 By Richard B. Holmes (auth.)

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1 is also valid , since But the reverse inequality value of (X, f + EXifi) over the set ~ K . i that X X I. _> 0 and so the cannot exceed the infinum of this function Hence the two values are equal and this means is a Lagrange multiplier vector. The upshot is that all questions vectors are reducible to questions about Lagrange multiplier about Sp(@). For instance, Lagrange multiplier vector fails to exist if and only if this latter condition is in turn equivalent vector y s Rn for which p'(@;y) the program is highly unstable obtain a satisfactory Corollary.

A), (5b both It follows consequently, t ~ ~}C ext (U(X*)). implies (w*-closures of course}. is a h o m e o m o r p h i s m Thus t ~ is 8t = i/2(v on a), that (U(X*)). spaces. III. that E ~ {~ 6t: 6t of the Now each U(X*)-extremal. so the Bipolar (the map theory as an element = E °° = c--o- ({@} KJ E) = c-o (E) is w * - c o m p a c t of the c l a s s i c a l in the x(t), x ~ X Therefore, E ° = U(X), role w~-topology. the d e s c r i p t i o n characterize at (If ~ v,a and in the to be p r e s e n t e d functional (U(X*)).

15. Polarity a) Let tion of A X be a real ics and was defined, provided Using the convention bers is + ~, defining A~X. that In 3d) the Minkowski A func- was a convex e-nbhd. that the infinum of the empty set of real num- we now expand the coverage of that definition by pA(@) = 0 and PA(X) = inf {t > 0: x ~ t A}, whenever x + e. PA(CX) = cPA(X ) cussed in (i) (2) for PA: X ÷ [0,~], c > 0. For any A C X, PA = 6 A O' e ~ A, 6A = p A° and is positively homogeneous: We also recall that polars were dis- §4.