Let be s be a positive constant, f € R, g € Rn, with g # 0, B € Rnxn be symmetric,
and D € Rnxn be a diagonal matrix with positive diagonal elements.
the function m : Rn Rn as m (d) == f + gT d + 1/1 Bd, Vd € Rn.
(1) Show that d. € Rn is an optimal solution of mind m(d) subject to ||Dd||2 < s if and only if
Il Dd, ||2 < A and there exists a real number l > 0 such that : = - D-1 9,
X/A - Dd. (2) = 0, and (B + XD2) is positive semidefinite.
Theorem 4.1 of the Textbook and the change of variable U = Dd.) (10 points)
(2) Show that the optimal solution of mind( (f + gTd) subject to
(3) (8 points) Let ds be given is in (2). Show that the optimal solution of min-> m(Tds)
subject to 112
if 9TD-2BD-29 0
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