Abstract
New general theorems of the alternative are presented. The constructive proofs based on the duality theory are given. From these results many well-known theorems of the alternative are obtained by simple substitutions. Computational applications of theorems of the alternative to solving linear systems, LP and NLP problems are given. A linear systems of possibly unsolvable equalities and inequalities are considered. With original linear system an alternative system is associated such that one and only one of these systems is consistent. If the original system is solvable then numerical method for solving this system consists of minimization of the residual of the alternative inconsistent system. From the results of this minimization the normal solution of the original system is determined.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chernikov, S.N. (1968) Linear Inequalities, Moscow, Nauka.
Razumikhin, B.S. (1975) Physical Models and the Methods of Equilibrium Theory in Programming and Economics, Moscow, Nauka, 1975. Physical Models and Equilibrium Methods in Programming and Economics, Dordrecht, Boston, D.Reidel Publishing, 1984.
Gale, D. (1960) The Theory of Linear Economic Models, New York, Toronto, London, McGraw-Hill Book Company.
Mangasarian, O.L. (1994) Nonlinear Programming, Philadelphia, SIAM.
Dax, A. (1993) “The relationship between theorems of the alternative, least norm problems, steepest descent directions, and degeneracy: A review,” Annals of Operation Research, vol. 46, pp. 11–60.
Giannessi, F. (2001) “Theorems of the Alternative and Optimization, ” Encyclopedia of Optimization, Dordrecht, Kluwer, vol. 5, pp. 437–444.
Broyden, C.G. (2001) “On Theorems of the Alternative, ” Optimization Methods and Software, vol. 16, pp.101–111.
Evtushenko, Yu.G. (1998) “Computation of Exact Gradients in Distributed Dynamic Systems, ” Optimization Methods and Software, vol. 9, pp. 45–75.
Eremin, I.I. (2001) Duality in Linear Optimization,Ekaterinburg, Ural Branch of RAS.
Golikov, A.I., and Evtushenko, Yu.G. (2000) “Search for Normal Solutions in Linear Programming Problems, ” Computational Mathematics and Mathematical Physics vol.40, No.12, pp. 1694–1714.
Golikov, A.I., and Evtushenko, Yu.G. (2002) “Two Parametric Families of LP Problems and Their Applications, ” Proceedings of the Steklov Institute of Mathematics, Suppl. 1, pp. S52–S66.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Kluwer Academic Publishers B.V.
About this chapter
Cite this chapter
Evtushenko, Y.G., Golikov, A.I. (2003). New perspective on the theorems of alternative. In: Di Pillo, G., Murli, A. (eds) High Performance Algorithms and Software for Nonlinear Optimization. Applied Optimization, vol 82. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0241-4_10
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0241-4_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7956-0
Online ISBN: 978-1-4613-0241-4
eBook Packages: Springer Book Archive