STOCHASTIC

(a) Suppose that A is fixed and that each b, has the marginal cumulative distribution
function G,. Show that P [A, x £ 6,] & fi, if and only if
x e {x| Atx g #„,}, where Kh = {mmy\ G,(y) S /?,}.
Hence the chance-constrained program
has the linear program
mine'*, s.t. P[Atx & bt] = i, i=l,...,w,
mine'*, Atx^Kp,, i=\,...,m,
as its deterministic equivalent.
(b) Show that KBl = G,~ 1 (/?,) if G, has an inverse.
Suppose now that A as well as b is random. Let w, = [Ai,b,] and z= (J^). Then
P{Ai S 6,} can be written as P[wi'z £ 0], Assume that >v, has the multivariate normal distribution
Wi ~ N(w,, V,), where w, is the mean vector and V, is the positive-semidefinite
variance-covariance matrix of the w,. Let 4i(y) denote the cumulative distribution of a
normally distributed random variable with mean 0 and variance 1 and Kg the /f-fractile of \p,
namely the Kf such that f//(Kt) = ft for any /? e (0,1).
(c) Show that P[w,' z g 0] £ 0, if and only if w,'z-Ktl(z'V, z)" 2 g 0.
(d) Show that (z'Vt z) 1 ' 2 is a convex function of z.
(e) Use the Cauchy-Schwaiz inequality [i.e., u, v e E", u'v S (u'u)(v'v)] to show that
z ot Vtzi (.z 0, vtz 0 y i2 (z'vtzyi 2 .
A function g is said to be subdifferentiable at x° if there exists a vector s(x°) e £" such that
g(x) a s(x°)'(*-*°)+#0t o ) for all xeE\
(f) Use (e) to show that (z'K,z)" 2 is a subdifferentiable function of z e E" having subgradient
vector
(g) The chance-constrained problem
has the deterministic equivalent
z'ViKz'Vtzy 2 if z'ViZ > 0,
0 otherwise.
minc'z, s.t. Pr[w,'z & 0] £ 0,, i=l,...,m,
minc'z, s.t. wt'z - K^z'Viz) 112 a 0, i=\,...,m. (1)
Suppose pt S 0.5 for all i. Show that the set of points that satisfy the constraint of (1)
form a convex set.
In chance-constrained programming models one can generally consider separately the
objective and the constraints. When the vector c is random one can consider numerous
objectives such as the expected utility criterion. In addition one can consider, as in the
Pyle-Turnovsky paper, the aspiration or fractile objectives. The aspiration model may be
written as
minP[c'AC > dol, s.t. Ax g b, x & 0
for a given aspiration level d0, while the fractile model may be written as
minrf, s.t. Plc'x ^]i«, Ax £ b, jiO
for a given fractile a e [0,1]. Suppose that c ~ N(c, V0).
(h) Show that the fractile model has as its deterministic equivalent
where K,= ^~*(a).
*(*) =
mine** + Ka(xV0xy 12 , s.t. Ax ^ b, x £ 0
MIND-EXPANDING EXERCISES 359