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Hence 1/1 is in S. ii) f is differentiable on S. P) - f' ('P)h ~ € Ill hill We apply Taylor theorem to the function L: given 17 > 0, there exists 50 > 0 such that Ill h Ill < 80 implies 39 General Methods . . e. p) h I < for all h in E, M ID h 10 where M is a ftxed number. ;;;K Jb Ill hIll dt a = K(b-a)lll hIll a Hence, l ( 'P ) is bounded, and continuous. Finally, the function f is differentiable on S and the differential f'( 'I' ) is l ('I') (10); it is obtained from (9) by differentiation under the integral sign.

Attainable vectors A vector u of Rn is locally constrained at the point x in F, if and only if, g is differentiable at x and u T 'V g i (x) is positive ( ~ 0) for all i such that gi (x) = 0. The set of numbers i verifying gi (x) = 0 will be noted N('x) (see (23), chapter 1). A vector u of Rn is attainable at the point x in F, if and only if. there exists an n-dimensional vector function w, t ~ 1/i (t), defined on [ 0, ttl, t 1 >O. such that 1/J (0) = g( 1/J (t) ~ 0, 1/J is differentiable at t = 0 and d 1/J /dt (0) = u.

E32 49 Kuhn-Tucker theorem for Problem P' If the function f has a local minimum at x, and if g and h satisfy the Kuhn-Tucker constraint qualification at x, then there exist a vector y of Rm and a vector z of RP such that -T -T y~O, y g(x) + z h(x) = 0, f'(x)- -T - -T - y g'(x)- z h'(x) 0 11-14 Sufficient Condition: Converse of Kuhn-Tucker Theorem in Convex-Concave Optimization Kuhn-Tucker conditions are, by themselves, not sufficient for a point to be optimal. But they become sufficient when some properties of convexity and concavity are prescribed.