX ANALYSIS 109 8.2 LAGRANGIAN RELAXATION AND
DUALITY ILL 8.2.1 PRIMAL-DUAL RELATIONS ILL 8.2.2 BACK TO THE PRIMAL.
RECOVERING PRIMAL SOLUTIONS . 113 8.3 TWO CONVEX NONDIFFERENTIABLE
FUNCTIONS 116 8.3.1 FINITE MINIMAX PROBLEMS 116 8.3.2 DUAL FUNCTIONS IN
LAGRANGIAN DUALITY 117 9 SOME METHODS IN NONSMOOTH OPTIMIZATION 119 9.1
WHY SPECIAL METHODS? 119 9.2 DESCENT METHODS 120 9.2.1 STEEPEST-DESCENT
METHOD 121 9.2.2 STABILIZATION. A DUAL APPROACH. THE E-SUBDIFFERENTIAL
124 9.3 TWO BLACK-BOX METHODS 126 9.3.1 SUBGRADIENT METHODS , 127 9.3.2
CUTTING-PLANES METHOD 130 10 BUNDLE METHODS. THE QUEST FOR DESCENT 137
10.1 STABILIZATION. A PRIMAL APPROACH 137 10.2 SOME EXAMPLES OF
STABILIZED PROBLEMS 140 10.3 PENALIZED BUNDLE METHODS 141 10.3.1 A TRIP
TO THE DUAL SPACE 144 10.3.2 MANAGING THE BUNDLE. AGGREGATION 147 10.3.3
UPDATING THE PENALIZATION PARAMETER. REVERSAL FORMS 150 10.3.4
CONVERGENCE ANALYSIS 154 11 APPLICATIONS OF NONSMOOTH OPTIMIZATION 161
11.1 DIVIDE TO CONQUER. DECOMPOSITION METHODS 161 11.1.1 PRICE
DECOMPOSITION 163 11.1.2 RESOURCE DECOMPOSITION 167 11.1.3 VARIABLE
PARTITIONING OR BENDERS DECOMPOSITION 169 11.1.4 OTHER DECOMPOSITION
METHODS 171 X TABLE OF CONTENTS 11.2 TRANSPASSING FRONTIERS 172 11.2.1
DYNAMIC BUNDLE METHODS 173 11.2.2 CONSTRAINED BUNDLE METHODS 177 11.2.3
BUNDLE METHODS FOR GENERALIZED EQUATIONS 180 12 COMPUTATIONAL EXERCISES
183 12.1 BUILDING PROTOTYPICAL NSO BLACK BOXES 183 12.1.1 THE FUNCTION
MAXQUAD 183 12.1.2 THE FUNCTION MAXANAL 184 12.2 IMPLEMENTATION OF SOME
NSO METHODS 185 12.3 RUNNING THE CODES 186 12.4 IMPROVING THE BUNDLE
IMPLEMENTATION 187 12.5 DECOMPOSITION APPLICATION 187 PART III NEWTON'S
METHODS IN CONSTRAINED OPTIMIZATION 13 BACKGROUND 197 13.1 DIFFERENTIAL
CALCULUS 197 13.2 EXISTENCE AND UNIQUENESS OF SOLUTIONS 199 13.3
FIRST-ORDER OPTIMALLY CONDITIONS 200 13.4 SECOND-ORDER OPTIMALITY
CONDITIONS 202 13.5 SPEED OF CONVERGENCE 203 13.6 PROJECTION ONTO A
CLOSED CONVEX SET 205 13.7 THE NEWTON METHOD 205 13.8 THE HANGING CHAIN
PROJECT I 208 NOTES 213 EXERCISES 214 14 LOCAL METHODS FOR PROBLEMS WITH
EQUALITY CONSTRAINTS . 215 14.1 NEWTON'S METHOD 216 14.2 ADAPTED
DECOMPOSITIONS OF R N 222 14.3 LOCAL ANALYSIS OF NEWTON'S METHOD 227
14.4 COMPUTATION OF THE NEWTON STEP 230 14.5 REDUCED HESSIAN ALGORITHM
235 14.6 A COMPARISON OF THE ALGORITHMS 243 14.7 THE HANGING CHAIN
PROJECT II 245 NOTES 250 EXERCISES 251 15 LOCAL METHODS FOR PROBLEMS
WITH EQUALITY AND INEQUALITY CONSTRAINTS 255 15.1 THE SQP ALGORITHM 256
15.2 PRIMAL-DUAL QUADRATIC CONVERGENCE 259 15.3 PRIMAL SUPERLINEAR
CONVERGENCE 264 TABLE OF CONTENTS XI 15.4 THE HANGING CHAIN PROJECT III
267 NOTES 270 EXERCISE 270 16 EXACT PENALIZATION 271 16.1 OVERVIEW 271
