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**Practical augmented Lagrangian methods for constrained optimization.**
*(English)*
Zbl 1339.90312

Fundamentals of Algorithms 10. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611973-35-8/pbk; 978-1-61197-336-5/ebook). xiii, 220 p. (2014).

The book is devoted to augmented Lagrangian methods, which are used for solving practical constrained optimization problems. The theoretical background as well as the development of practically usable algorithms and the description of practical problems are presented. The idea of using the augmented Lagrangian function is motivated by problems, which occur when we use a non-augmented Lagrangian penalty. It can happen that this penalty, a modest infeasibilty may be considered as being worse than almost feasible solution with a poor objective function value. In this situation, a very cautious increasing of the penalty parameter is recommended, which leads to a large number of subproblems of the Lagrangian penalty algorithm. The usage of Augmented Lagrangian tries to avoid this problem.

The book describes the optimality conditions and presents a model augmented Lagrangian algorithm. An appropriate approach to solve its subproblems from the point of view of global optimization is proposed. Both constrained and uncostrained subproblems are considered. A further part of the book provides necessary information about the Fortran subroutine Algencan for solving constrained minimization problems with the aid of an augmented Lagrangian function. The authors study especially an adequate choice of subroutines, making a good choice of algorithmic variants and parametric values. Some practical examples are summarized in the concluding chapter.

The book describes the optimality conditions and presents a model augmented Lagrangian algorithm. An appropriate approach to solve its subproblems from the point of view of global optimization is proposed. Both constrained and uncostrained subproblems are considered. A further part of the book provides necessary information about the Fortran subroutine Algencan for solving constrained minimization problems with the aid of an augmented Lagrangian function. The authors study especially an adequate choice of subroutines, making a good choice of algorithmic variants and parametric values. Some practical examples are summarized in the concluding chapter.

Reviewer: Karel Zimmermann (Praha)