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Unformatted text preview: z 0, 1T z = 1}.
Hints. Consider the problem of minimizing (1/2) y − x 2 subject to y
2
partial Lagrangian
L(y, ν ) = (1/2) y − x 2 + ν (1T y − 1),
2
leaving the constraint y 0, 1T y = 1. Form the 0 implicit. Show that y = (x − ν 1)+ minimizes L(y, ν ) over y 0. 7.11 Conformal mapping via convex optimization. Suppose that Ω is a closed bounded region in C with
no holes (i.e., it is simply connected). The Riemann mapping theorem states that there exists a
conformal mapping ϕ from Ω onto D = {z ∈ C  z  ≤ 1}, the unit disk in the complex plane.
(This means that ϕ is an analytic function, and maps Ω onetoone onto D.)
One proof of the Riemann mapping theorem is based on an inﬁnite dimensional optimization
problem. We choose a point a ∈ int Ω (the interior of Ω). Among all analytic functions that map
∂ Ω (the boundary of Ω) into D, we choose one that maximizes the magnitude of the derivative at
a. Amazingly, it can be shown that this function is a conform...
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 Fall '13
 F.Borrelli
 The Aeneid

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