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

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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 ) / bardbl g bardbl 2 2 (which is twice Polyak’s optimal step size value) we have bardbl x + x bardbl 2 < bardbl x x bardbl 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 finding a point in the intersection C negationslash = of convex sets C 1 , . . . , C m . To do this, we use alternating projections to find 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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