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Definition. The perturbation program 12c) is the mapping function of the ordinary convex p: R n + [-~,+=] defined by p(y) z P(YI'''''Yn ) = inf {f(x): fi (x) ! ,n}. Thus p is the value of a new ordinary convex program differing from the original only in that the original been perturbed. Intuitively we think of for a given level y for some p(y) of resource expenditure. although in practice unlikely, In any event, constraint that p levels have as the optimal payoff It is possible, may assume the value -~.

B) F o r m u l a fact such z g(y) L 0, the ray P A o ( Y ) = inf Similarly, recall _ < t x , y for and hence A° (I) t o g e t h e r is always w * - c l o s e d the h i g h l y Corollary. with useful (Bipolar Theorem) 14d) provides and convex. JA). Proof. have Since at least that closed half-space taking into 26. is closed Let {A } @. Show On the other it must proves and convex, the also reverse be a family hand, we if any contain A °°" inclusion.

The highlights gramming. In this section we have only touched upon a few of of the theory of Lagrange multipliers A most conspicuous omission is the interplay between this theory and that of dual programs. economic interpretation in convex pro- Also missing is the very pretty in terms of "equilibrium prices". For these ideas, and for a more detailed study of ordinary convex programs, we must refer the reader to the literature, in particular [20, 44, 54, 55, 7o3. ~14. Conjugate Functions In this section we introduce the second technical device for analyzing optimization problems - the "conjugate" of a (generally convex) in its modern form, has been developed function.

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