Theorems of the Alternative and Optimization

In a general format a theorem of the alternative (TA) claims that, between two given propositions, say S and S ^∗, one and only one is true. In mathematics S and S ^∗ are, in general, systems of equalities or inequalities. A TA for linear algebraic systems was established as early as 1873 by P. Gordan [11]; then there was the celebrated Farkas lemma in 1902 [7] (cf. also Farkas lemma; Farkas lemma: Generalizations); indeed, such a lemma does not appear as a TA, but an obvious reformulation shows it as a TA. Some further important TA were established in 1915 by E. Stiemke [22], in 1936 by T.S. Motzkin [19], in 1951 by M.L. Slater [21], in 1956 by A.W. Tucker and in 1956 by R.J. Duffin (see [17]). Subsequently, due mainly to the development of the optimization theory, there has been a blooming of TAs; they have been extended to not necessarily algebraic systems, to systems in an infinite-dimensional space, to systems in a complex space, and even to systems for point-to-set maps. TA...