Numerical methods and optimization: an introduction
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2014
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| Series: | Numerical analysis and scientific computing
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 397 S. graph. Darst. |
| ISBN: | 1466577770 9781466577770 |
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MARC
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| 100 | 1 | |a Butenko, Sergiy G. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Numerical methods and optimization |b an introduction |c Sergiy Butenko ; Panos M. Pardalos |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2014 | |
| 300 | |a XVI, 397 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Numerical analysis and scientific computing | |
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Record in the Search Index
| _version_ | 1804150452862844928 |
|---|---|
| adam_text | Contents
Basics 1
Preliminaries
3
1.1
Sets and Functions
........................ 3
1.2
Fundamental Theorem of Algebra
............... 6
1.3
Vectors and Linear (Vector) Spaces
.............. 7
1.3.1
Vector norms
....................... 10
1.4
Matrices and Their Properties
................. 12
1.4.1
Matrix addition and scalar multiplication
....... 12
1.4.2
Matrix multiplication
.................. 13
1.4.3
The transpose of a matrix
................ 14
1.4.4
Triangular and diagonal matrices
............ 15
1.4.5
Determinants
....................... 16
1.4.6
Trace of a matrix
..................... 17
1.4.7
Rank of a matrix
..................... 18
1.4.8
The inverse of a nonsingular matrix
.......... 18
1.4.9
Eigenvalues and eigenvectors
.............. 19
1.4.10
Quadratic forms
..................... 22
1.4.11
Matrix norms
....................... 24
1.5
Preliminaries from Real and Functional Analysis
....... 25
1.5.1
Closed and open sets
................... 26
1.5.2
Sequences
......................... 26
1.5.3
Continuity and differentiability
............. 27
1.5.4
Big
О
and little
о
notations
............... 30
1.5.5
Taylor s theorem
..................... 31
Numbers and Errors
37
2.1
Conversion between Different Number Systems
........ 39
2.1.1
Conversion of integers
.................. 40
2.1.2
Conversion of fractions
.................. 42
2.2
Floating Point Representation of Numbers
.......... 44
2.3
Definitions of Errors
....................... 45
2.4
Round-off Errors
......................... 47
2.4.1
Rounding and chopping
................. 47
2.4.2
Arithmetic operations
.................. 48
2.4.3
Subtractive cancellation and error propagation
.... 49
xi
Xl)
II Numerical Methods for Standard Problems
53
3
Elements of Numerical Linear Algebra
55
3.1
Direct Methods for Solving Systems uf Linear Equations
. . 57
3.1.1
Solution of triangular systems of linear equations
. . . 57
3.1.2
Gaussian elimination
................... 59
3.1.2.1
Pivoting strategies
............... (¡2
3.1.3
Gauss-Jordan met hot! and matrix inversion
...... 03
3.1.4
Triangular factorization
................. 00
3.2
Iterativi»
Methods for Solving Systems of Linear Equations
. 09
3.2.1
Jacobi method
...................... 70
3.2.2
Gauss-Seidel method
................... 72
3.2.3
Application: input-output models in economics
.... 7
і
3.3
Overdetcrinined Systems and Least Squares Solution
.... 75
3.3.1
Application: linear regression
.............. 70
3.4
Stability of a Problem
...................... 77
3.5
Computing Eigenvalues and Eigenvectors
...........
7S
3.5.1
The power method
.................... 7!)
3.5.2
Application: ranking methods
..............
SO
4
Solving Equations
87
4.1
Fixed Point Method
......................
8S
4.2
Bracketing Met bods
....................... 92
4.2.1
Bisection method
..................... 93
4.2.1.1
Convergence of the bisection method
.... 93
4.2.1.2
Intervals with multiple roots
......... 95
4.2.2
Hegula-falsi method
................... 90
4.2.3
Modified regula-falsi method
..............
