bv_cvxbook_extra_exercises

m 7 where the functions fi rn r are dierentiable

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: ons (6) can be used to derive a simple algorithm for (5). Using the T T eigenvalue decomposition A = n αi qi qi , of A, we make a change of variables yi = qi x, i=1 and write (5) as n n 2 minimize i=1 βi yi i=1 αi yi + 2 subject to y T y ≤ 1 T where βi = qi b. The transformed optimality conditions (6) are y 2 ≤ 1, λ ≥ − αn , ( α i + λ ) yi = − β i , i = 1, . . . , n, λ(1 − y 2 ) = 0, if we assume that α1 ≥ α2 ≥ · · · ≥ αn . Give an algorithm for computing the solution y and λ. 4.7 Connection between perturbed optimal cost and Lagrange dual functions. In this exercise we explore the connection between the optimal cost, as a function of perturbations to the righthand sides of the constraints, p⋆ (u) = inf {f0 (x) | ∃x ∈ D, fi (x) ≤ ui , i = 1, . . . , m}, (as in §5.6), and the Lagrange dual function g (λ) = inf (f0 (x) + λ1 f1 (x) + · · · + λm fm (x)) , x with domain restricted to λ 0. We assume the problem is convex. We consider a problem with inequality constraints only, for simplicity. We ha...
View Full Document

Ask a homework question - tutors are online