STOCHASTIC

where V is the gradient operator and \(e) is a Lagrange multiplier.
(e) Show that as g->0 these conditions converge to those for problem (B); hence
*(«)-•*((>).
Assuming that each x,(e) > 0, the conditions in (d) simplify to
VxW[x(e),e] - X(e)e = 0, e'x(fi) = 1.
(f) Show that a total differentiation of these n +1 equations yields the matrix equations
VxxW[x(e),e\ -e
-e' 0
Vex(£)
8X(e)
-V*,iV[x(e),el
where V,,, denotes the Hessian operator. Hence X(B) is differentiable if the Hessian matrix
on the left-hand side is nonsingular.
(g) Let the (« +1) x (« +1) matrix
A -e
B = _-e< 0
where A is an n x n negative-definite matrix. Show that B ~' exists and may be partitioned as
A~ l +X6~ 1 Y -0- l Y'
-0- l Y 0- 1
where X= -A~ l e, Y = -e'A' 1 , and 0 = -e'A-'e. [Hint: Utilize the fact that B~
exists if and only if 6 i= 0, which it is since A ~' is negative definite.]
(h) Let A(e) = V„ W[x{e),e]. Utilize (f) to show that x(e) is differentiable and that
V.*(e) = A(e) _, A(e)-l ee t A(a- l j Vx,fV[x(e),e].
e'A («)->«
(i) Show that Ve x(0) exists. [Hint: Show that
lira V,, W[x(e), s] = V„ Z(x) and that lim V« W[x(e), e]
exists. Hence \im^0^ex(e) = V£^(0) exists.]
The expression in (g) may be used to verify that x(e) is also twice differentiable utilizing
the following result.
(j) Suppose B(y) is an m x m matrix whose elements are functions of a single variable y,
i.e.,
AnOO - blm(y)
B(y) =
ft-i 00 bmm(y) .
Assume that each bu is a differentiable function of y in some neighborhood N(y°, S),
for some y° and 0 and that \B{y)\ jt 0 for all y e N(y°, d). Show that
U