Theory of Suboptimal Decisions: Decomposition and Aggregation / Edition 1

by A.A. Pervozvanskii, V.G. Gaitsgori

Overview

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were. thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various Isciences has changed drastically in recent years: measure theory is used (non­ trivially) in regional and theoretical economics; algebraic geom. eJry interacts with I physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and rpathematical programminglprofit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Product Details

ISBN-13:	9789027724014
Publisher:	Springer Netherlands
Publication date:	05/31/1988
Series:	Mathematics and its Applications , #12
Edition description:	1988
Pages:	384
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

1: The Perturbation Method in Mathematical Programming.- 1.1. Formulation and peculiarities of problems.- 1.2. Perturbations in linear programs.- 1.3 Nonlinear programs: perturbations in objective functions.- 1.4. Necessary and sufficient conditions for an extremum. Quasiconvex and quasilinear programs.- 1.5. Perturbations in nonconvex programs.- 2: Approximate Decomposition and Aggregation for Finite Dimensional Deterministic Problems.- 2.1. Perturbed decomposable structures and two-level planning.- 2.2. Aggregation of activities.- 2.3 Weakly controllable input-output characteristics.- 2.4. Input-output analysis.- 2.5. Aggregation in optimization models based on input-output analysis.- 2.6. Aggregation in the interregional transportation problem with regard to price scales.- 2.7. Optimization of discrete dynamic systems.- 2.8. Control of weakly dynamic systems under state variable constraints.- 3: Singular Programs.- 3.1. Singularity and regularization in quasiconvex problems.- 3.2. The auxiliary problem in the singular case.- 3.3. An approximate aggregation of Markov chains with incomes.- 3.4. An approximation algorithm for Markov programming.- 3.5. An iterative algorithm for suboptimization.- 3.6. An artificial introduction of singular perturbations in compact inverse methods.- 4: The Perturbation Method in Shastic Programming.- 4.1. One- and two-stage problems.- 4.2. Optimal control problems with small random perturbations.- 4.3. Discrete dynamic systems with weak or aggregatable controls. An asymptotic shastic maximum principle.- 4.4. Sliding planning and suboptimal decomposition of operative control in a production system.- 4.5. Sliding planning on an infinite horizon.- 4.6. Control of weakly dynamic systems under random disturbances.- 5: Suboptimal Linear Regulator Design.- 5.1. The LQ problem. Suboptimal decomposition.- 5.2. Loss of controllability, singularity, and suboptimal aggregation.- 5.3. Examples of suboptimal regulator synthesis.- 5.4. Control of oscillatory systems.- 5.5. LQG problems.- 6: Nonlinear Optimal Control Problems.- 6.1. The maximum principle and smooth solutions.- 6.2. The general terminal problem.- 6.3. Difference approximations.- 6.4. Weak control (nonuniqueness of the reduced solution).- 6.5. Aggregation in a singular perturbed problem.- Related Literature.

Editorial Reviews

'Therefore, this book can be recommended to everybody interested in mathematical systems analysis.' Operating Research 36:6 1992

From the Publisher