bv_cvxbook_extra_exercises

# If you use matlab version 70 which is lled with bugs

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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 one-to-one 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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