Modern mathematical methods of optimization:
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin
Akad.-Verl.
1993
|
| Series: | Mathematical topics
1 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 415 S. |
| ISBN: | 3055014529 |
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MARC
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| 245 | 1 | 0 | |a Modern mathematical methods of optimization |c ed. by Karl-Heinz Elster |
| 264 | 1 | |a Berlin |b Akad.-Verl. |c 1993 | |
| 300 | |a 415 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical topics |v 1 | |
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Contents
Introduction
15
1 Modern
numerical methods and software in optimization
23
1.1
The linearization method in constrained optimization
. 23
1.1.1
Introduction
. 23
1.1.2
The essence of the method, its formulation and convergence con¬
ditions
. 24
1.1.3
Exact penalty functions
. 27
1.1.4
Selection of
N
and construct!
vity
of convergence conditions
. . 29
1.1.5
Local analysis
., . 33
1.1.6
A linearization method with accelerated convergence
. 37
1.1.7
Some questions of implementation
. 39
1.2
Superlineariy convergent methods of nonlinear programming
. 43
1.2.1
Introduction
. 43
1.2.2
Local theory
. 43
1.2.3
Global theory
. 47
1.3
Optimization problems in the presence of noise
. 53
1.3.1
Sources and types of noise
. 53
1.3.2
Behaviour of standard methods in the presence of noise
. 54
1.3.3
Convergence of methods under random noise
. 55
1.3.4
Optimal methods
. 56
1.3.5
Generalizations
. 57
1.3.6
Implementable algorithms
. 59
1.3.7
Constrained minimization
. 60
g Contents
1.3.8
Conclusion
. 61
1.4
Software
for mathematical programming problems
. 61
References
. 65
2
Optimal methods-of convex programming and polynomial methods
of linear programming
75
2.1
Local methods of convex programming. Optimal methods
. 75
2.1.1
Local methods
. 75
2.1.2
Complexity of convex problems
. 76
2.1.3
Cutting plane methods
. 77
2.1.4
Method of centres
. 78
2.1.5
Method of circumscribed ellipsoids
. 79
2.1.6
Method of inscribed ellipsoids
. 79
2.1.7
Constrained problems
. 81
2.2
Polynomial algorithms in
Бпеаг
programming
. . . •. 82
2.2.1
Algebraic statement of the problem
. 82
2.2.2
Bit posing. Method of ellipsoids
. 83
2.2.3
Karmarkar's method
. 84
2.2.4
The projection method
. 86
2.2.5
The methods of
Renegar
and Vaidya
. 89
2.2.6
Dual algorithms
. 91
2.3
Optimal methods for the solution of large-scale convex programming
problems
. 97
2.3.1
Smooth spaces
. 98
2.3.2
General convex problems
. 99
2.3.3
Smooth convex problems
. 101
2.3.4
Problems with regular minimum
. 107
References
.
Ill
Additional References
. 114
3
Decomposition of optimization problems
116
3.1
Decomposition approach in optimization
. 116
3.1.1
Typical approaches
. 117
Contents
g
3.1.2
Generalizations
. 121
3.2
Convex problems with bordering structure
. 123
3.2.1
Statement of the problem
. 123
3.2.2
Primal decomposition with allocation of centralized resources
. 125
3.2.3
Simultaneous application of primal and dual decomposition
. . 128
3.3
A general approach to the construction of decomposition (block) meth¬
ods of convex programming
. 132
3.3.1
Statement of the problem
. 132
3.3.2
Description of the method
. 134
3.3.3
Discussion of the method
. 136
3.3.4
Decomposition method for the analysis of
monotonie
mappings
138
References
. 142
4
Nonsmooth functions
146
4.1
Nonsmooth analysis and directional derivatives
. 146
4.1.1
Introduction
. 146
4.1.2
Approximation of functions
. 147
4.1.3
Dini
derivatives
. 150
4.1.4
Clarke derivatives
. 152
4.1.5
Approximation of sets
. 155
4.1.6
Clarke's cone
. 158
4.2
Quasidifferentiable functions
. 159
4.3
Lexicographically smooth functions
. 164
4.3.1
The class of lexicographically smooth functions
. 164
4.3.2
Lexicographic derivatives and their properties
. 165
4.3.3
Fundamental theorem of lexicographic differential calculus
. . . 168
4.3.4
Examples of lexicographic derivatives
. 170
4.3.5
Some theorems from analysis
. 173
References
. 174
5
Improper problems of mathematical programming
178
5.1
Introduction
. 178
5.2
Duality
. 180
10 Contents
5.2.1
Realization of duality for improper linear programming problems
181
5.2.2
Duality for improper convex programming problems of the first
kind
. 187
5.2.3
On duality of improper problems in infinite-dimensional spaces
188
5.3
Infinite-dimensional linear programming problems with, a duality gap
. 191
5.3.1
Approximation of an infinite-dimensional LP problem by cones
191
