B the function with a r mn f x max ap x b p is

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Unformatted text preview: ubly stochastic and that diag(Y ) = Sλ. ij (c) Use the results in parts (a) and (b) to show that if f is convex and symmetric and X ∈ Sn , then f (λ(X )) = sup f (diag(V T XV )) V ∈V where V is the set of n × n orthogonal matrices. Show that this implies that f (λ(X )) is convex in X . 2.22 Convexity of nonsymmetric matrix fractional function. Consider the function f : Rn×n × Rn → R, defined by f (X, y ) = y T X −1 y, dom f = {(X, y ) | X + X T ≻ 0}. When this function is restricted to X ∈ Sn , it is convex. Is f convex? If so, prove it. If not, give a (simple) counterexample. 9 2.23 Show that the following functions f : Rn → R are convex. (a) f (x) = − exp(−g (x)) where g : Rn → R has a convex domain and satisfies ∇2 g ( x ) ∇g ( x ) T ∇g ( x ) 1 0 for x ∈ dom g . (b) The function with A ∈ R m×n f (x) = max { AP x − b | P is a permutation matrix} , b ∈ Rm . 2.24 Convex hull of functions. Suppose g and h are convex functions, bounded below, with dom g = dom h = Rn . The convex hull function of g and h is define...
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