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**Parallel optimization. Theory, algorithms, and applications.**
*(English)*
Zbl 0945.90064

Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press. xxviii, 539 p. £61.50 (1997).

The book focuses on parallel optimization methods for large-scale constrained optimization problems. The material presented in this book leads to implementable parallel algorithms. The book is divided into three parts.

Part I deals with theoretical foundation, on which the algorithms presented in the next part are developed. The theory of generalized distances, generalized projections are studied and the use of these concepts in solving linear programming problems via proximal minimization is investigated. The authors further deal with the theory of penalty and barrier methods as well as the properties of augmented Lagrangians.

Part II is devoted to developing of algorithms. The algorithms can be divided into three main groups: iterative projection algorithms, model decomposition algorithms, and interior point algorithms. The theory of generalized projections is used for developing iterative algorithms for the solution of convex feasibility problems. The theory of generalized distances and generalized projections are further used to develop methods for solving linearly constrained optimization problems. The theory of penalty methods and augmented Lagrangians are used to derive model decomposition algorithms. Finally interior point algorithms for solving linear and quadratic programming problems are presented.

Part III discusses applications from real world domains, where parallel algorithms may be useful. In this context the following real world problems are considered: matrix estimation, image reconstruction from projections, radiation therapy treatment planning, transportation and multicommodity flow problems as well as some problems connected with planning under uncertainty.

The final two chapters of this part are devoted to parallel implementation and testing of the algorithms on large-scale problems.

The book contains also an extensive list of relevant references, which may serve as a guide for further studies in this field of science.

Part I deals with theoretical foundation, on which the algorithms presented in the next part are developed. The theory of generalized distances, generalized projections are studied and the use of these concepts in solving linear programming problems via proximal minimization is investigated. The authors further deal with the theory of penalty and barrier methods as well as the properties of augmented Lagrangians.

Part II is devoted to developing of algorithms. The algorithms can be divided into three main groups: iterative projection algorithms, model decomposition algorithms, and interior point algorithms. The theory of generalized projections is used for developing iterative algorithms for the solution of convex feasibility problems. The theory of generalized distances and generalized projections are further used to develop methods for solving linearly constrained optimization problems. The theory of penalty methods and augmented Lagrangians are used to derive model decomposition algorithms. Finally interior point algorithms for solving linear and quadratic programming problems are presented.

Part III discusses applications from real world domains, where parallel algorithms may be useful. In this context the following real world problems are considered: matrix estimation, image reconstruction from projections, radiation therapy treatment planning, transportation and multicommodity flow problems as well as some problems connected with planning under uncertainty.

The final two chapters of this part are devoted to parallel implementation and testing of the algorithms on large-scale problems.

The book contains also an extensive list of relevant references, which may serve as a guide for further studies in this field of science.

### MSC:

90C30 | Nonlinear programming |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

65K05 | Numerical mathematical programming methods |

90C51 | Interior-point methods |