5.3.2
Infinite linear programming problems over
Et». 193
5.4
Correction of systems of linear equations and inequalities using addi¬
tional information
. 195
5.5
Correction of improper minimax problems
. 198
5.5.1
Solvability sets
. 198
5.5.2
Correction according to a convex criterion
. 199
5.5.3
Correction with respect to a concave-convex criterion
. 201
5.6
Regularization of improper linear problems
. 203
5.7
Correction of improper convex programming problems by means of aug¬
mented Lagrangians
. 206
5.7.1
Correction on the basis of a modification of the Lagrangian with
respect to dual variables
. 206
5.7.2
Correction on the basis of a symmetric modification of the La¬
grangian
. 209
References
. 211
6
Optimization in order scales
212
6.1
Introduction
. 212
6.2
Conceptions and mechanisms of choice
. 213
6.3
Choice functions
. 214
6.3.1
Choice and choice function
. 214
6.3.2
Partial and complete choice functions
. 214
6.3.3
Characteristic features of a choice function
. 215
6.4
Binary relations
. 216
6.4.1
Measurement of the quality of a solution
. 216
6.4.2
Classes of binary relations
. 217
6.4.3
Indicators
. 219
6.5
Binary relations and choice functions
. 220
Contents \\
6.5.1
Classes
of choice functions
.220
6.5.2
Characterizing properties of certain choice functions
.222
6.5.3
Decomposition of choice functions
.223
6.6
Optimization with respect to a binary relation
.224
6.6.1
Optimal elements
. 224
6.6.2
Comparison of variants
. 225
6.6.3
Correcting choice functions
. 227
6.7
Generalized mathematical programming
. 227
6.7.1
The development of models
. 227
6.7.2
Mathematical programming in order scales
. 228
6.7.3
Solution methods
. 229
6.7.4
Concluding remarks
. 231
References
. 231
7
Multiobjective optimization
233
7.1
Duality in multiobjective optimization
. 233
7.1.1
Strong and weak duality
. 234
7.1.2
Construction of dual problems
. 235
7.1.3
Duality for multiobjective location problems
. 239
7.1.4
Duality in integer multiobjective programming
. 240
7.2
Duality of choice in vector optimization problems
. 242
7.2.1
Primal and dual choice functions
. 242
7.2.2
Dualization of preferences and alternatives
. 243
7.2.3
Duality and Pareto optimality
. 244
7.2.4
Pareto optima and polar correspondences
. 246
7.2.5
Dual generalized criteria
. 248
7.2.6
An application
. 250
7.2.7
Duality for asymmetric budget sets
. 251
7.3
Multicriteria optimization problems involving importance-ordered criteria254
7.3.1
Basic definitions
. 254
7.3.2
Main kinds of importance
. 257
7.3.3
Problems involving homogeneous importance-ordered criteria
. 260
12 Contents
7.3.4
Convex
problems with homogeneous criteria ordered with re¬
spect to importance
. 262
7.3.5
Multicriteria problems with inhomogeneous criteria ordered with
respect to importance
. 264
7.4
Efficiency estimation of decision rules in discrete nraltiobjective problems267
7.4.1
Construction of solutions
. 268
7.4.2
Efficiency indicators of decision rules in multicriteria problems
270
7.4.3
Efficiency estimates for twocriterial problems
. 273
7.4.4
Some efficiency estimates for the general case
. 275
7.5
On the dialogue procedures of decision making in practical multicriteria
models of economy
. 277
7.5.1
Statement of the problem and optimization procedures
. 277
7.5.2
An example of applying the dialogue systems described
. 279
7.5.3
A program package of multicriteria optimization
. 280
7.5.4
A method of interactive multicriteria optimization
. 281
References
. 282
8
Discrete optimization
288
8.1
Mathematical models and some applied problems
. 288
8.1.1
The general problem of ILP
. 289
8.1.2
The multidimensional knapsack problem
. 289
8.1.3
The onedimensional knapsack problem
. 290
8.1.4
The problems of packing, partitioning and covering a set
. . . . 290
8.1.5
The travelling salesman problem
. 291
8.1.6
The fixed-charge problem
. 292
8.1.7
Consideration of problem specifics
. 293
8.1.8
The quadratic assignment problem
. 294
8.1.9
Problems of projecting mining enterprises
. 295
8.1.10
Selection of optimal technologies
. 295
8.1.11
Other models and applications
. 296
8.2
Methods of discrete optimization. Complexity of discrete problems
. . 297
8.2.1
General survey
. 297
8.2.2
Behaviour of methods
. 298
Contents 13
8.2.3 Elements
of complexity theory
.299
8.2.4
Complexity of discrete optimization problems
.301
8.2.5
A way out of the situation
. 302
8.3
Effective approximation methods of discrete optimization
.304
8.3.1
Polynomial approximation schemes
.304
8.3.2
