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11_hw4 - MS&E 211 Fall 2011 Linear and Nonlinear...

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MS&E 211 Fall 2011 Linear and Nonlinear Optimization Nov 1, 2011 Prof. Yinyu Ye Homework Assignment 4: Due 6:00pm Thursday, Nov 10 No late homework accepted! Problem 1 [20 points] a) [10 points] Consider the function f : R + R defined by f ( x ) = { 0 x = 0 x ln x x > 0 Is this function continuous? Is it convex? Does it have a minimizer on the positive real line? Justify your answer. b) [10 points] An entropy optimization problem that is frequently used in information science has the following general form: 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. Assume f ( x ) is defined as in Part (a). What are the KKT conditions for this problem? Some motivating background. According to the principle of maximum entropy, if noth- ing is known about a probability distribution except that it belongs to a certain class, then the distribution with the largest entropy should be chosen as the default. This is because maximizing entropy minimizes the amount of prior information built into the distribution.
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