hw3-s14-sol - Math 408 Homework Set 3 Solutions This homework set will focus on the optimization problem 1 Q minn xT Hx g T x xR 2 nn n where H R is

hw3-s14-sol - Math 408 Homework Set 3 Solutions This...

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Math 408 Homework Set 3 Solutions This homework set will focus on the optimization problemQminxRn12xTHx+gTx ,whereHRn×nis symmetric andgRn.(1) Each of the following functions can be written in the formf(x) =12xTHx+gTxwithHsymmetric.For each of these functions what areHandg(a)f(x) =x21-4x1+ 2x22+ 7 (c)f(x) =x21-2x1x2+12x22-8x2 (d)f(x) = (2x1-x2)2+ (x2-x3)2+ (x3-1)2 (e)f(x) =x21+ 16x1x2+ 4x2x3+x22 (2) Fora, bR, consider the matrixH=2a0a2b0b2.(a) Compute the eigenvalues ofHas functions ofaandb. (b) For what values ofaandbisHpositive definite.

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