16.2 THE LAGRANGIAN . 274 16.3 THE AUGMENTED LAGRANGIAN 275 16.4
NONDIFFERENTIABLE AUGMENTED FUNCTION 279 NOTES 284 EXERCISES 285 17
GLOBALIZATION BY LINE-SEARCH 289 17.1 LINE-SEARCH SQP ALGORITHMS 291
17.2 TRUNCATED SQP 298 17.3 FROM GLOBAL TO LOCAL 307 17.4 THE HANGING
CHAIN PROJECT IV 316 NOTES 320 EXERCISES 321 18 QUASI-NEWTON VERSIONS
323 18.1 PRINCIPLES 323 18.2 QUASI-NEWTON SQP 327 18.3 REDUCED
QUASI-NEWTON ALGORITHM 331 18.4 THE HANGING CHAIN PROJECT V 340 PART IV
INTERIOR-POINT ALGORITHMS FOR LINEAR AND QUADRATIC OPTIMIZATION 19
LINEARLY CONSTRAINED OPTIMIZATION AND SIMPLEX ALGORITHM 353 19.1
EXISTENCE OF SOLUTIONS 353 19.1.1 EXISTENCE RESULT 353 19.1.2 BASIC
POINTS AND EXTENSIONS 355 19.2 DUALITY 356 19.2.1 INTRODUCING THE DUAL
PROBLEM 357 19.2.2 CONCEPT OF SADDLE-POINT 358 19.2.3 OTHER FORMULATIONS
362 19.2.4 STRICT COMPLEMENTARITY 363 19.3 THE SIMPLEX ALGORITHM 364
19.3.1 COMPUTING THE DESCENT DIRECTION 364 19.3.2 STATING THE ALGORITHM
365 19.3.3 DUAL SIMPLEX 367 19.4 COMMENTS 368 XII TABLE OF CONTENTS 20
LINEAR MONOTONE COMPLEMENTARITY AND ASSOCIATED VECTOR FIELDS 371 20.1
LOGARITHMIC PENALTY AND CENTRAL PATH 371 20.1.1 LOGARITHMIC PENALTY 371
20.1.2 CENTRAL PATH 372 20.2 LINEAR MONOTONE COMPLEMENTARITY 373 20.2.1
GENERAL FRAMEWORK 374 20.2.2 A GROUP OF TRANSFORMATIONS 377 20.2.3
STANDARD FORM 378 20.2.4 PARTITION OF VARIABLES AND CANONICAL FORM 379
20.2.5 MAGNITUDES IN A NEIGHBORHOOD OF THE CENTRAL PATH . . 380 20.3
VECTOR FIELDS ASSOCIATED WITH THE CENTRAL PATH 382 20.3.1 GENERAL
FRAMEWORK 383 20.3.2 SCALING THE PROBLEM 383 20.3.3 ANALYSIS OF THE
DIRECTIONS 384 20.3.4 MODIFIED FIELD 387 20.4 CONTINUOUS TRAJECTORIES
389 20.4.1 LIMIT POINTS OF CONTINUOUS TRAJECTORIES 389 20.4.2 DEVELOPING
AFFINE TRAJECTORIES AND DIRECTIONS 391 20.4.3 MIZUNO'S LEMMA 393 20.5
COMMENTS 393 21 PREDICTOR-CORRECTOR ALGORITHMS 395 21.1 OVERVIEW 395
21.2 STATEMENT OF THE METHODS 396 21.2.1 GENERAL FRAMEWORK FOR
PRIMAL-DUAL ALGORITHMS 396 21.2.2 WEIGHTING AFTER DISPLACEMENT 397
21.2.3 THE PREDICTOR-CORRECTOR METHOD 397 21.3 A SMALL-NEIGHBORHOOD
ALGORITHM 398 21.3.1 STATEMENT OF THE ALGORITHM. MAIN RESULT 398 21.3.2
ANALYSIS OF THE CENTRALIZATION MOVE 398 21.3.3 ANALYSIS OF THE AFFINE
STEP AND GLOBAL CONVERGENCE . 399 21.3.4 ASYMPTOTIC SPEED OF CONVERGENCE
401 21.4 A PREDICTOR-CORRECTOR ALGORITHM WITH MODIFIED FIELD 402 21.4.1
PRINCIPLE 402 21.4.2 STATEMENT OF THE ALGORITHM. MAIN RESULT 404 21.4.3
COMPLEXITY ANALYSIS 404 21.4.4 ASYMPTOTIC ANALYSIS 405 21.5 A
LARGE-NEIGHBORHOOD ALGORITHM 406 21.5.1 STATEMENT OF THE ALGORITHM. MAIN
RESULT 406 21.5.2 ANALYSIS OF THE CENTERING STEP 407 21.5.3 ANALYSIS OF
THE AFFINE STEP 408 21.5.4 ASYMPTOTIC CONVERGENCE 408 21.6 PRACTICAL
