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Unformatted text preview: he investment gain with our mixed portfolio x. The
optimality condition is that, for each asset we invest in, the expected value of this ratio is one,
and for each asset we do not invest in, the expected value cannot exceed one. Very roughly
speaking, this means our portfolio does as well as any of the assets that we choose to invest
in, and cannot do worse than any assets that we do not invest in.
Hint. You can start from the simple criterion given in §4.2.3 or the KKT conditions. (b) In this part we will derive the dual of the logoptimal investment problem. We start by writing
the problem as
minimize − m πj log yj
j =1
subject to y = P T x, x 0, 1T x = 1.
Here, P has columns p1 , . . . , pm , and we have the introduced new variables y1 , . . . , ym , with
the implicit constraint y ≻ 0. We will associate dual variables ν , λ and ν0 with the constraints
y = P T x, x 0, and 1T x = 1, respectively. Deﬁning νj = νj /ν0 for j = 1, . . . , m, show that
˜
the dual problem can be written as
m
ν
maximize
j =1 πj log(˜j /πj )
subject to P ν 1,
˜ with variable ν . The objective here is the (...
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 Fall '13
 F.Borrelli
 The Aeneid

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