bv_cvxbook_extra_exercises

Suppose f rm r is convex and a rmn b rm c rn and

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Unformatted text preview: ive of log determinant. Show that f (X, t) = nt log t−t log det X , with dom f = Sn ×R++ , ++ is convex in (X, t). Use this to show that g (X ) = n(tr X ) log(tr X ) − (tr X )(log det X ) λi =n n n n log i=1 i=1 λi − log λi , i=1 where λi are the eigenvalues of X , is convex on Sn . ++ 2.7 Pre-composition with a linear fractional mapping. Suppose f : Rm → R is convex, and A ∈ Rm×n , b ∈ Rm , c ∈ Rn , and d ∈ R. Show that g : Rn → R, defined by g (x) = (cT x + d)f ((Ax + b)/(cT x + d)), dom g = {x | cT x + d > 0}, is convex. 2.8 Scalar valued linear fractional functions. A function f : Rn → R is called linear fractional if it has the form f (x) = (aT x + b)/(cT x + d), with dom f = {x | cT x + d > 0}. When is a linear fractional function convex? When is a linear fractional function quasiconvex? 2.9 Show that the function f (x) = is convex on {x | x 2 < 1}. Ax − b 2 2 1 − xT x 2.10 Weighted geometric mean. The geometric mean f (x) = ( as shown on page 74. Extend the proof to show that n f (x) = x αk , k k xk )1/n with dom f = Rn is concave, +...
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