High Performance Algorithms and Software for Nonlinear Optimization

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