Large-scale optimization - problems and methods (Q5930540)

The book is a monograph, which describes various approaches to the dimension-reduction problem. This problem is an important tool, which makes it possible to manage large-scale optimization problems. The material of the book is divided into four chapters. In Chapter 1, the aggregation in both linear and nonlinear convex optimization problems is studied. Special attention is devoted to aggregation in dynamic optimization problems and in integer programming. Chapter 2 deals with the procedure of iterative aggregation and its application to input-output model, block-separable mathematical programming and hierarchical problems of optimal control. The iterative aggregation constructs an iterative process, at every step of which a macroproblem is solved that is simpler than the original problem because of its lower dimension. The convergence of the solutions of simpler aggregated problems to the solution of the original problem is investigated. The issues of loss of accuracy in the process of disaggregation is analyzed. Chapter 3 is an introduction to block integer programming. Various appropriate decomposition approaches are described (e.g. Lagrangian relaxation, Benders method, cross decomposition, a modification of the Dantzig-Wolfe decomposition method, the method of sequential analysis of variants, probabilistic method and some others). Special attention is devoted to combined methods presented in the concluding paragraph of this chapter. In Chapter 4, block problems with a special feature are considered. A production-transportation problem is formalized as a linear programming problem with the block structure. Its special feature consists in the fact that the coupling variables do not enter the binding constraints. This special feature makes possible to use a two-level iterative decomposition with respect to primal-dual variables instead the laborious three-level algorithms, which are used in the general case. This approach is then extended to a special general class of block programming problems.