Hierarchical optimization and mathematical physics
The main subject of the book is optimal control of hierarchical systems including systems described by partial differential equations (PDEs). The proposed method is based on iterative aggregation when the coordinator optimizes few macro-variables, and relatively small subproblems of optimal control are solved at the lower level. The methods are justified and the corresponding algorithms are constructed using the theory of convex functions, duality theorems, necessary conditions of optimal control, and some results on ordinary differential equations (ODEs) and PDEs. The first chapter ``The main model and construction of the decomposition method'' presents an introduction. The Kuhn-Tucker condition and duality theory are introduced. The production planning problem is formulated and its solution by means of aggregation is considered. The problem is generalized for the case of stochastic parameters, and a special method to reduce the stochastic problem to a deterministic one is analyzed. The contents of Chapter 2 is well described by its title ``Generalization of the decomposition approach to mathematical programming and classical calculus of variations''. Let us only specify, that under ``mathematical programming'' linear and quadratic problems with block diagonal structure as well as finite-dimensional block separable problems are covered. Hierarchical optimization of systems described by ODEs and PDEs is considered in Chapters 3 and 4 which form approximately 1/3 of the volume of the book. The problem of optimal control with block separable structure and problems described by PDEs are considered. The method is extended to: two-level nonlinear systems, linear hierarchical systems with distributed parameters and quadratic objective, and block separable systems with large number of binding constraints. The possibilities of generalization of the method to the case of non-separable functional are discussed. Some numerical examples are included. The last Chapter 5 (Appendix) presents known decomposition methods: Danzig-Wolfe, Kornai-Liptak, parametric decomposition, iterative aggregation, Lagrangian decomposition in optimal control of block systems. Each chapter is finished by comments and references to publications of mainly 1960-1980.
scientific article; zbMATH DE number 1414335