MIT6_235S10_hw03

# MIT6_235S10_hw03 - 6.253 Convex Analysis and Optimization...

This preview shows pages 1–2. Sign up to view the full content.

6.253: Convex Analysis and Optimization Homework Prof. Dimitri P. Bertsekas Spring 2010, M.I.T. Problem 1 (a) Show that a nonpolyhedral closed convex cone need not be retractive, by using as an example the cone C = { ( u,v,w ) ( u,v ) w } , the recession direction d = (1 , 0 , 1), and the corresponding asymptotic sequence { ( k, | k, k 2 + k ) } . (This is the, so-called, second order cone, which plays an important role in conic programming; see Chapter 5.) (b) Verify that the cone C of part (a) can be written as the intersection of an inFnite number of closed halfspaces, thereby showing that a nested set sequence obtained by intersection of an inFnite number of retractive nested set sequences need not be retractive. Problem 2 Let C be a nonempty convex set in R n , and let M be a nonempty aﬃne set in R n . Show that M rin ( C ) = is a necessary and suﬃcient condition for the existence of a hyperplane H containing M , and such that rin ( C ) is contained in one of

This preview has intentionally blurred sections. Sign up to view the full version.

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

{[ snackBarMessage ]}

### Page1 / 3

MIT6_235S10_hw03 - 6.253 Convex Analysis and Optimization...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online