Numerical optimization:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
[2006]
|
| Edition: | Second Edition |
| Series: | Springer series in operations research and financial engineering
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | xxii, 664 Seiten Illustrationen, Diagramme |
| ISBN: | 0387303030 0387987932 9780387303031 9781493937110 |
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MARC
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| 100 | 1 | |a Nocedal, Jorge |0 (DE-588)121513238 |4 aut | |
| 245 | 1 | 0 | |a Numerical optimization |c Jorge Nocedal ; Stephen J. Wright |
| 250 | |a Second Edition | ||
| 264 | 1 | |a New York, NY |b Springer |c [2006] | |
| 300 | |a xxii, 664 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer series in operations research and financial engineering | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
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Record in the Search Index
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| adam_text |
Contents
Preface
Preface to the Second Edition
1
Mathematical Formulation
Example: A Transportation Problem
Continuous versus Discrete Optimization
Constrained and Unconstrained Optimization
Global and Local Optimization
Stochastic and Deterministic Optimization
Convexity
Optimization Algorithms
Notes and References
2
2.1
vîii
Recognizing a Local Minimum
Nonsmooth Problems
2.2
Two Strategies: Line Search and Trust Region
Search Directions for Line Search Methods
Models for Trust-Region Methods
Scaling
Exercises
Line Search Methods
3.1
The Wolfe Conditions
The Goldstein Conditions
Sufficient Decrease and Backtracking
3.2
3.3
Convergence Rate of Steepest Descent
Newton's Method
Quasi-Newton
3.4
Eigenvalue Modification
Adding a Multiple of the Identity
Modified Cholesky Factorization
Modified Symmetric Indefinite Factorization
3.5
Interpolation
Initial Step Length
A Line Search Algorithm for the Wolfe Conditions
Notes and References
Exercises
Trust-Region Methods
Outline of the Trust-Region Approach
4.1
The Cauchy Point
Improving on the Cauchy Point
The Dogleg Method
Two-Dimensional Subspace Minimization
4.2
Reduction Obtained by the Cauchy Point
Convergence to Stationary Points
4.3
Contents ix
The Hard Case
Proof of Theorem
Convergence of Algorithms Based on Nearly Exact Solutions
4.4
4.5
Scaling
Trust Regions in Other Norms
Notes and References
Exercises
5
5.1
Conjugate Direction Methods
Basic Properties of the Conjugate Gradient Method
A Practical Form of the Conjugate Gradient Method
Rate of Convergence
Preconditioning
Practical Preconditioners
5.2
The Fletcher-Reeves Method
The
Quadratic Termination and Restarts
Behavior of the Fletcher-Reeves Method
Global Convergence
Numerical Performance
Notes and References
Exercises
6
6.1
Properties of the BFGS Method
Implementation
6.2
Properties of SRI Updating
6.3
6.4
Global Convergence of the BFGS Method
Superlinear
Convergence Analysis of the SRI Method
Notes and References
Exercises
x Contents
7
7.1
Local Convergence of Inexact Newton Methods
Line Search Newton-CG Method
Trust-Region Newton-CG Method
Preconditioning the Trust-Region Newton-CG Method
Trust-Region Newton-Lanczos Method
7.2
Limited-Memory BFGS
Relationship with Conjugate Gradient Methods
General Limited-Memory Updating
Compact Representation of BFGS Updating
Unrolling the Update
7.3
7.4
7.5
Notes and References
Exercises
8
8.1
Approximating the Gradient
Approximating a Sparse Jacobian
Approximating the Hessian
Approximating a Sparse Hessian
8.2
An Example
The Forward Mode
The Reverse Mode
Vector Functions and Partial Separability
Calculating Jacobians of Vector Functions
Calculating Hessians: Forward Mode
Calculating Hessians: Reverse Mode
Current Limitations
Notes and References
Exercises
9
9.1
9.2
Interpolation and Polynomial Bases
Updating the Interpolation Set
Contents
A Method Based on Minimum-Change Updating
9.3
Coordinate Search Method
