Table Of ContentHierarchical Optimization and Mathematical Physics
Applied Optimization
Volume 37
Series Editors:
Panos M. Pardalos
University of Florida, U.S.A.
Donald Hearn
University of Florida, U.S.A.
The titles published in this series are listed at the end of this volume.
Hierarchical Optimization
and Mathematical Physics
by
Vladimir Tsurkov
Computing Center,
Russian Academy of Sciences,
Moscow, Russia
....
' '
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4613-7112-0 ISBN 978-1-4615-4667-2 (eBook)
DOI 10.1007/978-1-4615-4667-2
Printed on acid-free paper
All Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
Softcoverreprint ofthe bardeover 1st edition 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Chapter 1. The Main Model and Constructions of the
Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . .... 1
§1. Necessary Knowledge from the Theory of Extremal
Problems .................................................. 2
§2. Branch Modeland Description of the Algorithm .......... 14
§3. Optimality Criterion and the Aggregated Problem ........ 20
§4. Local Monotonicity with Respect to the Functional and
Numerical Computation .................................. 27
§5. Modification of the Main Model .......................... 33
§6. Random Parameters in the Branch Model ................ 47
Comments and References to Chapter 1 . . . . . . . . . . . . . . . . . . . . ... 59
Chapter 2. Generalization of the Decomposition
Approach to Mathematical Programming and
Classical Calculus of Variations .................. 62
§1 . Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 62
§2. Quadratic Programming .................................. 69
§3. Mathematical Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 76
§4. Classical Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . ... 84
Comments and References to Chapter 2 . . . . . . . . . . . . . . . . . . . . ... 97
Chapter 3. Hierarchical Systems of Mathematical
Physics .......................................... 99
§1. Construction of the Method for Block Separable
Problems of Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . ... 99
§2. Analytical Examples .................................... 113
Vl
§3. Block Problems of Optimal Control with Partial
Differential Equations ................................... 125
§4. Linear-Quadratic Optimal Control Problems of Block
Type .................................................... 146
Comments and References to Chapter 3................... . . 155
Chapter 4. Effectiveness of Decomposition ................ 158
§1. Nonlinear Two-level Statements ......................... 158
§2. Models of Hierarchical Systems with Distributed
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 4
§3. Block Separable Problems with Large Number of
Binding Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 189
§4. Nonseparable F\mctionals ................................ 200
§5. Results of Numerical Computation ...................... 212
Comments and References to Chapter 4 . . . . . . . . . . . . . . . . . . . . . 224
Chapter 5. Appendix. The Main Approaches in
Hierarchical Optimization ....................... 226
§1. Dantzig-Wolfe Principle ... 226
§2. Kornai-Liptak Principle ... 235
§3. Parametrie Decomposition ............................... 256
§4. Iterative Aggregation .................................... 262
§5. The Use of Lagrange Functional in Block Dynamical
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 276
Comments and References to Chapter 5 . . . . . . . . . . . . . . . . . . . . . 297
Index .............................................................. 304
Vll
Preface
This book should be considered as an introduction to a special dass of
hierarchical systems of optimal control, where subsystems are described by
partial differential equations of various types. Optimization is carried out
by means of a two-level scheme, where the center optimizes coordination for
the upper level and subsystems find the optimal solutions for independent
local problems. The main algorithm is a method of iterative aggregation.
The coordinator solves the problern with macrovariables, whose number is
less than the number of initial variables. This problern is often very simple.
On the lower level, we have the usual optimal control problems of math
ematical physics, which are far simpler than the initial statements. Thus,
the decomposition (or reduction to problems ofless dimensions) is obtained.
The algorithm constructs a sequence of so-called disaggregated solutions that
are feasible for the main problern and converge to its optimal solutionunder
certain assumptions (e .g., under strict convexity of the input functions).
