Finitely Additive Measures and Relaxations of Extremal Problems

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Springer Science & Business Media, Sep 30, 1996 - Language Arts & Disciplines - 244 pages
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This monograph constructs correct extensions of extremal problems, including problems of multicriteria optimization as well as more general cone optimization problems. The author obtains common conditions of stability and asymptotic nonsensitivity of extremal problems under perturbation of a part of integral restrictions for finite and infinite systems of restrictions. Features include individual chapters on nonstandard approximation of finitely additive measures by indefinite integrals and constructions of attraction sets. Professor Chentsov illustrates abstract settings by providing examples of problems of impulse control, mathematical programming, and stochastic optimization.
 

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Contents

Relaxations of Extremal Problems cogent discussion and examples
1
12 Relaxations of Integral Constraints
2
13 Constraints Imposed on the Vector Integrand of a Control
7
14 Asymptotic Effects in a Cone Optimization Problem an Example
13
Compactificators in Extremal Problems
19
22 Some Notions from Topology
20
23 Compactificator of an Extremal Problem with Constraints of Asymptotic Character
26
24 Some Notions of Set Theory Products of Topological Spaces
33
55 Strong Boundedness Conditions and Relaxations of Admissible Sets II
113
Relaxations of Attainable Sets
121
62 Limit Representation of Attainable Sets
122
63 Asymptotic Nonsensitivity and Relaxations of Attainable Sets Case of Positive Measures
125
64 Asymptotic Nonsensitivity and Relaxations of Attainable Sets Case of Alternating Measures
131
65 Attainability Domains of Controllable Systems and Their Relaxations
135
Asymptotic Values and Their Generalized Representation
139
72 Example
140

25 The Problem of Asymptotic Optimization in a Preordered Space and Its Extension
39
26 Extension of the Cone Optimization Problem
45
Finitely Additive Measures on a Semialgebra of Sets
55
32 Semialgebras and Algebras of Sets
57
33 Finitely Additive Measures
59
34 Measures and Integral Representation of Linear Functionals
65
35 Compactification in the Class of TwoValued Measures
71
36 Extension of the Cone Optimization Problem in the Class of TwoValued Measures
72
Finitely Additive Measures and Indefinite Integrals
77
42 Weakly Absolutely Continuous Finitely Additive Measures
79
43 Density Properties
83
44 Examples
89
Admissible Sets and Their Relaxations
97
52 Asymptotics of Perturbed Constraints and Its Limit Representation Case of Positive Measures
100
53 Asymptotics of Perturbed Constraints and Its Limit Representation Case of Alternating Measures
104
54 Strong Boundedness Conditions and Relaxations of Admissible Sets I
108
73 Generalized Extremal Problem and Asymptotic Values
143
74 Value Asymptotics for Strongly Bounded Problems
151
75 On an Extremal Problem with InequalityType Constraints
159
Asymptotic Efficiency
167
82 Generalized Problem of Cone Optimization I
168
83 Generalized Problem of Cone Optimization II
177
Issues of Computational Stability
185
92 Constraints on the Vector Integrand
186
93 Computational Stability
193
94 A Problem of the Extremal Choice of Probability Density
194
95 Extension in the Class of Vector Finitely Additive Measures
198
Conclusion
231
References
235
Index
243
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