Conjugate Duality and Optimization

Kuhn-Tucker condition (0, 0) e dK(x, y) into a more explicit and familiar form. Writing where -k(y) = /0(x) + ylfl(x] + • • • + ymfm(x) and P is the nonnegative orthant in R, we see from the example following Theorem 20 that the condition 0 e dyK(x, y) holds if and only if the vector On the other hand, for y e P we have by Theorem 20, assuming for example that f{ is a continuous function for i = 1, • • • , m (as follows from Corollary 8 A if X = jR"). Moreover, for ft ^ 0 and ff finite one has trivially the relation The abstract Kuhn-Tucker condition is therefore equivalent to conditions (10.2) and (10.5), assuming /) is continuous for i = 1, • • • , m. If/ is differentiate, (10.5) reduces of course to Example 6. (Stochastic programming.) The given problem is and the chosen representation involves where h is measurable on X x S (relative to the Borel structure on X) and convex in the X argument, and C is a nonempty closed subset of X. In order to apply the theory of integral functionals in § 9 at its fullest, we shall assume X is a separable, reflexive Banach space (whose dual is identifiable with V under the pairing <x, y», and where the measure space is complete and has, of course, a(S) = 1. (In the pairing Formula (10.3) thus allows us to write the condition 0 e BxK(x, y) as is normal to P at y. This means that 66 R. TYRRELL ROCKAFELLAR between U and 7, we take the norm topology on Y, but the Mackey topology on U; see the end of § 7.) We shall assume further that h(x, s) is finite and lowersemicontinuous in .v for all s, and h(u(s),s) is summable in 5 for every ueU. (If X = /?", the latter assumption can be weakened to the summability of /j(x, s) in 5 for each x e X.) These assumptions imply via Theorem 22 that F is a closed convex function on X x t/, and indeed F(x, u) is finite and continuous in u in the Mackey topology for each x. The Lagrangian function can be calculated as