Thus, we bridge the gap between two disciplines: optimization theory
of large-scale systems and mathematical physics. The first motivation was
a special model of branch planning, where the final product obeys a preset
assortment relation. The ratio coefficient is maximized. Constraints are given
in the form of linear inequalities with block diagonal structure of the part
of a matrix that corresponds to subsystems. The central coordinator assem
bles the final production from the components produced by the subsystems.
Therefore, the binding constraints of the initial matrix are specific: their
submatrices are diagonal. This structure suggests a special decomposition
algorithm, where variables from various blocks are aggregated. Here, all the
difficulties related tothelarge number of dimensions, i.e., the large amount of
subsystems and components they produce, are reduced to a simple aggregated
problern of the upper level, which consists in finding the minimal element of
a large-dimension matrix.
Substantiation of the decomposition scheme is based on the duality princi
ples of linear programming. Local monotonicity with respect to the functional
of the iterative process is important. This scheme of iterative aggregation
is generalized to a wide dass of hierarchical problems, for which duality
principles hold. It is a question of block separable problems of mathematical
programming, calculus of variations, and of optimal control. We go on to
viii
consider systems with subsystems described by partial differential equations
and consider models of distribution of energy resources, propagation of heat,
oscillation damping, etc. The approach is generalized for nonseparable prob
lems: for example, the so-called systems with cross-connection, when, e.g.,
there is a heat exchange between subsystems.
The format of the book is as follows:
Necessary knowledge from the theory of extremal problems is given in
Chapter 1. This is mostly related to duality theory and parametric pro
gramming, which are used to justify the decomposition method. Here, the
main model of branch planning is described, and the iterative algorithm is
constructed. Moreover, we study the properties of the problem's solution
with aggregated variables of the upper level.
In Chapter 2, the scheme of iterative aggregation is generalized toblock
separable problems of linear, quadratic, convex mathematical programming
and classical calculus of variations. Particular attention is given to the
criterion of optimality of disaggregated solutions (c ondition of termination of
the iterative process) and local monotonicity with respect to the functional.
The main mathematical techniques are the duality theorems and Kuhn
Thcker theory. The necessary conditions for optimality in the form of Euler
equations for the classical calculus of variations are used in the justification.
In Chapter 3, we state hierarchical problems of optimal control, where
subsystems are described by ordinary and partial differential equations. il
lustrative examples are given, where all intermediary problems in the iterative
process are solved analytically. Particular emphasis is placed on block linear
quadratic problems of optimal control. Here, we propose a reduction method
for the systems of linear algebraic equations that are finally obtained after
application of the Pontryagin maximum principle.
In Chapter 4, we demonstrate the efficiency of the decomposition using
iterative aggregation. At first, this is done for nonlinear block statements, for
which the direct application of the maximum principle leads to intractable
problems. Here, model hierarchical control problems are stated, where the
subsystems are described by equations of mathematical physics. Their phys
ical meaning relates to the optimal distribution of resources over subsystems.
Use of Fourier series leadseither to efficient decomposition or to a reduction in
the systems of algebraic equations. The results of the numerical computations
testify to the fast convergency and efficiency of the decomposition algorithm.
To relate the algorithm of iterative aggregation to other approaches as
ix
well as to areas of application, we give in the Appendix a brief survey of
methods in hierarchical optimization. Theseare schemes based on Dantzig
Wolfe and Kornai-Liptak decomposition, the method of Lagrangian function,
parametric decomposition, other approaches to iterative aggregation, etc.
The reader is assumed to be acquainted with mathematical programming,
optimal control, and with mathematical physics. References to additional
literature are given in comments to the chapters.
In the book, two numbers are used to identify each formulas and theorems.
The first number denotes the number of the section of this chapter, the second
is the number of the formula or theorem. This should be specially noted when
a reference to a theorem or a formula from another chapter is made. In this
book, we use common mathematical symbols and notations. Sometimes,
their meaning is additionally cleared from the text. In particular, by [1 : N],
we denote the set of sequential integers from 1 to N inclusive.
