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Unformatted text preview:  p(0) = 1, p(t) ≤ 1 for α ≤ t ≤ β }, where
p( t ) = a 1 + a 2 t + · · · + a k t k − 1 ,
convex?
1.2 Set distributive characterization of convexity. [7, p21], [6, Theorem 3.2] Show that C ⊆ Rn is
convex if and only if (α + β )C = αC + βC for all nonnegative α, β .
1.3 Composition of linearfractional functions. Suppose φ : Rn → Rm and ψ : Rm → Rp are the
linearfractional functions
φ(x) = Ax + b
,
cT x + d ψ (y ) = Ey + f
,
gT y + h with domains dom φ = {x  cT x + d > 0}, dom ψ = {y  g T x + h > 0}. We associate with φ and
ψ the matrices
Ab
Ef
,
,
Td
c
gT h
respectively.
Now consider the composition Γ of ψ and φ, i.e., Γ(x) = ψ (φ(x)), with domain
dom Γ = {x ∈ dom φ  φ(x) ∈ dom ψ }.
Show that Γ is linearfractional, and that the matrix associated with it is the product
E
gT f
h A
cT b
d . 1.4 Dual of exponential cone. The exponential cone Kexp ⊆ R3 is deﬁned as
Kexp = {(x, y, z )  y > 0, yex/y ≤ z }.
∗
Find the dual cone Kexp . We are not worried here about the ﬁne details of wh...
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 Fall '13
 F.Borrelli
 The Aeneid

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