EE364a Homework 3 solutions

EE364a , Winter 2007-08 Prof. S. BoydEE364a Homework 3 , .. , fn:R Rbe given continuous functions. Weconsider the problem of approximatingf0as a linear combination off1, .. , fn. Forx Rn, we say thatf=x1f1+ +xnfnapproximatesf0with tolerance >0 overthe interval [0, T] if|f(t) f0(t)| for 0 t T. Now we choose a fixed tolerance >0 and define theapproximation widthas the largestTsuch thatfapproximatesf0over the interval [0, T]:W(x) = sup{T||x1f1(t) + +xnfn(t) f0(t)| for 0 t T}.Show thatWis show thatWis quasiconcave we show that the sets{x|W(x) }areconvex for all . We haveW(x) if and only if x1f1(t) + +xnfn(t) f0(t) for allt [0, ). Therefore the set{x|W(x) }is an intersection of infinitely manyhalfspaces (two for eacht), hence a convex of Gaussian cumulative distribution cumulative distribu-tion function of a Gaussian random variable,f(x) =1 2 Zx e t2/2dt,is log-concave.]

a component parallel to a and a component orthogonal to a: c = aλ+ ˆc, with aT ˆc= 0. • If λ > 0, the problem is unbounded below. Choose x = −ta, and let t go to infinity: cTx = −tcTa = −tλaTa → −∞ and aTx−b = −taTa−b ≤ 0 for large t, so x is feasible for large t. …

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