STOCHASTIC

dZISxj = E„[df(x,p)ldxJ]. [Hint: Utilize the dominated convergence theorem along
with the mean-value theorem.]
A function c(x): D-+ R is said to be subdifferentiable on an open convex set AT 0. Show that there exists a 1° S 0 such that
for all Xe[X°,\), Z{Xx + (\ -l)x) < Z(x). [Hint: Let ^(A) = Z[lx + (\ -k)x] and
utilize (b).]
(d) Suppose x e K and Z{x) > Z{x*) = mm{Z(x) \ x s K}. Show that there exists an x
such that VZW'jt > Z(x)'x.
(e) Show that
Z(x*) = mm{Z(x)\xeK}
if and only if
VZ(x')'x* = mm{VZ(,x*)'x\xeK}.
MIND-EXPANDING EXERCISES
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