ASPECTS 408 21.7 COMMENTS 409 TABLE OF CONTENTS XIII 22 NON-FEASIBLE
ALGORITHMS 411 22.1 OVERVIEW 411 22.2 PRINCIPLE OF THE NON-FEASIBLE PATH
FOLLOWING 411 22.2.1 NON-FEASIBLE CENTRAL PATH 411 22.2.2 DIRECTIONS OF
MOVE 412 22.2.3 ORDERS OF MAGNITUDE OF APPROXIMATELY CENTERED POINTS 413
22.2.4 ANALYSIS OF DIRECTIONS 415 22.2.5 MODIFIED FIELD 418 22.3
NON-FEASIBLE PREDICTOR-CORRECTOR ALGORITHM 419 22.3.1 COMPLEXITY
ANALYSIS 420 22.3.2 ASYMPTOTIC ANALYSIS 422 22.4 COMMENTS 422 23
SELF-DUALITY 425 23.1 OVERVIEW 425 23.2 LINEAR PROBLEMS WITH INEQUALITY
CONSTRAINTS 425 23.2.1 A FAMILY OF SELF-DUAL LINEAR PROBLEMS 425 23.2.2
EMBEDDING IN A SELF-DUAL PROBLEM 427 23.3 LINEAR PROBLEMS IN STANDARD
FORM 429 23.3.1 THE ASSOCIATED SELF-DUAL HOMOGENEOUS SYSTEM 429 23.3.2
EMBEDDING IN A FEASIBLE SELF-DUAL PROBLEM 430 23.4 PRACTICAL ASPECTS 431
23.5 EXTENSION TO LINEAR MONOTONE COMPLEMENTARITY PROBLEMS. . 433 23.6
COMMENTS 434 24 ONE-STEP METHODS 435 24.1 OVERVIEW 435 24.2 THE
LARGEST-STEP SETHOD 436 24.2.1 LARGEST-STEP ALGORITHM 436 24.2.2
LARGEST-STEP ALGORITHM WITH SAFEGUARD 436 24.3 CENTRALIZATION IN THE
SPACE OF LARGE VARIABLES 437 24.3.1 ONE-SIDED DISTANCE 437 24.3.2
CONVERGENCE WITH STRICT COMPLEMENTARITY 441 24.3.3 CONVERGENCE WITHOUT
STRICT COMPLEMENTARITY 443 24.3.4 RELATIVE DISTANCE IN THE SPACE OF
LARGE VARIABLES. 444 24.4 CONVERGENCE ANALYSIS 445 24.4.1 GLOBAL
CONVERGENCE OF THE LARGEST-STEP ALGORITHM . . 445 24.4.2 LOCAL
CONVERGENCE OF THE LARGEST-STEP ALGORITHM . . . 446 24.4.3 CONVERGENCE
OF THE LARGEST-STEP ALGORITHM WITH SAFEGUARD 447 24.5 COMMENTS 450 XIV
TABLE OF CONTENTS 25 COMPLEXITY OF LINEAR OPTIMIZATION PROBLEMS WITH
INTEGER DATA 451 25.1 OVERVIEW 451 25.2 MAIN RESULTS 452 25.2.1 GENERAL
HYPOTHESES 452 25.2.2 STATEMENT OF THE RESULTS 452 25.2.3 APPLICATION
453 25.3 SOLVING A SYSTEM OF LINEAR EQUATIONS 453 25.4 PROOFS OF THE
MAIN RESULTS 455 25.4.1 PROOF OF THEOREM25.1 455 25.4.2 PROOF OF THEOREM
25.2 455 25.5 COMMENTS 456 26 KARMARKAR'S ALGORITHM 457 26.1 OVERVIEW
457 26.2 LINEAR PROBLEM IN PROJECTIVE FORM 457 26.2.1 PROJECTIVE FORM
AND KARMARKAR POTENTIAL 457 26.2.2 MINIMIZING THE POTENTIAL AND SOLVING
(PLP) 458 26.3 STATEMENT OF KARMARKAR'S ALGORITHM 459 26.4 ANALYSIS OF
THE ALGORITHM 460 26.4.1 COMPLEXITY ANALYSIS 460 26.4.2 ANALYSIS OF THE
POTENTIAL DECREASE 460 26.4.3 ESTIMATING THE OPTIMAL COST 461 26.4.4
PRACTICAL ASPECTS 462 26.5 COMMENTS 463 REFERENCES 465 INDEX 485 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
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| id | DE-604.BV021997851 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:11:01Z |
| indexdate | 2024-07-09T20:49:00Z |
| institution | BVB |
| isbn | 354035445X 9783540354451 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015212500 |
