Duality Theorems and Theorems of the Alternative
نویسنده
چکیده
It is shown, in a completely general setting, that a theorem of the alternative is logically equivalent to a duality theorem linking two constrained optimization problems. A standard technique for proving a duality theorem linking two constrained optimization problems is to apply an appropriate theorem of the alternative, sometimes called a transposition theorem (see, e.g., Mangasarian [2], Rockafellar [4], Stoer and Witzgall [6]). In the context of linear problems in finite dimensions it has been observed also by some (e.g., Balinski and Tucker [l]) that, conversely, from such a duality result a theorem of the alternative follows. In this note we extract the essence of the logical arguments involved in such derivations and thus exhibit the extreme generality of these techniques of proof. In particular, we show the simple, yet basic, logical principle that a duality theorem is actually equivalent to a theorem of the alternative. No linear space structure is needed. Let X and Y be arbitrary nonempty sets, and let / and g be arbitrary extended-real-valued functions defined on X and Y', respectively. For each a e (—°o, +oo] consider the statements (Ia) 3x £ X such that f(x) < a, (IIa) 3y e Y such that g(y) > a. The following statement is an abstract theorem of the alternative involving the pairs (/, X) and (g, Y): (fl) Va £ (-oo, +<*>], exactly one of (I ), (lla) holds. We do not assert anything here as to the validity of (tf), but merely introduce it as a logical statement. Consider now the two abstract optimization problems inf /(%) and sup g(y). xeX yeY Received by the editors September 13, 1974. AMS (MOS) subject classifications (1970). Primary 90C99; Secondary 15A39.
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