Aggregation in Large-Scale Optimization

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Springer Science & Business Media, Sep 30, 2003 - Mathematics - 291 pages
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When analyzing systems with a large number of parameters, the dimen sion of the original system may present insurmountable difficulties for the analysis. It may then be convenient to reformulate the original system in terms of substantially fewer aggregated variables, or macrovariables. In other words, an original system with an n-dimensional vector of states is reformulated as a system with a vector of dimension much less than n. The aggregated variables are either readily defined and processed, or the aggregated system may be considered as an approximate model for the orig inal system. In the latter case, the operation of the original system can be exhaustively analyzed within the framework of the aggregated model, and one faces the problems of defining the rules for introducing macrovariables, specifying loss of information and accuracy, recovering original variables from aggregates, etc. We consider also in detail the so-called iterative aggregation approach. It constructs an iterative process, atĚ every step of which a macroproblem is solved that is simpler than the original problem because of its lower dimension. Aggregation weights are then updated, and the procedure passes to the next step. Macrovariables are commonly used in coordinating problems of hierarchical optimization.
 

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Contents

Aggregated Problem and Bounds for Aggregation
1
11 Definitions and Preliminary Results
2
12 A Posteriori Bounds
6
13 A Priori Bounds
8
14 Some Generalizations
10
142 A Priori Bounds
17
15 Illustrative Example and Numerical Indications
19
2 The Generalized Transportation Problem
25
8 Distributed Parameter Systems
135
9 Numerical Indications
147
References to Chapter 2
157
Consistent Aggregation in Parametric Optimization
163
2 Parametric Linear Programming
166
21 The Base Problem
167
22 Aggregation of Variables
170
23 Constraint Reduction
173

21 The Aggregated Problem
26
22 The Error Bounds
30
23 Numerical Indications
35
3 Variable Aggregation in Nonlinear Programming
38
4 Sharpening Localization by Valid Inequalities in Integer Programming
44
5 The Generalized Assignment Problem
51
References to Chapter 1
58
Iterative AggregationDecomposition in Optimization Problems
61
1 Linear Aggregation in FiniteDimensional Problems
62
2 Problems with Constraints of Special Structure
72
3 Decomposition of the Macroproblem in the BlockSeparable Case
92
4 Adaptive Clustering in Variable Aggregation
104
5 Aggregation of Constraints
112
6 Aggregation of Controls in Dynamic Problems with Mixed Constraints
121
7 Aggregation of State Variables in LinearQuadratic Problems
129
24 The Aggregated Problem in the Standard Form and the Proof of Its Consistency
177
25 The Degenerate Case
187
26 Various Forms of the Base Problem
189
27 An Example of Varying One Element in the Constraint Matrix
198
3 Parametric Convex Programming
202
32 Parameters in Convex Constraints
217
4 Special Classes of Problems
233
42 LinearConvex Problems in Banach Spaces
245
43 Optimal Control of Linear Systems under Terminal Constraints
256
44 Weaker Consistency Conditions Integer Linear Programming
264
441 A QuasiConsistency Theorem
272
442 Using the Cuts of the Primal Algorithm
278
Comments and References to Chapter 3
284
Index
289
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