Theory of linear optimization
著者
書誌事項
Theory of linear optimization
（Inverse and ill-posed problems series）
VSP, 2002
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注記
Bibliography: p[241]-248
内容説明・目次
内容説明
This monographis devoted tothe basic component of the theory of linear optimization problems: systems of linear inequalities. Such an approach is exact in both a historical and methodological sense.
In the first two chaptersdeal witheconomic interpretation of models, theorems and approaches. The other chapters are dedicated to less traditional problems of linear optimization, such as contradictory problems and duality, lexicographic problems and duality, piecewise linear problems and duality, and more. The bookalso covers some general methods for calculating processes for certain problems of linear optimization: the problem of stability and correctness.
目次
Contents
Introduction
Finite systems of linear inequalities
Basic definitions
The structure of polyhedrons
Bounded polyhedrons
A parametric representation of polyhedrons
The Farkas-Minowski theorem on dependent inequalities
Attainability theorem for inequalities-implications of second kind
A refined formulation of the Farkas-Minowski theorem
Conditions of compatibility of a finite system of linear inequalities
The cleaning theorem
Separability of nonintersecting polyhedrons
The Fourier elimination method
Linear programming
Setting of the problem of linear programming and some of its properties
Economic interpretation of the linear programming problem
Duality: informative approach
The duality theorem
The optimality conditions
Informative interpretations of optimality conditions
Matrix plays and duality
The theorem of marginal values
The method of exact penalty functions in linear programming
LP problems with several criterion functions
Inconsistent problems of linear programming
Classification of improper problems of linear programming (IP LP)
Informative interpretation of improper problems of linear programming
Methods of correction of improper problems of linear programming: general approach
Duality: the main theorem
Special realizations of duality
The duality problem for l-problems
Problems of successive linear programming and duality
The scheme of duality formation in linear successive programming
Solvability conditions for lexicographic optimisation problems
The duality theorem
Reduction of lexicographic optimisation problems to systems of linear inequalities
Lexicographic duality for improper LP problems - a special case
Duality for improper LP problems in lexicographic interpretation
Symmetric duality for the Pareto optimisation problem
Stability and well-posedness of linear programming problems
Necessary definitions and auxiliary results
Stability of the linear programming problem
Well-posedness of linear programming problems
The Tikhonov regularization of linear programming problems
Methods of projection in linear programming
Fejer mappings and their properties
Basic constructions of Fejer mappings for algebraic polyhedrons
Decomposition and parallelizing of Fejer processes
Randomization of Fejer processes
Fejer processes and inconsistent systems of linear inequalities
Fejer processes for regularized LP problems
Piecewise linear functions and problems of disjunctive programming
Introductory considerations
?-extensions of linear functional space
The problem on the saddle point of the disjunctive Lagrange function
Piecewise linear functions and systems of piecewise linear inequalities
The problem of piecewise linear programming
Duality for improper problems of piecewise linear programming
The method of exact penalty functions for the problem of piecewise linear programming
Questions of polyhedral separability
Appendix. Elements of convex analysis and convex programming
Convex sets and convex functions
Subdifferentiability of convex functions
The problem of convex programming
The theorem of marginal values
The penalty function method for problems of nonlinear programming
An estimate of deviation with respect to the argument in the asymptotic penalty method
Notations and abbreviations
Bibliography
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