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Unformatted text preview: ons (6) can be used to derive a simple algorithm for (5). Using the
T
T
eigenvalue decomposition A = n αi qi qi , of A, we make a change of variables yi = qi x,
i=1
and write (5) as
n
n
2
minimize
i=1 βi yi
i=1 αi yi + 2
subject to y T y ≤ 1
T
where βi = qi b. The transformed optimality conditions (6) are y 2 ≤ 1, λ ≥ − αn , ( α i + λ ) yi = − β i , i = 1, . . . , n, λ(1 − y 2 ) = 0, if we assume that α1 ≥ α2 ≥ · · · ≥ αn . Give an algorithm for computing the solution y and λ.
4.7 Connection between perturbed optimal cost and Lagrange dual functions. In this exercise we explore
the connection between the optimal cost, as a function of perturbations to the righthand sides of
the constraints,
p⋆ (u) = inf {f0 (x)  ∃x ∈ D, fi (x) ≤ ui , i = 1, . . . , m},
(as in §5.6), and the Lagrange dual function
g (λ) = inf (f0 (x) + λ1 f1 (x) + · · · + λm fm (x)) ,
x with domain restricted to λ 0. We assume the problem is convex. We consider a problem with
inequality constraints only, for simplicity.
We ha...
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 Fall '13
 F.Borrelli
 The Aeneid

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