1
15.093J/2.098J Optimization Methods
Assignment 6 Solutions
Exercise 6.1
These
functions
are
twice
differentiable;
therefore,
we
can
check
the
convexity
using
the
Hessian
matrices.
A
function
f
(
x
)
is
convex
if
its
Hessian
matrix
is
positive
semidefinite
for
all
x
.
The
definiteness
of
matrices
can
be
determined
by
using
the
definition,
eigenvalues
or
principal
minors.
i)
f
(
x
1
, x
2
) =
x
2
+ 2
x
1
x
2
−
10
x
1
+ 5
x
2
1
2
2
H
(
x
1
, x
2
) =
is
indefinite
(using
principal
minors),
thus
f
is
neither
concave
nor
convex.
2
0
2
2
ii)
f
(
x
1
, x
2
, x
3
) =
−
x
1
−
3
x
2
−
2
x
2
+ 4
x
1
x
2
+ 2
x
1
x
3
+ 4
x
2
x
3
⎛
3
⎞
−
2
4
2
H
(
x
1
, x
2
, x
3
) =
⎝
4
−
6
4
⎠
is
indefinite
(using
principal
minors
again),
thus
f
is
neither
convex
2
4
−
4
nor
concave.
−
(
x
1
+
x
2
)
iii)
f
(
x
1
, x
2
) =
x
1
e
H
(
x
1
, x
2
) =
x
1
−
2
x
1
−
1
e
−
(
x
1
+
x
2
)
is
not
a
definite
matrix
for
all
x
, thus
f
is
neither
convex
nor
x
1
−
1
x
1
concave.
1
iv)
f
(
x
1
, x
2
, x
3
) =
−
∑
3
i
=1
ln(
x
i
) +
x
1
+
e
x
2
for
x
i
>
0
We
can
prove
this
function
to
be
convex
by
computing
its
Hessian
matrix
and
showing
that
it
is
a
positive
1
definite
matrix.
The
second
way
is
to
show
that
f
is
the
sum
of
convex
functions,
namely
−
ln(
x
),
x
, and
x
e
.
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 Spring '10
 Bertsekas
 Numerical Analysis, Optimization, Convex function, hessian matrix

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