Pattern-Search Methods
9.4
9.5
9.6
Notes and References
Exercises
10
10.1
10.2
10.3
The Gauss-Newton Method
Convergence of the Gauss-Newton Method
The Levenberg-Marquardt Method
Implementation of the Levenberg-Marquardt Method
Convergence of the Levenberg-Marquardt Method
Methods for Large-Residual Problems
10.4
Notes and References
Exercises
11
11.1
Newton's Method for Nonlinear Equations
Inexact Newton Methods
Broyden's Method
Tensor Methods
11.2
Merit Functions
Line Search Methods
Trust-Region Methods
11.3
Motivation
Practical Continuation Methods
Notes and References
Exercises
12
Local and Global Solutions
xîi
Smoothness
12.1
A Single Equality Constraint
A Single Inequality Constraint
Two Inequality Constraints
12.2
12.3
12.4
Relating the Tangent Cone and the First-Order Feasible Direction Set
A Fundamental Necessary Condition
Farkas' Lemma
Proof of Theorem
12.5
Second-Order Conditions and Projected Hessians
12.6
12.7
12.8 Lagrange
12.9
Notes and References
Exercises
13
Linear Programming
13.1
Optimality Conditions
The Dual Problem
13.2
Bases and Basic Feasible Points
Vertices of the Feasible Polytope
13.3
Outline
A Single Step of the Method
13.4
13.5
Pricing and Selection of the Entering Index
Starting the Simplex Method
Degenerate Steps and Cycling
13.6
13.7
13.8
Notes and References
Exercises
Contents
14
14.1
Outline
The Central Path
Central Path Neighborhoods and Path-Following Methods
14.2
Corrector and Centering Steps
Step Lengths
Starting Point
A Practical Algorithm
Solving the Linear Systems
14.3
Other Path-Following Methods
Potential-Reduction Methods
Extensions
14.4
Notes and References
Exercises
15
15.1
15.2
15.3
Simple Elimination using Linear Constraints
General Reduction Strategies for Linear Constraints
Effect of Inequality Constraints
15.4
Merit Functions
Filters
15.5
15.6
Nonmonotone
Notes and References
Exercises
16
16.1
Properties of Equality-Constrained QPs
16.2
Factoring the Full KKT System
Schur-Complement Method
Null-Space Method
xîv
16.3
CG Applied
The Projected
16.4
Optimality Conditions for Inequality-Constrained Problems
Degeneracy
16.5
Specification of the Active-Set Method for Convex QP
Further Remarks on the Active-Set Method
Finite Termination of Active-Set Algorithm on Strictly Convex QPs
Updating Factorizations
16.6
Solving the Primal-Dual System
Step Length Selection
A Practical Primal-Dual Method
16.7
Cauchy Point Computation
Subspace Minimization
16.8
Notes and References
Exercises
17
17.1
Motivation
Algorithmic Framework
Convergence of the Quadratic Penalty Method
111 Conditioning and Reformulations
17.2
A Practical ix Penalty Method
A General Class of Nonsmooth Penalty Methods
17.3
Motivation and Algorithmic Framework
Properties of the Augmented Lagrangian
17.4
Bound-Constrained Formulation
Linearly Constrained Formulation
Unconstrained Formulation
17.5
Notes and References
Exercises
Contents
18
18.1
SQP Framework
Inequality Constraints
18.2
IQPandEQP
Enforcing Convergence
18.3
Handling Inconsistent Linearizations
Full
Reduced-Hessian
Merit Functions
Second-Order Correction
18.4
18.5
A Relaxation Method for Equality-Constrained Optimization
SiiQP (Sequential lx Quadratic Programming)
Sequential Linear-Quadratic Programming (SLQP)
A Technique for Updating the Penalty Parameter
18.6
18.7
Rate of Convergence
18.8
Notes and References
Exercises
19
19.1
19.2
19.3
Primal vs. Primal-Dual System
Solving the Primal-Dual System
Updating the Barrier Parameter
Handling Nonconvexity and Singularity
Step Acceptance: Merit Functions and Filters
Quasi-Newton
Feasible Interior-Point Methods
19.4
19.5
An Algorithm for Solving the Barrier Problem
Step Computation
Lagrange
xvi
Description of a Trust-Region Interior-Point Method
19.6
19.7
Failure of the Line Search Approach
Modified Line Search Methods
Global Convergence of the Trust-Region Approach
19.8 Superlinear
19.9
Notes and References
Exercises
A Background Material
A.
