Math 164: Homework #8, due on Wednesday, May 27
[1] (read Appendix B5) Consider the problem
minimize f(x) = 1
2xTQx −cTx.
(a) Write the first-order necessary condition. When does a stationary point of fexist ?
(b) Under what conditions on Qdoes a local minimizer exist ?
(c) If Qis a positive definite matrix, prove that Newton’s method will determine the
minimizer of fin one iteration, regardless of the starting point.
[2] Consider the problem
min
xkAx −bk2
2,
where Ais an m×nmatrix with m≥n, and bis a vector of length m. Assume that the
rank of Ais equal to n.
(a) Write down the first-order necessary condition for optimality. Is this also a sufficient
condition ?
(b) Write down the optimal solution in closed form.
[3] Compute a basis matrix for the null space of the matrix Aand express the point xas
x=p+q, where pis in the null space of Aand qis in the range space of AT:
A=
1 1 1 1
1−1−1 1
0 1 0 1
,x=
1
3
1
2
.
[4] Consider the problem
minimize f(x) = x2
1+x2
1x2
3+ 2x1x2+x4
2+ 8x2
subject to 2x1+ 5x2+x3= 3.
(a) Determine which of the following points are stationary points:
(i) (0,0,2)T; (ii) (0,0,3)T; (iii) (1,0,1)T
(b) Determine whether each stationary point is a local minimizer, a local maximizer or
a saddle point.
[5] Solve the problem
maximize f(x) = x1x2x3
subject to x1
a1
+x2
a2
+x3
a3
= 1 (a1, a2, a3>0)
1