Theory of Linear Optimization
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Theory of Linear Optimization

Theory of Linear Optimization

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About the Book

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.

Table of Contents:
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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Product Details
  • ISBN-13: 9789067643535
  • Publisher: Brill
  • Publisher Imprint: VSP International Science Publishers
  • Language: English
  • Weight: 550 gr
  • ISBN-10: 906764353X
  • Publisher Date: 01 Jan 2002
  • Binding: Hardback
  • No of Pages: 258


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