Polynomial approximately feasible algorithms
.308
8.4
The knapsack problem and its extensions
.310
8.4.1
The knapsack problem
. 310
8.4.2
The multiple-choice knapsack problem
. 313
8.4.3
The rmiltistep knapsack problem
. 315
8.4.4
Linear generalizations
. 316
8.4.5
Nonlinear generalizations
. 316
8.4.6
A special problem
. 318
8.5
Dual approach in integer programming
. 318
8.5.1
Duality in ILP
.318
8.5.2
Generalized labelling methods
.323
References
.326
9
Some problems of mathematical programming in infinite-dimensional
spaces
337
9.1
Duality assertions and optimality conditions of general fractional opti¬
mization problems
.337
9.1.1
Introduction
. 337
9.1.2
Conjugate functions
. 338
9.1.3
Duality assertions and optimality conditions
. 340
9.1.4
Comparison of conjugation concepts
. 345
9.2
Optimal control and duality in control problems
. 347
9.2.1
Fundamental problems of optimal control
. 347
9.2.2
Variational problems
. 349
9.2.3
Pontryagin's maximum principle
. 350
9.2.4
Dual problems for optimal control problems
. 351
9.3
Numerical analysis and solution methods for semi-infinite programming
problems
. 353
14 Contents
9.3.1
Introduction
. 353
9.3.2
Theoretical
background
of numerical methods
. 355
9.3.3
Numerical methods for semi-infinite problems
. 359
9.4
Proximity-space methods in optimization with constraints
. 362
9.4.1
Introduction
. 362
9.4.2
Topologica!
background
. 363
9.4.3
Optimization problems treated with tolerance
. 365
9.4.4
Compactification of optimization problems
. 368
References
. 369
10
Optimization and mathematical economics
375
10.1
Price regulation in the presence of queues and quantity rationing
. . . 375
10.1.1
Demand and fixed price equilibrium in the presence of queues
. 376
10.1.2
Demand and fixed price equilibrium under quantity rationing
. 379
10.1.3
Price regulation
. 381
10.1.4
Concluding remarks
. 383
10.2
Processes of finding equilibrated states
. 383
10.2.1
Equilibrated states
. 384
10.2.2
First method of searching equilibrated states: movement in the
space of weighted coefficients
. 388
10.2.3
Second approach to finding equilibrated states: movement in
the original space
. 390
10.3
Local estimates and the selection problem
. 395
10.3.1
Selection functions
. 396
10.3.2
The problem of multicriteria optimization
. 397
10.3.3
Regularization of contradictory requirements
. 398
10.3.4
The problem of
quasilinear
programming
. 400
10.4
Computation of equilibria for a class of piecewise linear models by a
sequence of linear programs
. 401
References
. 407
Index
411 |
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| id | DE-604.BV007728507 |
| illustrated | Not Illustrated |
| indexdate | 2025-02-10T13:12:29Z |
| institution | BVB |
| isbn | 3055014529 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005079704 |
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| physical | 415 S. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Akad.-Verl. |
| record_format | marc |
| series | Mathematical topics |
| series2 | Mathematical topics |
| spelling | Modern mathematical methods of optimization ed. by Karl-Heinz Elster Berlin Akad.-Verl. 1993 415 S. txt rdacontent n rdamedia nc rdacarrier Mathematical topics 1 Mathematical optimization Optimierung (DE-588)4043664-0 gnd rswk-swf Optimierung (DE-588)4043664-0 s DE-604 Elster, Karl-Heinz Sonstige oth Mathematical topics 1 (DE-604)BV008671507 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005079704&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Modern mathematical methods of optimization Mathematical topics Mathematical optimization Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4043664-0 |
| title | Modern mathematical methods of optimization |
| title_auth | Modern mathematical methods of optimization |
| title_exact_search | Modern mathematical methods of optimization |
| title_full | Modern mathematical methods of optimization ed. by Karl-Heinz Elster |
| title_fullStr | Modern mathematical methods of optimization ed. by Karl-Heinz Elster |
| title_full_unstemmed | Modern mathematical methods of optimization ed. by Karl-Heinz Elster |
| title_short | Modern mathematical methods of optimization |
| title_sort | modern mathematical methods of optimization |
| topic | Mathematical optimization Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Mathematical optimization Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005079704&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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