hw3 - EE364b Prof. S. Boyd EE364b Homework 3 1. Minimizing...

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EE364b Prof. S. Boyd EE364b Homework 3 1. Minimizing a quadratic. Consider the subgradient method with constant step size α , used to minimize the quadratic function f ( x ) = (1 / 2) x T Px + q T x , where P 0. For which values of α do we have x ( k ) x , for any x (1) ? What value of α gives fastest asymptotic convergence? 2. Step sizes that guarantee moving closer to the optimal set. Consider the subgradient method iteration x + = x αg , where g ∂f ( x ). Show that if α < 2( f ( x ) f ) / b g b 2 2 (which is twice Polyak’s optimal step size value) we have b x + x b 2 < b x x b 2 , for any optimal point x . This implies that dist ( x + ,X ) < dist ( x,X ). (Methods in which successive iterates move closer to the optimal set are called ejer monotone . Thus, the subgradient method, with Polyak’s optimal step size, is F´ ejer monotone.) 3. A variation on alternating projections. We consider the problem of ±nding a point in the intersection C n = of convex sets C 1 ,..., C m . To do this, we use alternating projections to ±nd a point in the intersection of the two sets C 1 × ··· × C m R mn and { ( z 1 ,...,z m ) R mn | z 1 = ··· = z m } ⊆ R mn . Show that alternating projections on these two sets is equivalent to the following it- eration: project the current point in R n onto each convex set, and then average the results. Draw a simple picture to illustrate this.
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This note was uploaded on 04/09/2010 for the course EE 360B taught by Professor Stephenboyd during the Fall '09 term at Stanford.

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hw3 - EE364b Prof. S. Boyd EE364b Homework 3 1. Minimizing...

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