# HW2 - f is convex if and only if its domain is convex and H...

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ELE 704 Optimization Homework 2 Due 6 March, 2007 1. Consider the problem of minimizing k Ax - b k 2 where A R m × n , x R n and b R m (a) Give a geometric interpretation of the problem. (b) Write a necessary condition for optimality. Is this also a sufficient condition? (c) Is the optimal solution unique? Why or why not? (d) Can you give a closed form solution of the optimal solution? Specify any assumptions that you may need. (e) Solve the problem for the following A and b A = 1 - 1 0 0 2 1 0 1 0 1 0 1 , b = 2 1 1 0 2. Which of the following sets are convex? Also give the geometric interpre- tation of these sets. (a) A slab, S = ' x R n | α a T x β for any α and β , (b) A rectangle, S = { x R n | α i x i β i , i = 1 , ..., n } , (c) A wedge, S = ' x R n | a T 1 x b 1 , a T 2 x b 2 , (d) The set of points closer to a point than a given set, S = n x R n | k x - x o k 2 ≤ k x - y k 2 , y ∈ T o for some set T ⊆ R n and x o R n outside T . 3. Prove that a twice differentiable function
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Unformatted text preview: f is convex if and only if its domain is convex and H ( x ) ” ∀ x ∈ dom f . Hint: Use the ﬁrst order condition for convexity. You may ﬁrst consider a function f : R → R and then extend the solution to f : R n → R . 4. Consider the following functions. First check these functions for con-vexity/concavity using Jensen’s inequality. Then verify your answer by plotting these functions in MATLAB. You may use the surf command. (a) f ( x ) = e x-1 on R , (b) f ( x 1 ,x 2 ) = x 1 x 2 on R 2 ++ , (c) f ( x 1 ,x 2 ) = 1 / ( x 1 x 2 ) on R 2 ++ , (d) f ( x 1 ,x 2 ) = x 1 /x 2 on R 2 ++ , (e) f ( x 1 ,x 2 ) = x 2 1 /x 2 on R × R ++ , (f) f ( x 1 ,x 2 ) = x α 1 x 1-α 2 where 0 ≤ α ≤ 1 R 2 ++ . 1...
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