bv_cvxbook_extra_exercises

# The gure shows for c 1 36 dom c c 5 45 4 35 u 3 25 2

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: mize tr(AX ) subject to tr X = r 0 X I, with variable X ∈ Sn , is equal to f (A). (b) Show that f is a convex function. (c) Assume A(x) = A0 + x1 A1 + · · · + xm Am , with Ak ∈ Sn . Use the observation in part (a) to formulate the optimization problem minimize f (A(x)), with variable x ∈ Rm , as an SDP. 4.18 An exact penalty function. Suppose we are given a convex problem minimize f0 (x) subject to fi (x) ≤ 0, i = 1, . . . , m with dual (14) maximize g (λ) subject to λ 0. (15) We assume that Slater’s condition holds, so we have strong duality and the dual optimum is attained. For simplicity we will assume that there is a unique dual optimal solution λ⋆ . For ﬁxed t > 0, consider the unconstrained minimization problem minimize f0 (x) + t max fi (x)+ , i=1,...,m (16) where fi (x)+ = max{fi (x), 0}. (a) Show that the objective function in (16) is convex. (b) We can express (16) as minimize f0 (x) + ty subject to fi (x) ≤ y, 0≤y i = 1. . . . , m (17) where the variables are x and y ∈ R. Find the Lagrange dual...
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