bv_cvxbook_extra_exercises

# Ak coskt f t k1 k0 we consider two approximations

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Unformatted text preview: han or equal to zero. (a) Show that (19) implies that t − ( x − bi ) T ( x − b j ) ≤ − ( y − a i ) T ( y − a j ) for i, j ∈ I, if (x, t) is feasible in (20), and I ⊆ {1, . . . , m} is the set of active constraints at x, t. (b) Suppose x, t are optimal in (20) and that λ1 , . . . , λm are optimal dual variables. Use the optimality conditions for (20) and the inequality in part a to show that m m t=t− i=1 λ i ( x − bi ) 38 2 2 ≤− i=1 λi ( y − a i ) 2 . 2 5 Approximation and ﬁtting 5.1 Three measures of the spread of a group of numbers. For x ∈ Rn , we deﬁne three functions that measure the spread or width of the set of its elements (or coeﬃcients). The ﬁrst function is the spread, deﬁned as φsprd (x) = max xi − min xi . i=1,...,n i=1,...,n This is the width of the smallest interval that contains all the elements of x. The second function is the standard deviation, deﬁned as 1 φstdev (x) = n n i=1 x2 − i 1 n 2 n xi i=1 1/2 . This is the statistical standard deviation of a random variable that takes the values x1 , . . . , xn , each with probability 1/n. The third...
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