Perturbation theory in mathematical programming and its applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chichester u.a.
Wiley
1994
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| Series: | Wiley interscience series in discrete mathematics and optimization
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 383 S. |
| ISBN: | 0471939358 |
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| 245 | 1 | 0 | |a Perturbation theory in mathematical programming and its applications |c Evgenij S. Levitin |
| 264 | 1 | |a Chichester u.a. |b Wiley |c 1994 | |
| 300 | |a XVIII, 383 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| adam_text | Contents
PREFACE xi
INTRODUCTION xiii
PART I FOUNDATIONS OF THEORY AND METHODS OF
FINITE DIMENSIONAL OPTIMIZATION 1
1. Basic Concepts, Problems and Fields of Applications of
Perturbation Theory 3
1.1. Basic Concepts of Perturbation Theory 3
1.1.1. Concepts of stability in a mathematical programming problem (MPP) 3
1.1.2. Perturbation of MPP 5
1.1.3. Classification of perturbations 6
1.1.4. Definitions of MPP stability with respect to perturbation 7
1.2. Basic Problems of Perturbation Theory. Counterexamples 9
1.3. Brief Review of Investigations of Perturbation Theory 18
1.4. Vector Optimization and Parametric Extremum Problems 23
1.5. Accuracy Estimate of Optimization Problem Solutions. Models of Decision
Making under Uncertainty Conditions 25
1.6. Two level Systems and Parametric Optimization 28
1.6.1. 1 vo level optimization 29
1.6.2. Economical equilibrium models (noncooperative) 30
1.6.3. Inter section cooperation models 31
1.7. Approximation by a Sequence of Extremum Problems and Convergence
of Numerical Methods 32
1.8. Dynamic Programming and the Problem of the Differentiability of the
Bellman Function 33
1.9. Mathematical Programming Problems with a Block Structure and
Related Perturbations 34
1.9.1. Parametrization and perturbation function in a block problem with binding
variables 35
1.9.2. Parametrization and perturbation function in a block problem with binding
constraints 37
1.9.3. Parametrization in the case of both binding constraints and binding variables 39
1.10. Nonsmooth Optimization and Perturbation Functions 41
1.11. Asymptotic Method of Constructing an Approximate Solution for
Optimization Problems with Small Parameters 45
1.12. Correction of Solutions of Optimization Problems With Small Variations of
Original Data 51
viii Contents
2. Perturbation Theory of Smooth Mathematical Programming
Problems (Main Results) 53
2.1. Theorem on Estimate of Distance to the Set Q(w) 53
2.2. Zero Approximation Theory 54
2.2.1. Properties of optimal solutions to the generating problem 54
2.2.2. Stability with respect to optimal value and solution set 57
2.2.3. Lipshitz property of the perturbation function 59
2.2.4. Stability of the set of optimal Lagrange s multipliers. Local stability of an
active index 59
2.2.5. On one property of approximate solutions to the perturbed problem 59
2.3. First Approximation Theory 60
2.3.1. First approximation problem for the perturbed problem and its properties 60
