Algebraic Hilbert Field Characterizations of Asymptotic Duality States and Optimal Paths to Infinity

摘要

Every finite subset of the following infinite set of inequalities has a solution, although there is no (real) solution to all these inequalities: x =or> n, for n = 0,1,2,3,... By the introduction of an 'infinitely large' quantity M these inequalities obtain a solution x = M in the field R(M) of the reals with M adjoined. It is shown that this solution is a special instance of the following general theorem: every set of linear inequalities in R sup n whose every finite subset has a solution, itself has a solution R((M) sup n). The authors give other results which relate R(M) - solutions to asymptotic solutions in the reals, and use their main result to give an algebraic characterization of asymptotic duality states in a duality theory developed earlier by Ben-Israel, Charnes, and Kortanek. (Author)