9S
4.3
Newton s Method
........................ 99
4.3.1
Convergence rate of Newton s method
......... 103
4.4
Secant. Method
.......................... 105
4.5
Solution of Nonlinear Systems
................. 100
4.5.1
Fixed point method for systems
............. 100
4.5.2
Newton s method for systems
.............. 107
5
Polynomial Interpolation
113
5.1
Forms of Polynomials
...................... 115
5.2
Polynomial Interpolation Methods
............... 116
5.2.1 Lagrange
method
..................... 117
5.2.2
The method of undetermined coefficients
....... 118
5.2.3
Newton s method
..................... 118
5.3
Theoretical Error of Interpolation and Chebyshev Polynomials
120
5.3.1
Properties of Chebyshev polynomials
.......... 122
XIII
6
Numerical Integration
127
6.1
Trapezoidal Rule
......................... 129
6.2
Simpson s Rule
.......................... 131
6.3
Precision and Error of Approximation
............. 132
6.4
Composite Rules
......................... 134
6.4.1
The composite trapezoidal rule
............. 134
6.4.2
Composite Simpson s rule
................ 135
6.5
Using Integrals to Approximate Sums
............. 137
7
Numerical Solution of Differential Equations
141
7.1
Solution of a Differential Equation
............... 142
7.2
Taylor Series and
Picard s
Methods
.............. 143
7.3
Euler s Method
.......................... 145
7.3.1
Discretization errors
................... 147
7.4
Runge-Kutta Methods
...................... 147
7.4.1
Second-order Runge-Kutta methods
.......... 148
7.4.2
Fourth-order Runge-Kutta methods
.......... 151
7.5
Systems of Differential Equations
............... 152
7.6
Higher-Order Differential Equations
.............. 155
III Introduction to Optimization
159
8
Basic Concepts
161
8.1
Formulating an Optimization Problem
............. 161
8.2
Mathematical Description
.................... 164
8.3
Local and Global Optimality
.................. 166
8.4
Existence of an Optimal Solution
............... 168
8.5
Level Sets and Gradients
.................... 169
8.6
Convex Sets, Functions, and Problems
............. 173
8.6.1
First-order characterization of a convex function
... 177
8.6.2
Second-order characterization of a convex function
. . 179
9
Complexity Issues
185
9.1
Algorithms and Complexity
................... 185
9.2
Average Running Time
..................... 189
9.3
Randomized Algorithms
..................... 190
9.4
Basics of Computational Complexity Theory
......... 191
9.4.1
Class
ЛЛР
......................... 193
9.4.2
V vs.
MV......................... 193
9.4.3
Polynomial time reducibility
.............. 194
9.4.4
ЛЛР-сотріеЇе
and
ЛЛР
-hard
problems
......... 195
9.5
Complexity of Local Optimization
............... 198
9.6
Optimal Methods for Nonlinear Optimization
......... 203
9.6.1
Classes of methods
.................... 203
9.6.2
Establishing lower complexity bounds for a class of
methods
.......................... 204
XIV
í).G.
Л
Defining an optimal method
............... -00
10
Introduction to Linear Programming
211
10.1
Formulating a Linear Programming Model
.......... 211
10.1.1
Defining the decision variables
............. 211
10.1.2
Formulating the objective function
........... 212
10.1.3
Specifying the constraints
................ 212
10.1.4
The complete linear programming formulation
.... 21 3
10.2
Examples of LP Models
..................... 213
10.2.1
A diet problem
...................... 213
10.2.2
A resource allocation problem
.............. 214
10.2.3
A scheduling problem
.................. 215
10.2.4
A mixing problem
.................... 217
10.2.5
A transportation problem
................ 219
10.2.
(i A production planning problem
............. 220
10.3
Practical Implications of Using LP Models
.......... 221
10.4
Solving Two-Variable LPs Graphically
............. 222
10.
Γι
Classification
of LPs
....................... 22!»
11
The Simplex Method for Linear Programming
235
11.1
The Standard Form of LP
................... 235
11.2
The Simplex Method
......................
2.Ч7
11.2.1
Step
1........................... 239
11.2.2
Step
2........................... 242
11.2.3
Recognizing opt
ini u
lit
y..................
244
11.2.4
Recognizing unbounded LPs
............... 244
11.2.5
Degeneracy and cycling
.................
24-r>
11.2.0
Properties of LP dictionaries and the simplex method
24!)
11.3
GtKHiietry of the Simplex Method
...............
2Г>1
11.4
The Simplex Method for a General LP
............ 254
11.4.1
The two-pha.se simplex method
............. 259
11.4.2
The big-
M
method
.................... 204
11.5
The Fundamental Theorem of LP
............... 200
11.
G
The Revised Simplex Method
.................. 200
11.7
Complexity of the Simplex Method
.............. 270
12
Duality and Sensitivity Analysis in Linear Programming
281
12.1
Defining the Dual LP
...................... 281
12.1.1
Forming the dual of a general LP
............ 284
12.2
Weak Duality and the Duality Theorem
............ 287
12.3
Extracting an Optimal Solution of the Dual LP from an
Optimal Tableau of the Primal LP
............... 289
12.4
Correspondence between the Primal and Dual LP Types
. .
