bv_cvxbook_extra_exercises

1 is the set a rk p0 1 pt 1 for t where p

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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 linear-fractional functions. Suppose φ : Rn → Rm and ψ : Rm → Rp are the linear-fractional 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 linear-fractional, 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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