hw4_2007 - MS&E 211 Linear and Nonlinear Optimization Prof...

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MS&E 211 Fall 2007 Linear and Nonlinear Optimization Nov 1, 2007 Prof. Yinyu Ye Homework Assignment 4: Due 3:15pm Friday, Nov 9 There is a homework collecting box outside of professor Ye’s office (Terman 316) for you to submit your homework. No late homework accepted! Problem 1 Let X , A , B and C be defined as: A = { x = ( x 1 , x 2 ) : x 1 . 2 } , B = { x = ( x 1 , x 2 ) : | x 1 | ≥ 1 } , C = { x = ( x 1 , x 2 ) : | x 2 | ≥ 1 } , X = A ( B C ). a) Plot the region defined by X . Is it convex? b) Let f ( x ) be defined as: f ( x ) = x 1 + x 2 . Find all global, local minimum points constrained on set X . Problem 2 Consider the function f : R + R defined by: f ( x ) = 0 x = 0 x ln x x > 0 a) Is this function continuous? Does it have a minimizer under non-negativity constraint? Justify your answer. (Note that f ( x ) is a convex function.) b) An entropy optimization problem that is frequently used in information science has the following general form:
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Minimize i I f ( x i ) subject to i I a i x i = 1 x i 0 , i I Here I = { 1 , 2 , ...N } is the index set. a i ’s are all positive. Assume f ( x ) is defined as above in Part (a). What are the KKT conditions for this problem?
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