EE364a Homework 3 solutions - Stanford Engineering …

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. This follows from the general result that theconvolution of two log-concave functions is log-concave.]

All feasible solutions are optimal. • The problem is feasible, and c is not in the range of AT (ˆc 6= 0). The problem is unbounded (p⋆ = −∞). To verify this, note that x = x 0 −tˆcis feasible for all t; as t goes to infinity, the objective value decreases unboundedly. In summary, p⋆ = +∞ b ∈ R(A) λTb c = ATλ for some λ

Loading..

Tags:

  Solutions, Homework

Information

Related search queries