Vectors and Matrices
Norms
Subspaces
Eigenvalues, Eigenvectors, and the Singular-Value Decomposition
Determinant and Trace
Matrix Factorizations: Cholesky,
Symmetric Indefinite Factorization
Sherman-Morrison-Woodbury Formula
Interlacing Eigenvalue Theorem
Error Analysis and
Conditioning and Stability
A.2 Elements of Analysis, Geometry, Topology
Sequences
Rates of Convergence
Topology of the Euclidean Space R"
Convex Sets in R"
Continuity and Limits
Derivatives
Directional Derivatives
Mean Value Theorem
Implicit Function Theorem
Order Notation
Root-Finding for Scalar Equations
В
References
Index |
| adam_txt |
Contents
Preface
Preface to the Second Edition
1
Mathematical Formulation
Example: A Transportation Problem
Continuous versus Discrete Optimization
Constrained and Unconstrained Optimization
Global and Local Optimization
Stochastic and Deterministic Optimization
Convexity
Optimization Algorithms
Notes and References
2
2.1
vîii
Recognizing a Local Minimum
Nonsmooth Problems
2.2
Two Strategies: Line Search and Trust Region
Search Directions for Line Search Methods
Models for Trust-Region Methods
Scaling
Exercises
Line Search Methods
3.1
The Wolfe Conditions
The Goldstein Conditions
Sufficient Decrease and Backtracking
3.2
3.3
Convergence Rate of Steepest Descent
Newton's Method
Quasi-Newton
3.4
Eigenvalue Modification
Adding a Multiple of the Identity
Modified Cholesky Factorization
Modified Symmetric Indefinite Factorization
3.5
Interpolation
Initial Step Length
A Line Search Algorithm for the Wolfe Conditions
Notes and References
Exercises
Trust-Region Methods
Outline of the Trust-Region Approach
4.1
The Cauchy Point
Improving on the Cauchy Point
The Dogleg Method
Two-Dimensional Subspace Minimization
4.2
Reduction Obtained by the Cauchy Point
Convergence to Stationary Points
4.3
Contents ix
The Hard Case
Proof of Theorem
Convergence of Algorithms Based on Nearly Exact Solutions
4.4
4.5
Scaling
Trust Regions in Other Norms
Notes and References
Exercises
5
5.1
Conjugate Direction Methods
Basic Properties of the Conjugate Gradient Method
A Practical Form of the Conjugate Gradient Method
Rate of Convergence
Preconditioning
Practical Preconditioners
5.2
The Fletcher-Reeves Method
The
Quadratic Termination and Restarts
Behavior of the Fletcher-Reeves Method
Global Convergence
Numerical Performance
Notes and References
Exercises
6
6.1
Properties of the BFGS Method
Implementation
6.2
Properties of SRI Updating
6.3
6.4
Global Convergence of the BFGS Method
Superlinear
Convergence Analysis of the SRI Method
Notes and References
Exercises
x Contents
7
7.1
Local Convergence of Inexact Newton Methods
Line Search Newton-CG Method
Trust-Region Newton-CG Method
Preconditioning the Trust-Region Newton-CG Method
Trust-Region Newton-Lanczos Method
7.2
Limited-Memory BFGS
Relationship with Conjugate Gradient Methods
General Limited-Memory Updating
Compact Representation of BFGS Updating
Unrolling the Update
7.3
7.4
7.5
Notes and References
Exercises
8
8.1
Approximating the Gradient
Approximating a Sparse Jacobian
Approximating the Hessian
Approximating a Sparse Hessian
8.2
An Example
The Forward Mode
The Reverse Mode
Vector Functions and Partial Separability
Calculating Jacobians of Vector Functions
Calculating Hessians: Forward Mode
Calculating Hessians: Reverse Mode
Current Limitations
Notes and References
Exercises
9
9.1
9.2
Interpolation and Polynomial Bases
Updating the Interpolation Set
Contents
A Method Based on Minimum-Change Updating
9.3
Coordinate Search Method
Pattern-Search Methods
9.4
9.5
9.6
Notes and References
Exercises
10
10.1
10.2
10.3
The Gauss-Newton Method
Convergence of the Gauss-Newton Method
The Levenberg-Marquardt Method