2i)0
12.5
Complementary Slackness
.................... 291
12.0
Economic Interpretation of the Dual LP
............ 294
XV
12.7
Sensitivity Analysis
....................... 296
12.7.1
Changing the objective function coefficient of a basic
variable
.......................... 302
12.7.2
Changing the objective function coefficient of
a nonbasic
variable
.......................... 303
12.7.3
Changing the column of
a nonbasic
variable
...... 305
12.7.4
Changing the right-hand side
.............. 305
12.7.5
Introducing a new variable
............... 307
12.7.6
Introducing a new constraint
.............. 308
12.7.7
Summary
......................... 310
13
Unconstrained Optimization
317
13.1
Optimality Conditions
...................... 317
13.1.1
First-order necessary conditions
............. 317
13.1.2
Second-order optimality conditions
........... 320
13.1.3
Using optimality conditions for solving optimization
problems
......................... 322
13.2
Optimization Problems with a Single Variable
........ 323
13.2.1
Golden section search
.................. 323
13.2.1.1
Fibonacci search
................ 325
13.3
Algorithmic Strategies for Unconstrained Optimization
. . . 327
13.4
Method of Steepest Descent
.................. 328
13.4.1
Convex quadratic case
.................. 330
13.4.2
Global convergence of the steepest descent method
. . 331
13.5
Newton s Method
........................ 333
13.5.1
Rate of convergence
................... 334
13.5.2
Guaranteeing the descent
................ 335
13.5.3
Levenberg-Marquardt method
.............. 335
13.6
Conjugate Direction Method
.................. 336
13.6.1
Conjugate direction method for convex quadratic prob¬
lems
............................ 337
13.6.2
Conjugate gradient algorithm
.............. 340
13.6.2.1
Non-quadratic problems
............ 341
13.7
Quasi-Newton
Methods
..................... 342
13.7.1
Rank-one correction formula
.............. 344
13.7.2
Other correction formulas
................ 345
13.8
Inexact Line Search
....................... 346
14
Constrained Optimization
351
14.1
Optimality Conditions
...................... 351
14.1.1
First-order necessary conditions
............. 351
14.1.1.1
Problems with equality constraints
...... 351
14.1.1.2
Problems with inequality constraints
..... 358
14.1.2
Second-order conditions
................. 363
14.1.2.1
Problems with equality constraints
...... 363
XVI
14.1.2.2
Problems with inequality constraints
.....
14.2
Duality
..............................
ЗОН
14.3
Projected Gradient Methods
.................. 371
14.3.1 Affine
scaling method for LP
.............. 374
14.4
Sequential Unconstrained Minimization
............ 377
14.4.1
Penalty function methods
................ 378
14.4.2
Barrier methods
..................... 379
14.4.3
Interior point methods
.................. 381
Notes and References
387
Bibliography
389
Index
391
|
| any_adam_object | 1 |
| author | Butenko, Sergiy G. Pardalos, Panos M. 1954- |
| author_GND | (DE-588)115385827 |
| author_facet | Butenko, Sergiy G. Pardalos, Panos M. 1954- |
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| bvnumber | BV041081353 |
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| id | DE-604.BV041081353 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:39:09Z |
| institution | BVB |
| isbn | 1466577770 9781466577770 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026058136 |
| oclc_num | 879359401 |
| open_access_boolean | |
| owner | DE-703 DE-634 DE-573 DE-861 |
| owner_facet | DE-703 DE-634 DE-573 DE-861 |
| physical | XVI, 397 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Numerical analysis and scientific computing |
| spelling | Butenko, Sergiy G. Verfasser aut Numerical methods and optimization an introduction Sergiy Butenko ; Panos M. Pardalos Boca Raton [u.a.] CRC Press 2014 XVI, 397 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Numerical analysis and scientific computing Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s Optimierung (DE-588)4043664-0 s DE-604 Pardalos, Panos M. 1954- Verfasser (DE-588)115385827 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026058136&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Butenko, Sergiy G. Pardalos, Panos M. 1954- Numerical methods and optimization an introduction Numerische Mathematik (DE-588)4042805-9 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4043664-0 |
| title | Numerical methods and optimization an introduction |
| title_auth | Numerical methods and optimization an introduction |
| title_exact_search | Numerical methods and optimization an introduction |
| title_full | Numerical methods and optimization an introduction Sergiy Butenko ; Panos M. Pardalos |
| title_fullStr | Numerical methods and optimization an introduction Sergiy Butenko ; Panos M. Pardalos |
| title_full_unstemmed | Numerical methods and optimization an introduction Sergiy Butenko ; Panos M. Pardalos |
| title_short | Numerical methods and optimization |
| title_sort | numerical methods and optimization an introduction |
| title_sub | an introduction |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Numerische Mathematik Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026058136&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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