# New perspective on the theorems of alternative

## 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.

## Keywords

theorems of the alternative duality theory alternative system normal solution inconsistent system steepest descent linear programming## Preview

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## References

- [1]Chernikov, S.N. (1968)
*Linear Inequalities*, Moscow, Nauka.Google Scholar - [2]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.Google Scholar - [3]Gale, D. (1960)
*The Theory of Linear Economic Models*, New York, Toronto, London, McGraw-Hill Book Company.Google Scholar - [4]Mangasarian, O.L. (1994)
*Nonlinear Programming*, Philadelphia, SIAM.zbMATHGoogle Scholar - [5]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.MathSciNetGoogle Scholar - [6]Giannessi, F. (2001) “Theorems of the Alternative and Optimization, ”
*Encyclopedia of Optimization*, Dordrecht, Kluwer, vol. 5, pp. 437–444.Google Scholar - [7]Broyden, C.G. (2001) “On Theorems of the Alternative, ”
*Optimization Methods and Software*, vol. 16, pp.101–111.MathSciNetzbMATHCrossRefGoogle Scholar - [8]Evtushenko, Yu.G. (1998) “Computation of Exact Gradients in Distributed Dynamic Systems, ”
*Optimization Methods and Software*, vol. 9, pp. 45–75.MathSciNetzbMATHCrossRefGoogle Scholar - [9]Eremin, I.I. (2001)
*Duality in Linear Optimization*,Ekaterinburg, Ural Branch of RAS.Google Scholar - [10]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.MathSciNetzbMATHGoogle Scholar - [11]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.MathSciNetGoogle Scholar