midsolns_ee227a - QUESTION BOOKLET EECS 227A Fall 2009...

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Unformatted text preview: QUESTION BOOKLET EECS 227A Fall 2009 Midterm Tuesday, Ocotober 20, 11:10-12:30pm DO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO • You have 80 minutes to complete the midterm. • The midterm consists of three problems, provided in the question booklet (THIS BOOKLET), that are in no particular order of difficulty. • Write your solution to each problem in the space provided in the solution booklet (THE OTHER BOOKLET). Try to be neat! If we can’t read it, we can’t grade it. • You may give an answer in the form of an arithmetic expression (sums, products, ratios, factorials) that could be evaluated using a calculator. Expressions like ( 8 3 ) or ∑ 5 k =0 (1 / 2) k are also fine. • A correct answer does not guarantee full credit and a wrong answer does not guarantee loss of credit. You should concisely explain your reasoning and show all relevant work. The grade on each problem is based on our judgment of your understanding as reflected by what you have written. • This is a closed-book exam except for one 8 . 5 ′′ × 11 ′′ page of notes. Problem 1: For all parts of this problem , let f be a convex and twice continuously differentiable function. Consider the steepest descent method x k +1 = x k − α k ∇ f ( x k ) . (a) Suppose that ∇ 2 f ( x ) ≻ 0 for all x ∈ R n . How many local minima does f have? How many global minima does f have? (b) State a convergence result for the steepest descent method. Be sure to specify explicitly how your step size is chosen. (c) Consider the condition ∃ γ > 0 such that bardbl∇ f ( x ) − ∇ f ( y ) bardbl 2 2 γ ≤ ( ∇ f ( x ) −∇ f ( y )) T ( x − y ) for all x,y ∈ R n . (1) Given a quadratic function f ( x ) = 1 2 x T Qx − b T x where Q is symmetric, state the weakest sufficient conditions on ( Q,b ) under which condition (1) guaranteed to hold. (d) Assume that there exists some x ∗ ∈ R n such that ∇ f ( x ∗ ) = 0. Assuming that condition (1) holds, prove that x k → x ∗ as long as there is some ǫ ∈ (0 , 2) such that the step sizes α k satisfy ǫ ≤ α k ≤ 2 − ǫ γ for all k = 1 , 2 ,... . Solution: (a) Under the given condition, any local minimum must be a global minimum. It is possible that f does not have a global minimum (e.g., f ( x ) = exp( − x ) for x ∈ R .) If it has a global minimum, then it must be unique since the assumption implies that f is strictly convex on R n ....
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midsolns_ee227a - QUESTION BOOKLET EECS 227A Fall 2009...

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