MIT6_235S10_mid_sol - 6.253: Convex Analysis and...

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6.253: Convex Analysis and Optimization Midterm Prof. Dimitri P. Bertsekas Spring 2010, M.I.T. Problem 1 State which of the following statements are true and which are false. You don’t have to justify your answers: 1. If X 1 , X 2 are convex sets that can be separated by a hyperplane, and X 1 is open, then X 1 and X 2 are disjoint. (8 points) TRUE 2. If f : R n R is a convex function that is bounded in the sense that for some γ > 0 , | f ( x ) | ≤ γ for all x R n , then the problem minimize f ( x ) subject to x R n , has a solution. (8 points) TRUE 3. The support function of the set { ( x 1 ,x 2 ) | | x 1 | + | x 2 | = 1 } is σ ( y ) = max { y 1 ,y 2 } . (8 points) FALSE 4. If f : R n ( −∞ , ] is a convex function such that ∂f x ) is nonempty for some x ¯ R n , then f is lower semicontinuous at ¯ x . (8 points) TRUE 5. If M = { ( u,w ) | u R , | u | ≤ w } , the dual function in the MC/MC framework corresponding to M is q ( µ ) = 0 for all
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MIT6_235S10_mid_sol - 6.253: Convex Analysis and...

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