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Unformatted text preview: wn. However, we do have (matrix) upper
26 bounds on the covariance of the random variables yi = AT z ∈ Rki , which is AT XAi . The problem
is to ﬁnd the covariance matrix for z , that is consistent with the known upper bounds on the
covariance of yi , that has the largest volume conﬁdence ellipsoid.
Derive the Lagrange dual of this problem. Be sure to state what the dual variables are (e.g.,
vectors, scalars, matrices), any constraints they must satisfy, and what the dual function is. If
the dual function has any implicit equality constraints, make them explicit. You can assume that
i=1 Ai Ai ≻ 0, which implies the feasible set of the original problem is bounded.
What can you say about the optimal duality gap for this problem? 4.3 The relative entropy between two vectors x, y ∈ Rn is deﬁned as
n xk log(xk /yk ).
k=1 This is a convex function, jointly in x and y . In the following problem we calculate the vector x
that minimizes the relative entropy with a given vector y , subje...
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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
- The Aeneid