MIT6_235S10_hw03 - 6.253 Convex Analysis and Optimization...

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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
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MIT6_235S10_hw03 - 6.253 Convex Analysis and Optimization...

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