2.3.2. First order differential expansions of the perturbation function 62
2.3.3. Differential properties of approximate, o(e), solutions to the perturbed problem 65
2.4. Second Approximation Theory 66
2.4.1. Second approximation problem for the perturbed problem and its properties 66
2.4.2. Second order differential expansions of the perturbation function 70
2.4.3. Differential properties of approximate, o(e2), solutions to the perturbed
problem 71
3. Examples of Studies in Parametric Optimization Problems.
Applications of Perturbation Theory 73
3.1. Dependence of the Optimal Value and the Solutions to a Linear
Programming Problem on All of its Coefficients 73
3.2. Investigation of Two Classes of Parametric Convex Programming Problems 77
3.2.1. Dependence of optimal value and solutions on the weight coefficients
in a criterion and on the right hand side of constraints 77
3.2.2. Dependence of the projection of the point £ onto the convex set Q(rj) on the
parameters w — (£,»?) 80
3.3. An Illustrative Example: Analysis of the Perturbation of a Two Dimensional
Problem with Quadratic Separable Functions 83
3.3.1. Properties of the generating problem 83
3.3.2. Stability 84
3.3.3. First approximation theory 84
3.3.4. Second approximation theory 87
3.4. A Small Perturbation of the Cauchy Bunyakovsky Inequality by Quadratic
Forms 93
3.5. Decomposition Method of Choice of Descent Direction in a Block Problem
with Binding Variables 95
3.6. Asymptotic Method of Constructing the Approximate Solution for the
Perturbation of a Block Problem with Binding Variables and Binding
Constraints 99
3.7. Asymptotic Method of Constructing the Approximate Solution for the
Perturbation of a Problem that Can Be Aggregated with Respect to a Group
of Variables 106
PART II BASIC PERTURBATION THEORY IN
FINITE DIMENSIONAL OPTIMIZATION 109
4. Elements of Convex and Nonconvex Analysis 111
4.1. Convex Sets and Convex Functions 111
Contents jx
4.1.1. Algebraic properties of convex sets 111
4.1.2. Topological properties of convex sets 113
4.1.3. Convex and quasiconvex, uniformly convex and quasiconvex, strongly convex
functions and their properties 113
4.1.4. Properties of quasiconvex function minima on a convex set 122
4.1.5. Projection operator onto a convex set 123
4.1.6. Separation theorems for convex sets 124
4.1.7. Properties of sublinear functions 125
4.1.8. Adjoint cones. Farkas theorem 128
4.2. Differential Properties (Continuity, Lipshitz Property, Directional
Differentiability) of Convex Functions 128
4.3. Definition and Properties of rj Subdifferentials at a Point, Supporting
Vectors, and the Function Adjoint to a Convex One 139
4.4. Calculation of a Set of ij Subdifferentials at a Point, a Set of Supporting
Vectors, and a Function Adjoint to a Convex One 147
4.5. Minimax Theorem. Criterion for Existence of a Saddle Point 156
4.6. Optimality Criteria in Convex Programming (Duality Theory, Optimality
Conditions in Differential Form) 159
4.7. Compatibility Conditions and Estimate of Distance to Solution Set
for a System of Convex Inequalities and Linear Equations.
