bv_cvxbook_extra_exercises

We assume a is full rank and m n a formulate the

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Unformatted text preview: ., the set of vectors p that satisfy the constraint is convex). (c) Show that the constraint ‘no more than half of the lamps are on’ is (in general) not convex. 3.8 Schur complements and LMI representation. Recognizing Schur complements (see §A5.5) often helps to represent nonlinear convex constraints as linear matrix inequalities (LMIs). Consider the function f (x) = (Ax + b)T (P0 + x1 P1 + · · · + xn Pn )−1 (Ax + b) where A ∈ Rm×n , b ∈ Rm , and Pi = PiT ∈ Rm×m , with domain dom f = {x ∈ Rn | P0 + x1 P1 + · · · + xn Pn ≻ 0}. This is the composition of the matrix fractional function and an aﬃne mapping, and so is convex. Give an LMI representation of epi f . That is, ﬁnd a symmetric matrix F (x, t), aﬃne in (x, t), for which x ∈ dom f, f (x) ≤ t ⇐⇒ F (x, t) 0. Remark. LMI representations, such as the one you found in this exercise, can be directly used in software systems such as CVX. 3.9 Complex least-norm problem. We consider the complex least ℓp -norm problem minimize xp subject to Ax = b, whe...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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