Theory of suboptimal decisions. Decomposition and aggregation. Transl. from the Russian.

*(English)*Zbl 0691.90075
Mathematics and Its Applications. Soviet Series 12. Dordrecht etc.: Kluwer Academic Publishers Group (ISBN 90-277-2401-6). xvii, 384 p. (1988).

[For a review of the Russian original (1979) see Zbl 0508.90057.]

This book is basically concerned with the use of perturbation theory in mathematical programming and optimal control theory. The theory is applied to many specific practical problems. Perturbation is used to solve complicated problems which are “close” to reduced problems which are simpler to solve; in particular, consideration is given to the simplification of problems by either (a) decomposition, which is the partitioning of a problem into subproblems which are easier to handle, and which give suboptimal solutions which are not too different from the solutions of the original problem or (b) aggregation, which is the replacement of a set of state variables by a single state variable, it being assumed that the state variables in the original set are similar to each other and so can be represented suboptimally by a single state variable which leads to solutions which are not too different from the solutions of the nonaggregated problem.

Chapter one is devoted to perturbation in mathematical programming problems, linear, convex, and nonconvex, with theorems for convergence as the perturbation approaches zero.

Chapter two deals with approximate decomposition and aggregation for various practical finite dimensional deterministic problems, that is problems of a mathematical programming nature.

In Chapter three consideration is given to problems where a gap exists between the perturbed and reduced solution.

Chapter four deals with perturbation in stochastic programming, and the remaining chapters deal with specific applications of decomposition and aggregation in optimal control problems.

This book is basically concerned with the use of perturbation theory in mathematical programming and optimal control theory. The theory is applied to many specific practical problems. Perturbation is used to solve complicated problems which are “close” to reduced problems which are simpler to solve; in particular, consideration is given to the simplification of problems by either (a) decomposition, which is the partitioning of a problem into subproblems which are easier to handle, and which give suboptimal solutions which are not too different from the solutions of the original problem or (b) aggregation, which is the replacement of a set of state variables by a single state variable, it being assumed that the state variables in the original set are similar to each other and so can be represented suboptimally by a single state variable which leads to solutions which are not too different from the solutions of the nonaggregated problem.

Chapter one is devoted to perturbation in mathematical programming problems, linear, convex, and nonconvex, with theorems for convergence as the perturbation approaches zero.

Chapter two deals with approximate decomposition and aggregation for various practical finite dimensional deterministic problems, that is problems of a mathematical programming nature.

In Chapter three consideration is given to problems where a gap exists between the perturbed and reduced solution.

Chapter four deals with perturbation in stochastic programming, and the remaining chapters deal with specific applications of decomposition and aggregation in optimal control problems.

Reviewer: M.A.Hanson

##### MSC:

90C30 | Nonlinear programming |

90C15 | Stochastic programming |

49K40 | Sensitivity, stability, well-posedness |

90C05 | Linear programming |

90C25 | Convex programming |

90C90 | Applications of mathematical programming |

93B05 | Controllability |

90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

49M27 | Decomposition methods |