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Tables of Contents for Modified Lagrangians and Monotone Maps in Optimization
Chapter/Section Title
Page #
Page Count
Preface
vii
1 Introduction to Convex Analysis
1
104
1 Convex Sets
1
6
2 Concave and Convex Functions
7
16
3 Basic Facts of Duality Theory for Convex Programming
23
17
4 Minimax Theorems and Dual Problems
40
10
Subdifferentials of Concave Functions
50
25
6 Conjugate Functions and Their Properties
75
30
2 Modified Lagrangian Functions for Convex Programming Problems
105
58
1 Weak Modified Lagrangian Functions
105
11
2 WMLFs and Duality Theorems
116
8
3 Modified Lagrangian Functions
124
2
4 Stability of Saddle Sets and MLFs
126
7
5 WMLFs and MLFs Generating Smooth Dual Problems
133
12
6 Equality Constraints, Examples, Economic Interpretation
145
18
3 Dual Methods
163
58
1 Gradient Methods of Minimization with Approximate Computeration of Gradients
164
17
2 Dual Gradient Method
181
6
3 Dual Transformed Gradient Method
187
10
4 Dual Methods as Applied to Problems with Inconsistent Constraints
197
10
5 Dual Methods in Linear Case
207
14
4 Monotone Maps
221
34
1 Structure of Maximal Monotone Maps
221
13
2 Equations Generated by Monotone Maps
234
8
3 Operations Preserving Maximal Monotonicity; Variational Inequalities
242
13
5 Gradient-Type Methods and Modification of a Monotone Map
255
74
1 Maps Satisfying Inverse Strong Monotonicity condition and Finite-Step Iterative Algorithms
255
12
2 Monotone Maps and Decreasing-Step Interactive Algorithms
267
13
3 Method of Modified Map
280
9
4 Modification of a Monotone Map and MLFs
289
13
5 Decomposition of Monotone Maps and a General Approach to Decomposition Methods of Convex Programming
302
27
6 Saddle Gradient Methods
329
38
1 Concave-Convex Functions Whose Saddle Sets Are Stable and Decreasing-Step Saddle Gradient Methods
329
10
2 Differentiable Concave-Convex Functions Whose Saddle Sets Are Stable with Respect to One Vector Variable
339
17
3 Constant-Step Saddle Gradient Methods
356
11
7 Modified Lagrangian Functions For Smooth Mathematical Programming Problems And Related Dual Methods
367
52
1 Auxiliary Results
367
13
2 MLFs for Smooth Equality-Constrained Problems and Related Dual Problems
380
19
3 MLFs for Smooth Inequality-Constrained Problems and for Smooth Convex Programming Problems
399
11
4 Dual Methods: Rate of Convergence
410
9
Bibliographic Comments
419
8
References
427
6
Index
433