bv_cvxbook_extra_exercises

Show that f is convex this composition rule subsumes

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online