Implementation of the Levenberg-Marquardt Method
Convergence of the Levenberg-Marquardt Method
Methods for Large-Residual Problems
10.4
Notes and References
Exercises
11
11.1
Newton's Method for Nonlinear Equations
Inexact Newton Methods
Broyden's Method
Tensor Methods
11.2
Merit Functions
Line Search Methods
Trust-Region Methods
11.3
Motivation
Practical Continuation Methods
Notes and References
Exercises
12
Local and Global Solutions
xîi
Smoothness
12.1
A Single Equality Constraint
A Single Inequality Constraint
Two Inequality Constraints
12.2
12.3
12.4
Relating the Tangent Cone and the First-Order Feasible Direction Set
A Fundamental Necessary Condition
Farkas' Lemma
Proof of Theorem
12.5
Second-Order Conditions and Projected Hessians
12.6
12.7
12.8 Lagrange
12.9
Notes and References
Exercises
13
Linear Programming
13.1
Optimality Conditions
The Dual Problem
13.2
Bases and Basic Feasible Points
Vertices of the Feasible Polytope
13.3
Outline
A Single Step of the Method
13.4
13.5
Pricing and Selection of the Entering Index
Starting the Simplex Method
Degenerate Steps and Cycling
13.6
13.7
13.8
Notes and References
Exercises
Contents
14
14.1
Outline
The Central Path
Central Path Neighborhoods and Path-Following Methods
14.2
Corrector and Centering Steps
Step Lengths
Starting Point
A Practical Algorithm
Solving the Linear Systems
14.3
Other Path-Following Methods
Potential-Reduction Methods
Extensions
14.4
Notes and References
Exercises
15
15.1
15.2
15.3
Simple Elimination using Linear Constraints
General Reduction Strategies for Linear Constraints
Effect of Inequality Constraints
15.4
Merit Functions
Filters
15.5
15.6
Nonmonotone
Notes and References
Exercises
16
16.1
Properties of Equality-Constrained QPs
16.2
Factoring the Full KKT System
Schur-Complement Method
Null-Space Method
xîv
16.3
CG Applied
The Projected
16.4
Optimality Conditions for Inequality-Constrained Problems
Degeneracy
16.5
Specification of the Active-Set Method for Convex QP
Further Remarks on the Active-Set Method
Finite Termination of Active-Set Algorithm on Strictly Convex QPs
Updating Factorizations
16.6
Solving the Primal-Dual System
Step Length Selection
A Practical Primal-Dual Method
16.7
Cauchy Point Computation
Subspace Minimization
16.8
Notes and References
Exercises
17
17.1
Motivation
Algorithmic Framework
Convergence of the Quadratic Penalty Method
111 Conditioning and Reformulations
17.2
A Practical ix Penalty Method
A General Class of Nonsmooth Penalty Methods
17.3
Motivation and Algorithmic Framework
Properties of the Augmented Lagrangian
17.4
Bound-Constrained Formulation
Linearly Constrained Formulation
Unconstrained Formulation
17.5
Notes and References
Exercises
Contents
18
18.1
SQP Framework
Inequality Constraints
18.2
IQPandEQP
Enforcing Convergence
18.3
Handling Inconsistent Linearizations
Full
Reduced-Hessian
Merit Functions
Second-Order Correction
18.4
18.5
A Relaxation Method for Equality-Constrained Optimization
SiiQP (Sequential lx Quadratic Programming)
Sequential Linear-Quadratic Programming (SLQP)
A Technique for Updating the Penalty Parameter
18.6
18.7
Rate of Convergence
18.8
Notes and References
Exercises
19
19.1
19.2
19.3
Primal vs. Primal-Dual System
Solving the Primal-Dual System
Updating the Barrier Parameter
Handling Nonconvexity and Singularity
Step Acceptance: Merit Functions and Filters
Quasi-Newton
Feasible Interior-Point Methods
19.4
19.5
An Algorithm for Solving the Barrier Problem
Step Computation
Lagrange
xvi
Description of a Trust-Region Interior-Point Method
19.6
19.7
Failure of the Line Search Approach
Modified Line Search Methods
Global Convergence of the Trust-Region Approach
19.8 Superlinear
19.9
Notes and References
Exercises
A Background Material
A.