| oclc_num | 73203995 |
| open_access_boolean | |
| owner | DE-706 DE-384 DE-20 |
| owner_facet | DE-706 DE-384 DE-20 |
| physical | XIV, 490 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Optimisation numérique Numerical optimization theoretical and practical aspects J. Frédéric Bonnans ... 2. ed. Berlin [u.a.] Springer 2006 XIV, 490 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Algorithmes Analyse numérique - Informatique Optimisation mathématique Datenverarbeitung Computer algorithms Mathematical optimization Numerical analysis Data processing Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Optimierung (DE-588)4043664-0 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Bonnans, J. Frédéric 1957- Sonstige (DE-588)122195507 oth HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015212500&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Numerical optimization theoretical and practical aspects Algorithmes Analyse numérique - Informatique Optimisation mathématique Datenverarbeitung Computer algorithms Mathematical optimization Numerical analysis Data processing Numerisches Verfahren (DE-588)4128130-5 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4043664-0 |
| title | Numerical optimization theoretical and practical aspects |
| title_alt | Optimisation numérique |
| title_auth | Numerical optimization theoretical and practical aspects |
| title_exact_search | Numerical optimization theoretical and practical aspects |
| title_exact_search_txtP | Numerical optimization theoretical and practical aspects |
| title_full | Numerical optimization theoretical and practical aspects J. Frédéric Bonnans ... |
| title_fullStr | Numerical optimization theoretical and practical aspects J. Frédéric Bonnans ... |
| title_full_unstemmed | Numerical optimization theoretical and practical aspects J. Frédéric Bonnans ... |
| title_short | Numerical optimization |
| title_sort | numerical optimization theoretical and practical aspects |
| title_sub | theoretical and practical aspects |
| topic | Algorithmes Analyse numérique - Informatique Optimisation mathématique Datenverarbeitung Computer algorithms Mathematical optimization Numerical analysis Data processing Numerisches Verfahren (DE-588)4128130-5 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Algorithmes Analyse numérique - Informatique Optimisation mathématique Datenverarbeitung Computer algorithms Mathematical optimization Numerical analysis Data processing Numerisches Verfahren Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015212500&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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