Linear optimization problems with inexact data.

*(English)*Zbl 1106.90051
New York, NY: Springer (ISBN 0-387-32697-9/hbk). xv, 214 p. (2006).

The authors consider a standard linear programming problem with inexact data, where the matrix, the right hand side and the objective vector belong to certain subsets \(A\), \(b\) and \(c\), respectively, expressing inexactness of the data. The book consists of six chapters.

In Chapter I basic notions on matrices and determinants, as well as on norms and basic numerical algebra are presented. Special attention is paid to such topics as symmetric matrices, generalized inverses and nonnegative matrices.

In Chapter II solvability and feasibility of systems of interval linear equations and inequalities are investigated. Weak and strong solvability and feasibility of linear systems of equality and inequality are studied separately. In this way, combining weak and strong solvability or feasibility of the above systems, the authors arrive to eight decision problems, each of them solvable by finite means.

Chapter III deals with an interval linear programming problem where \(A\) is an interval matrix and \(b\) and \(c\) are interval vectors. The main topics of this chapter are computation and properties of the exact lower and upper bounds of the range of the optimal value of the problem with data varying independently of each other in the prescribed intervals.

By generalizing linear programming problems with interval data, the authors obtain problems with \(A\), \(b\) and \(c\) being compact convex sets. Such problems are studied in Chapter IV. In comparison with Chapter III, \(A\), \(b\) and \(c\) are not necessarily matrix or vector intervals. Such a family of linear programming problems is called a linear programming problem with set coefficients. The interest of the authors is focused on the case where \(A\), \(b\) and \(c\) are either compact convex set or, in particular, convex polytopes.

In Chapter V the authors propose a new general approach to fuzzy single and multicriteria linear programming problems. A unified concept of this approach is the concept of a fuzzy relation, particularly fuzzy extension of the usual inequality or equality relations.

In Chapter VI the authors investigate another class of optimization problems, the special structure of which makes it possible to find global optimal solutions. These problems form a special class of the so-called max-separable optimization problems. Functions occurring in these problems both as objective functions and in the constraints can be treated as “linear” with respect to a pair of semigroup operations. Properties of such optimization problems with interval data are presented as well.

In Chapter I basic notions on matrices and determinants, as well as on norms and basic numerical algebra are presented. Special attention is paid to such topics as symmetric matrices, generalized inverses and nonnegative matrices.

In Chapter II solvability and feasibility of systems of interval linear equations and inequalities are investigated. Weak and strong solvability and feasibility of linear systems of equality and inequality are studied separately. In this way, combining weak and strong solvability or feasibility of the above systems, the authors arrive to eight decision problems, each of them solvable by finite means.

Chapter III deals with an interval linear programming problem where \(A\) is an interval matrix and \(b\) and \(c\) are interval vectors. The main topics of this chapter are computation and properties of the exact lower and upper bounds of the range of the optimal value of the problem with data varying independently of each other in the prescribed intervals.

By generalizing linear programming problems with interval data, the authors obtain problems with \(A\), \(b\) and \(c\) being compact convex sets. Such problems are studied in Chapter IV. In comparison with Chapter III, \(A\), \(b\) and \(c\) are not necessarily matrix or vector intervals. Such a family of linear programming problems is called a linear programming problem with set coefficients. The interest of the authors is focused on the case where \(A\), \(b\) and \(c\) are either compact convex set or, in particular, convex polytopes.

In Chapter V the authors propose a new general approach to fuzzy single and multicriteria linear programming problems. A unified concept of this approach is the concept of a fuzzy relation, particularly fuzzy extension of the usual inequality or equality relations.

In Chapter VI the authors investigate another class of optimization problems, the special structure of which makes it possible to find global optimal solutions. These problems form a special class of the so-called max-separable optimization problems. Functions occurring in these problems both as objective functions and in the constraints can be treated as “linear” with respect to a pair of semigroup operations. Properties of such optimization problems with interval data are presented as well.

Reviewer: Maxim Todorov (Puebla)

##### MSC:

90C06 | Large-scale problems in mathematical programming |

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

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

90C29 | Multi-objective and goal programming |