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By Richard B. Holmes (auth.)

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Ln) that vectors for the -~p(@). be a Lagrange multiplier vector. K i ~ {x s X: fi(x) ~ 0}, p(@) = value of (X, f + E~K. ) 1 = value of (X, f + ~%ifi ) <_ f(x) + EkiYi, Therefore, p(@) <_ p(y) Conversely, assume that definition of p, Then k ~ -~p(@). < 0). p(@) <_ p(~) fi (x) <--Yi vi" which proves Yi >- 0 =)p'(@;y) -h i = (-k)-e i < p'(@;ei) Yi = fi (~)" + ZkiYi, if Then <_ 0, h i >__ 0 => -I s ~p(@). (because, so by 9a-Remark Now choose any + Zkiy i that ~ s X 3), and put by 39 p(@) < inf {f(x): fi(x) !

And such that ~ ~-nbhd. x o + tx ~ ~ i V whenever ~ > 0 0 < t < ~ and is again such a f(x o + tx) < f(Xo) x ~ V. But x ~ Ko and also. ), and for sufficiently x + r(t)/t x e V f(x o + tx) e A such that < f(Xo), s V small for small t. t > 0. This shows x ° + tx E A('~ ~ l ( ' ~ . . ~ n _ l and so x° is not a solution By that ~ dom (f) after all, qed. §17. An Application As one illustration sider here the so-called variations". equation "simplest In particular, of 16a) tial equation. minimizing of the Dubovitskii-Milyutin leads defined along which gravity), and the profile solutions of these cycloids, and catenaries.

F set Now t o < 0, assume when (f) C (f**) (f), dom such {tot+ < X , Y o > for o t h e r w i s e since otherwise t o = -i. Then t = f(x). that a contradiction convex such g Then (dom (f)). Then < f(Xo). Yo E X* Hence set. and > : (x,t) the sup in for given to Young's With e epi (f)}. be contradicted. (2) would x e dom In fact be infinite. (f), So we may the sup is a t t a i n e d > X* x s dora ( f ) } = f * ( y o ) , inequality. endowed is an o r d e r - r e v e r s i n g 24. f** = c--o (f) ~ sup only is p r o p e r But now C o r o l l a r X.