1. Find the gradient (or Jacobian) and hessian of the following functions.
(a) f(x) = kxk44
with x 2 Rn.
(b) f(x) = ha; xi2 with x 2 Rn.
(c) f(x) = 1
2xTAx where A is an n n symmetric data matrix and x 2 Rn.
(d) f(x) = 1
2ky Axk22
where y 2 Rm, A 2 Rmn are data, and x 2 Rn.
(e) f(x) = 1
2kA xxT k2
F where A is an n n symmetric data matrix and x 2 Rn.
2. Answer the following problems
(a) Find all the critical points of the 2-dimensional function f(x1; x2) = (x21
1)2 + x22
.
(b) Find all the critical points of the 2-dimensional function f(x1; x2) = (x21
1)2+(x22
1)2.
(c) Compute the gradient rf(x) and Hessian r2f(x) of the Rosenbrock function
f(x) = 100(x2 x21
)2 + (1 x1)2
Prove that x? = [1; 1]T is the local minimizer of this function.
(d) Show that the function f(x) = 8×1 + 12×2 + x21
2×22
has only one stationary point,
and that it is neither a local maximum or local minimum.
(e) Show that the 2-dimensional function f(x1; x2) = (x2 x21
)2 x21
has only one sta-
tionary point, which is neither a local maximum nor a local minimum.
2
0 comments