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ELE 704 Optimization
HW4
Due 20 March 2007
1. Prove that
0
„
m
I
„
H
(
x
)
„
M
I
where
H
(
x
) is the Hessian matrix of a strongly convex function
f
(
x
) and
m
is the smallest and
M
is the largest eigenvalues of
H
(
x
).
2. Write the MATLAB code which performs the following line search algo
rithms for the function you used for HW3
(a) Exact Line Search
(b) Bisection Line Search
(c) Backtracking Line Search.
Try to be as much ﬂexible as possible. Write your code as a MAT
LAB function which can be called from another (possibly main) code
with some parameters. This means you may also need to write your
quadratic function, its gradient and its Hessian as a function also.
3. (a) Choose an arbitrary initial point. By assuming that the descent
direction is
d
=
∇
f
(
x
(0)
) using the MATLAB command ”surfc”
ﬁrst draw the cost surface then add the curve
h
(
α
) =
f
(
x
(0)
+
α
d
)
on top of the cost function.
(b) For
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