By Richard B. Holmes (auth.)

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N~n ~ ~f(Xo). Proof. Immediate from the optimality criterion lld) and the Corollary in b) above. Corollary. ,fn becomes: ~ h i K 0 are smooth, such that then the solvability ~ifi(Xo ) = 0 For x ~ (xl,x2) f(x) = x~ + 2x~ Show that f e) Remark. programs ((Xl,X2): x I K 0~. Use convex program K = (x: x I K 0, x 2 ~ 2, 3Xl+2X 2 ~ i0~. It is possible for which conditions the solutions. 4x I - 6x 2. in d) to solve the ordinary where = @. +XnVfn(Xo) Exercise 21. If all the to generalize of Kuhn-Tucker the class of convex type still characterize In one direction we may adjoin a finite number of affine constraints (constraints of the form ~(x)j = ~j, where Zj 35 is a continuous corresponding In another linear functional); additional direction, A is a compact ~ f (x) < 0, is then that for some <_ 0, on measure >~ space x, the function regularity assumption x, ~ ~ A} < 0.

For the ordinary Then by d)~ Let convex program de- -~- E aq(@). If p is the for the original program, we see that and hence that aq(@) = ap(7). -~ e ap(~), and so the right-hand f(x) = p(y) and In addition inequality by examining inequality Therefore, follows by noting that f(x) = P(Y). y < p' (@;y), whenever p program 12c), is the perturbation X function for the ordinary is a Lagrange multiplier Taking in particular y vector, and to be the jth unit vector in convex y ~ R n. R n, we may state that -X.

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. Let rivatives Among a ring descends in its second be continuous r that surface joining are included (the shape to of revolution. straight with continuous arguments. of curves problem are respectively and third differen- in least time subject of a minimal three problems F: R 3 + R 1 such programs the brachistochone Euler Euler is essentially we con- of the abstract over a class of smooth R 2.