bv_cvxbook_extra_exercises

32 hello world in cvx use cvx to verify the optimal

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Unformatted text preview: T x = (y1 − y2 )x1 + (y2 − y3 )(x1 + x2 ) + (y3 − y4 )(x1 + x2 + x3 ) + · · · + (yn−1 − yn )(x1 + x2 + · · · + xn−1 ) + yn (x1 + x2 + · · · + xn ) and assume that the components of y are sorted in nonincreasing order. 11 (b) Show that x satisfies xT y ≤ SC (y ) for all y if and only if sk (x) ≤ sk (a ), k = 1, . . . , n − 1, sn (x ) = s n (a ), where sk denotes the function sk (x) = x[1] + x[2] + · · · + x[k] . When these inequalities hold, we say the vector a majorizes the vector x. (c) Conclude from this that the conjugate of SC is given by ∗ SC (x) = 0 if x is majorized by a +∞ otherwise. ∗ Since SC is the indicator function of the convex hull of C , this establishes the following result: x is a convex combination of the permutations of a if and only if a majorizes x. 12 3 Convex optimization problems 3.1 Minimizing a function over the probability simplex. Find simple necessary and sufficient conditions for x ∈ Rn to minimize a differentiable convex function f over the probability simplex, {x | 1T x = 1, x 0}. 3.2...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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