bv_cvxbook_extra_exercises

# Show that f is convex this composition rule subsumes

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Unformatted text preview: . Suppose f (x) = h(g1 (x), g2 (x), . . . , gk (x)) where h : Rk → R is convex, and gi : Rn → R. Suppose that for each i, one of the following holds: • h is nondecreasing in the ith argument, and gi is convex • h is nonincreasing in the ith argument, and gi is concave • gi is aﬃne. Show that f is convex. (This composition rule subsumes all the ones given in the book, and is the one used in software systems such as CVX.) You can assume that dom h = Rk ; the result ˜ also holds in the general case when the monotonicity conditions listed above are imposed on h, the extended-valued extension of h. 2.3 Logarithmic barrier for the second-order cone. The function f (x, t) = − log(t2 −xT x), with dom f = {(x, t) ∈ Rn × R | t > x 2 } (i.e., the second-order cone), is convex. (The function f is called the logarithmic barrier function for the second-order cone.) This can be shown many ways, for example by evaluating the Hessian and demonstrating that it is positive semideﬁnite. In this exercise you establish convexity of f...
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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 Berkeley.

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