Vectors and Matrices
Norms
Subspaces
Eigenvalues, Eigenvectors, and the Singular-Value Decomposition
Determinant and Trace
Matrix Factorizations: Cholesky,
Symmetric Indefinite Factorization
Sherman-Morrison-Woodbury Formula
Interlacing Eigenvalue Theorem
Error Analysis and
Conditioning and Stability
A.2 Elements of Analysis, Geometry, Topology
Sequences
Rates of Convergence
Topology of the Euclidean Space R"
Convex Sets in R"
Continuity and Limits
Derivatives
Directional Derivatives
Mean Value Theorem
Implicit Function Theorem
Order Notation
Root-Finding for Scalar Equations
В
References
Index |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Nocedal, Jorge Wright, Stephen J. 1960- |
| author_GND | (DE-588)121513238 (DE-588)121513300 |
| author_facet | Nocedal, Jorge Wright, Stephen J. 1960- |
| author_role | aut aut |
| author_sort | Nocedal, Jorge |
| author_variant | j n jn s j w sj sjw |
| building | Verbundindex |
| bvnumber | BV021724241 |
| callnumber-first | Q - Science |
| callnumber-label | QA402 |
| callnumber-raw | QA402.5 |
| callnumber-search | QA402.5 |
| callnumber-sort | QA 3402.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 160 SK 870 SK 905 |
| classification_tum | MAT 910f MAT 912f |
| ctrlnum | (OCoLC)68629100 (DE-599)BVBBV021724241 |
| dewey-full | 519.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.6 |
| dewey-search | 519.6 |
| dewey-sort | 3519.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| edition | Second Edition |
| format | Book |
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| id | DE-604.BV021724241 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:24:19Z |
| indexdate | 2025-02-24T09:01:13Z |
| institution | BVB |
| isbn | 0387303030 0387987932 9780387303031 9781493937110 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014937869 |
| oclc_num | 68629100 |
| open_access_boolean | |
| owner | DE-29T DE-355 DE-BY-UBR DE-20 DE-703 DE-1051 DE-91G DE-BY-TUM DE-384 DE-634 DE-83 DE-19 DE-BY-UBM DE-11 DE-188 DE-706 DE-739 DE-898 DE-BY-UBR DE-573 DE-M347 DE-824 DE-945 DE-4325 |
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| physical | xxii, 664 Seiten Illustrationen, Diagramme |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in operations research and financial engineering |
| spelling | Nocedal, Jorge (DE-588)121513238 aut Numerical optimization Jorge Nocedal ; Stephen J. Wright Second Edition New York, NY Springer [2006] xxii, 664 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Springer series in operations research and financial engineering Hier auch später erschienene, unveränderte Nachdrucke 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 Wright, Stephen J. 1960- (DE-588)121513300 aut Erscheint auch als Online-Ausgabe 978-0-3874-0065-5 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014937869&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nocedal, Jorge Wright, Stephen J. 1960- Numerical optimization 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 |
| title_auth | Numerical optimization |
| title_exact_search | Numerical optimization |
| title_exact_search_txtP | Numerical optimization |
| title_full | Numerical optimization Jorge Nocedal ; Stephen J. Wright |
| title_fullStr | Numerical optimization Jorge Nocedal ; Stephen J. Wright |
| title_full_unstemmed | Numerical optimization Jorge Nocedal ; Stephen J. Wright |
| title_short | Numerical optimization |
| title_sort | numerical optimization |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Numerisches Verfahren Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014937869&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT nocedaljorge numericaloptimization AT wrightstephenj numericaloptimization |