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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 leastnorm 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.
 Fall '13
 F.Borrelli
 The Aeneid

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