bv_cvxbook_extra_exercises

Since y is even you can take the sine coecients in

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Unformatted text preview: function is the average absolute deviation from the median of the values: n φaamd (x) = (1/n) i=1 |xi − med(x)|, where med(x) denotes the median of the components of x, defined as follows. If n = 2k − 1 is odd, then the median is defined as the value of middle entry when the components are sorted, i.e., med(x) = x[k] , the k th largest element among the values x1 , . . . , xn . If n = 2k is even, we define the median as the average of the two middle values, i.e., med(x) = (x[k] + x[k+1] )/2. Each of these functions measures the spread of the values of the entries of x; for example, each function is zero if and only if all components of x are equal, and each function is unaffected if a constant is added to each component of x. Which of these three functions is convex? For each one, either show that it is convex, or give a counterexample showing it is not convex. By a counterexample, we mean a specific x and y such that Jensen’s inequality fails, i.e., φ((x + y )/2) > (φ(x) + φ(y ))/2. 5.2 Minimax...
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