STOCHASTIC

The problem in (b)-(c) hints that the functions involved determine whether the Kuhn-
Tucker conditions are necessary, not the shape of the feasible region. Suppose d is a
direction vector in E", A(x) is the set of binding constraints at x, i.e., those gt(x) = 0, and
F is the feasible region. Let D(x)= [d\3o > 0: a £ r £ 0^-x + xde F], i.e., the set of
feasible directions at x and @(x) = {d\Vgj{x)'d^Q, ie A(x)}, the set of outward-pointing
gradients of the active constraints.
(d) Show that if x* is optimal for the nonlinear programming problem, then
Vf(x*)'d S 0 VrfeZ)(x*) or D(x*) (the closure of D).
(e) Show that D(x) (x*~) = D(x*), i.e., the set of outward-pointing gradients
of the active constraints equals the set of feasible directions.
(f) Show that the Kuhn-Tucker conditions are necessary under the constraint qualification.
[Hint: Utilize Farkas' lemma, i.e., the statement q'x S 0 for all x such that
Ax g 0 is equivalent to the statement that there exist u £0 such that q + A'u = 0.]
(g) Show the following assumptions are sufficient to guarantee that the constraint
qualification is satisfied:
(i) all constraint functions are linear; or
(ii) all constraint functions are convex or pseudo-convex and the constraint set has a
nonempty interior; or
(iii) the gradients of all binding constraints are linearly independent.
14. Suppose the primal problem is {minf(x) | gs (x) g 0, j = 1,..., m); then the Wolfe dual
problem is
{max/!(x,A) =/W+X^W|A £ 0, Vxf(x)+-£XjVxgj(x) = 0},
where/and the g} are differentiable functions.
(a) (Weak duality) Suppose x 1 is feasible for the primal problem, (x 2 , A 2 ) is feasible
for the dual problem, and/and the gj are convex at x 2 . Show that/Cx: 1 ) g h(x 2 ,X 2 ).
(b) (Wolfe's duality theorem) Suppose/and the gt are convex functions. Let x 1 solve
the primal problem and suppose the gs satisfy a constraint qualification. Show that there
exists aA'e£" such that (x 1 , A 1 ) solve the dual problem and/^c 1 ) = h(x l ,k l ).
(c) (Mangasarian's converse duality theorem) Suppose/and the#j are convex functions.
Let x 1 solve the primal problem and suppose the gt satisfy a constraint qualification.
Show that if ix 2 ,X 2 ) solve the dual problem and h(x,A 2 ) is strictly convex at x 2 , then
x 2 = x 1 a-ndfix 1 ) = h(x 2 ,X 2 ).
(d) (Huard's converse duality theorem) Suppose (x 2 , A 2 ) solve the dual problem and
/and the gs are convex at x 2 . Show that if h(x, A 2 ) is twice continuously differentiable at
x 2 and the nx n Hessian matrix ^„^xh{x 2 ,X 2 ) is nonsingular, then x 2 solves the primal
problem and/(x 2 ) = h(x 2 , A 2 ).
15. Prove the following properties of convex sets.
(a) Show that the intersection of a finite or infinite number of convex sets in £" is a
convex set.
(b) A necessary and sufficient condition for a set C to be convex is that for each integer
/Mil:
x\...,x m e C\
Ai,...,Am S 0 | => XiX 1 + ••• + Xmx m e C;
Ai + - + Am = 1 J
that is, all convex combinations of points of C belong to C.
11 PART 1 MATHEMATICAL TOOLS