STOCHASTIC

(c) Show that the dual of problem PO is
DO: max £ H/(0)«/,
J=I
s.t.
«, - a ZP/UJ g Hi) (1 § i g A?), (1)
»i-«I;/«^iW + '.(*-0 (lg/SAf-l; *>/), (2)
«i - a I?J'«J ^ £W + CiO'-Ar) (2 g / £ JV; A; < /). (3)
(d) Show that problem DO is equivalent to the following problem:
N
Dl: max £ 7ij(0)uJt
J = I
s.t.
"i - « Y.PJ'UJ £ £(') (1 S i g JV), (1')
j
u,-w,^-cj (1^/Ss/V-l), (2')
u,-v,^cdi (2gigJV), (3')
w, - wi + i g 0 (1 S / g JV-1; wN = 0), (4')
f i - v,-1 g 0 (2 S i g TV; «! = 0), (5')
Wi-«Ift'«^iW + C«: (ISf'S/V-l, £>0, (6')
J
v,-«Y.Pj k uj ^ L(k)-cdk (2g/gJV, k iis assumed to be true only for k = i+l, etc.]
(f) Show that the dual of problem D2 is
PI: min £ L(i)xt> + £ (-c.0* + E (C„J>, d")
1=1 (=1 1=2
+ z[i(*)+c*]*+"5: [iw-c,,*]**,
* = 2 k = l
S.t.
xt' - « I />,**»* + Ji-«I Pi"yk + r,-« £ />,'