Johannes Jahn
Mathematical
Vector Optimization
in Partially Ordered
Linear Spaces
Verlag Peter Lang
/Frankfurt am Main • Bern • New York
Table of Contents
Part I.
1
2
3
Convex Analysis
Linear Spaces
1
1
1.1
Linear Spaces and Convex Sets
1.2
Partially Ordered Linear Spaces
1.3
Topological Linear Spaces
19
1.4
Some Examples
30
9
Mappings on Linear Spaces
2 .1 Convex Mappings
35
35
2.2
41
Differentiable Mappings
Some Fundamental Theorems
57
3.1
Zorn's Lemma and the Hahn-Banach Theorem
57
3.2
3.3
Separation Theorems
A Theorem of James
67
78
3.4
3.5
Two Theorems of Krein-Rutman
Tangent Cones and a Theorem of Ljusternik
84
87
Part II.
Theory of Vector Optimization
4
Optimality Notions
5
Scalarization
5.1
5.2
6
1
Necessary Conditions for Optimal Elements
of a Set
Sufficient Conditions for Optimal Elements
of a Set
Existence Theorems
99
99
109
109
125
139
fable of Contents
7
Generalized Multiplier Rule
7.1
Necessary Conditions for Minimal and Weakly
Minimal Elements
7.2
Sufficient Conditions for Minimal and Weakly
168
7.2.1
168
Generalized Quasiconvex Mappings
Sufficiency of the Generalized
Multiplier Rule
Duality
8.1 A General Duality Principle
8.2
Duality Theorems for Abstract Optimization
8.3
Problems
Specialization to Abstract Linear Optimization
Problems
Part III.
9
152
Minimal Elements
7.2.2
8
152
Mathematical Applications
176
185
185
188
198
207
Vector Approximation
207
9.1
Introduction
208
9.2
Simultaneous Approximation
210
9.3
Generalized Kolmogorov Condition
213
9.4
Nonlinear Chebyshev Vector Approximation
216
9.5
Linear Chebyshev Vector Approximation
224
9.5.1
Duality Results
226
9.5.2
An Alternation Theorem
232
-fj
Table of Contents
10
Cooperative n Player Differential Games
243
10.1
Basic Remarks on the Cooperation Concept
243
10.2
A Maximum Principle
245
10.2.1
Necessary Conditions for Optimal and
Weakly Optimal Controls
10.2.2
Sufficient Conditions for Optimal
and Weakly Optimal Controls
10.3
248
262
A Special Cooperative n Player
Differential Game
274
References
285
List of Symbols
305
Subject Index
307