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Introduction
to
Convex
Constrained
Optimization
March 4, 2004
c
±
2004
Massachusetts
Institute
of
Technology.
1
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Overview
•
Nonlinear
Optimization
•
Portfolio
Optimization
•
An
Inventory
Reliability
Problem
•
Further
concepts
for
nonlinear
optimization
•
Convex
Sets
and
Convex
Functions
•
Convex
Optimization
•
Pattern
Classiﬁcation
•
Some
Geometry
Problems
•
On
the
Geometry
of
Nonlinear
Optimization
•
Classiﬁcation
of
Nonlinear
Optimization
Problems
•
Solving
Separable
Convex
Optimization
via
Linear
Optimization
•
Optimality
Conditions
for
Nonlinear
Optimization
•
A
Few
Concluding
Remarks
2
Nonlinear
versus
Linear
Optimization
Recall
the
basic
linear
optimization
model:
2
T
LP
:
minimize
x
c x
s.t.
T
a
1
x
≤
b
1
,
·
=
·
≥
.
.
.
T
a
m
x
≤
b
m
,
n
x
∈±
.
In
this
model,
all
constraints
are
linear
equalities
or
inequalities,
and
the
objective
function
is
a
linear
function.
In
contrast,
a
nonlinear
optimization
problem
can
have
nonlinear
functions
in
the
constraints
and/or
the
objective
function:
NLP
:
minimize
x
f
(
x
)
s.t.
g
1
(
x
)
≤
0
,
·
=
·
≥
.
.
.
g
m
(
x
)
≤
0
,
n
x
,
n
²
n
²
In
this
model,
we
have
f
(
x
):
± →±
and
g
i
(
x
,i
=1
,...,m
.
Below
we
present
several
examples
of
nonlinear
optimization
models.
3
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±
±
±
±
±
3
Portfolio
Optimization
Portfolio
optimization
models
are
used
throughout
the
ﬁnancial
investment
management
sector.
These
are
nonlinear
models
that
are
used
to
determine
the
composition
of
investment
portfolios.
Investors
prefer
higher
annual
rates
of
return
on
investing
to
lower
an
nual
rates
of
return.
Furthermore,
investors
prefer
lower
risk
to
higher
risk.
Portfolio
optimization
seeks
to
optimally
trade
oﬀ
risk
and
return
in
invest
ing.
We
consider
n
assets,
whose
annual
rates
of
return
R
i
are
random
vari
ables,
i
=1
,...,n
.
The
expected
annual
return
of
asset
i
is
µ
i
,
i
,
andsoi
fw
einv
e
s
taf
ra
c
t
ion
x
i
of
our
investment
dollar
in
asset
i
,
the
expected
return
of
the
portfolio
is:
n
T
µ
i
x
i
=
µ x
i
=1
where
of
course
the
fractions
x
i
must
satisfy:
n
T
x
i
=
e x
.
0
i
=1
and
x
≥
0
.
(Here,
e
is
the
vector
of
ones,
e
=(1
,
1
,...,
1)
T
.
The
covariance
of
the
rates
of
return
of
assets
i
and
j
is
given
as
Q
ij
=COV(
i, j
)
.
can
think
of
the
Q
ij
values
as
forming
a
matrix
Q
,
whereby
the
variance
of
portfolio
is
then:
n
n
n
n
COV(
i, j
)
x
i
x
j
=
Q
ij
x
i
x
j
=
x
T
Qx.
i
=1
j
=1
i
=1
j
=1
It
should
be
noted,
by
the
way,
that
the
matrix
Q
will
always
by
SPD
(symmetric
positivedeﬁnite).
4
±
²
²
The
“risk”
in
the
portfolio
is
the
standard
deviation
of
the
portfolio:
STDEV
=
x
T
Qx
,
and
the
“return”
of
the
portfolio
is
the
expected
annual
rate
of
return
of
the
portfolio:
T
RETURN
=
µ
x.
Suppose
that
we
would
like
to
determine
the
fractional
investment
values
x
1
,...,x
n
in
order
to
maximize
the
return
of
the
portfolio,
subject
to
meet
ing
some
prespeciﬁed
target
risk
level.
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This note was uploaded on 12/04/2011 for the course ESD 15.094 taught by Professor Jiesun during the Spring '04 term at MIT.
 Spring '04
 JieSun

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