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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.
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- Fall '13
- The Aeneid