Hoffman s Theorem 168
4.7.1. Compatibility criterion for a system of convex inequalities and linear
inhomogeneous equations 168
4.7.2. Incompatibility criteria for a system of strict convex inequalities, nonstrict
affine inequalities and linear inhomogeneous equations 171
4.7.3. Estimate of distance to the solution set of a system of nonstrict convex
inequalities and linear equations 174
4.7.4. Estimate of the distance to the solution set of a parametric system of linear
inhomogeneous inequalities and equations (Hoffman s theorem) 175
4.8. Upper Convex, Fine Convex and Fine Linear Approximations of
Functions Depending on Parameters 177
4.9. Locally Convex Functions. Classes £(Z,Zo), F^Zgo) and Q(Zzo) of
Functions 182
4.10. Strict Differentiability with a Second Order Remainder.
Second Order Approximation at a Point 193
4.11. Estimate of Distance to the Set of Solutions of a System of Convex
Inequalities and Nonlinear Equations 194
5. Optimality Criteria in Generating Problem 201
5.1. Definition of Necessary and Sufficient Optimality Conditions of First
and Higher Orders 202
5.2. Necessary First Order Minimum Condition 207
5.3. Sufficient First Order Optimality Condition 213
5.4. Local Minimum Criteria for Nonconvex Problems 216
5.5. Criteria of Higher Order Minimum 231
5.6. Necessary and Sufficient Second Order Optimality Conditions 241
5.7. Examples of Studying an Extremum Problem for a Local Minimum 243
5.8. Local Minimum Criteria Without Regularity Condition 255
x Contents
PART III PERTURBATION THEORY IN
MATHEMATICAL PROGRAMMING 259
6. General Perturbation Theory 261
6.1. Estimate of Distance to the Set Q (w) 262
6.2. General Assumptions 265
6.3. Stability with Respect to Optimal Value and to Solution Set 267
6.4. Lipshitz Property of the Perturbation Function 270
6.5. Stability of the Set of Optimal Lagrange Multipliers. Local Stability of
the Active Index /0 €/(wo,Xo) Properties of Approximate
Solutions of the Perturbed Problem 271
6.6. First Approximation Problem 276
6.7. Main Results of First Approximation Theory 284
6.8. Proofs of Theorems of First Approximation Theory 290
6.9. Second Approximation Problem for the Perturbed Problem 301
6.10. Main Results of the Quadratic Theory 304
6.11. Proofs of Theorems of Quadratic Theory 306
6.12. General Theory for the Unbounded Set of Optimal Solutions to the
Generating Problem 318
7. Perturbation Theory for Smooth Mathematical
Programming Problems 323
7.1. First Approximation Theory for Problems with Functions J,fo,
and gj Continuously Differentiable with Respect to {wjc} 323
7.2. Second Approximation Theory for Problems with Functions
J,f , and gj Twice Differentiable with Respect to {wjc} 328
8. Perturbation Theory for Convex Programming
Problems 329
8.1. Properties of Optimal Solutions to the Generating Problem and its Dual 329
8.2. Perturbation of a Convex Programming Problem and its Dual 331
8.3. Stability with Respect to Optimal Value and Solution Set in the Perturbed
and Dual Problems 335
8.4. First and Second Approximation Theory for the Perturbed Problem 338
8.5. First Approximation Theory for the Dual Perturbed Problem. Differential
Expansion for the Dual Gap 343
9. Theory of Minimax Perturbations Under Bound Constraints 349
9.1. Problem Posing, Notation and Assumptions 349
9.2. The Zero, First and Second Approximation Theories 354
NOTES AND BIBLIOGRAPHICAL COMMENTS 359
REFERENCES 367
INDEX 381
|
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| language | English |
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| series2 | Wiley interscience series in discrete mathematics and optimization |
| spelling | Levitin, Evgenij S. Verfasser (DE-588)10428904X aut Perturbation theory in mathematical programming and its applications Evgenij S. Levitin Chichester u.a. Wiley 1994 XVIII, 383 S. txt rdacontent n rdamedia nc rdacarrier Wiley interscience series in discrete mathematics and optimization Mathematische programmering gtt Programmation (mathématiques) ram Storingen gtt critère optimalité inriac optimisation paramètre inriac programmation convexe inriac programmation mathématique inriac théorie perturbation inriac Perturbation (Mathematics) Programming (Mathematics) Störungstheorie (DE-588)4128420-3 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Störungstheorie (DE-588)4128420-3 s Optimierung (DE-588)4043664-0 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006433243&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Levitin, Evgenij S. Perturbation theory in mathematical programming and its applications Mathematische programmering gtt Programmation (mathématiques) ram Storingen gtt critère optimalité inriac optimisation paramètre inriac programmation convexe inriac programmation mathématique inriac théorie perturbation inriac Perturbation (Mathematics) Programming (Mathematics) Störungstheorie (DE-588)4128420-3 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4128420-3 (DE-588)4043664-0 |
| title | Perturbation theory in mathematical programming and its applications |
| title_auth | Perturbation theory in mathematical programming and its applications |
| title_exact_search | Perturbation theory in mathematical programming and its applications |
| title_full | Perturbation theory in mathematical programming and its applications Evgenij S. Levitin |
| title_fullStr | Perturbation theory in mathematical programming and its applications Evgenij S. Levitin |
| title_full_unstemmed | Perturbation theory in mathematical programming and its applications Evgenij S. Levitin |
| title_short | Perturbation theory in mathematical programming and its applications |
| title_sort | perturbation theory in mathematical programming and its applications |
| topic | Mathematische programmering gtt Programmation (mathématiques) ram Storingen gtt critère optimalité inriac optimisation paramètre inriac programmation convexe inriac programmation mathématique inriac théorie perturbation inriac Perturbation (Mathematics) Programming (Mathematics) Störungstheorie (DE-588)4128420-3 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Mathematische programmering Programmation (mathématiques) Storingen critère optimalité optimisation paramètre programmation convexe programmation mathématique théorie perturbation Perturbation (Mathematics) Programming (Mathematics) Störungstheorie Optimierung |
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