# Let be s be a positive constant, f € R, g € Rn, with g # ...

## Transcribed Text

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. Define 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. (Hint: use 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 ||Dd||2 < is D-29 (3) (8 points) Let ds be given is in (2). Show that the optimal solution of min-> m(Tds) subject to 112 1, T* = if 9TD-2BD-29 0 min 1) otherwise

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