Recursive Methods
Introduction to Dynamic Optimization Nr. 1
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An introduction to dynamic optimization, focusing on the sequence problem and the bellman equation. It outlines the principle of optimality, which allows the use of the bellman equation to find the value function and optimal plan. The document also discusses why this approach is beneficial, including intuition, notation, analysis, and computation. It then presents a proof of theorem 4.3, which shows that the value function defined by the sequence problem is the same as the solution to the bellman equation. Finally, the document introduces the bellman equation as a fixed point problem and discusses the needed tools, including basic real analysis, the contraction mapping theorem, and the theorem of the maximum.
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2021/2022
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Recursive
Methods
Introduction
to
Dynamic
Optimization
Nr.
1
Outline
Today’s
Lecture
housekeeping:
ps#
and
recitation
day/
theory
general
web
page
fi
nish
Principle
of
Optimality:
Sequence
Problem
(for
values
and
plans)
solution
to
Bellman
Equation
begin
study
of
Bellman
equation
with
bounded
and
continuous
F
tools:
contraction
mapping
and
theorem
of
the
maximum
Introduction
to
Dynamic
Optimization
Nr.
2
Principle
of
Optimality
IDEA:
use
BE
to
fi
nd
value
function
V
∗
and
optimal
plan
x
∗
Thm
V
∗
de
fi
ned
by
SP
V
∗
solves
FE
Thm
V
solves
FE
and
V
V
∗
Thm
x
∗
x
0
is
optimal
V
∗
x
∗ t
F
x
∗ t
, x
∗ t
βV
∗
x
∗ t
Thm
x
∗
x
0
satis
fi
es
V
∗
x
∗ t
F
x
∗ t
, x
∗ t
βV
∗
x
∗ t
and
x
∗
is
optimal
Introduction
to
Dynamic
Optimization
Nr.
4
Why is this Progress?
intuition:
breaks
planning
horizon
into
two:
‘now’
and
‘then’
notation:
reduces
unnece
s
sary
notation
(especially
with
uncertainty)
analysis:
prove
existence,
uniqueness
and
properties
of
optimal
policy
(e.g.
continuity,
montonicity,
etc...)
computation:
associated
numerical
algorithm
are
powerful
for
many
applications
Introduction
to
Dynamic
Optimization
Nr.
5
Continue
this
way
V
x
0
n
X
t
=
β
t
F
x
t
, x
t
β
n
V
x
n
for
all
x
x
0
n
X
t
=
β
t
F
x
∗ t
, x
∗ t
β
n
V
x
∗ n
for
some
x
∗
x
0
Since
β
T
V
x
T
taking
the
limit
n
on
both
sides
of
both
expres
sions
we
conclude
that:
V
x
0
u
x
for
all
x
x
0
V
x
0
u
x
∗
for
some
x
∗
x
0
Introduction
to
Dynamic
Optimization
Nr.
7
Bellman
Equation
as
a
Fixed
Point
de
fi
ne
operator
T
f
x
max
y
∈
Γ
(
x
)
F
x,
y
βf
y
V
solution
of
BE
V
fi
xed
point
of
T
[i.e.
T V
V
]
Bounded
Returns:
if
k
F
k
< B
and
F
and
are
continuos:
T
maps
continuous
bounded
functions
into
continuous
bounded
functions
bounded
returns
T
is
a
Contraction
Mapping
unique
fi
xed
point
many
other
bonuses
Introduction
to
Dynamic
Optimization
Nr.
8