Lecture 1
Introduction to Optimization
UCSD Math 171A: Numerical Optimization
Philip E. Gill
(
pgill@ucsd.edu
)
Monday, January 5th, 2009
UCSD Center for Computational Mathematics
Slide 1/42, Monday, January 5th, 2009
What is optimization?
Webster’s dictionary:
Optimization
“
the process or method for making something
(design, system, decision) as fully perfect, functional or eﬀective as
possible
.”
UCSD Center for Computational Mathematics
Slide 2/42, Monday, January 5th, 2009
How do we optimize something?
Formulate a mathematical model of a given situation or resource
for which “optimizing” means minimizing or maximizing a function:
f
(
x
,
y
,
z
, . . .
)
the
”objective function”
subject to restrictions on the values that
x
,
y
,
z
, . . . , can take.
For example,
x
may need to be
nonnegative
, i.e.,
x
≥
0.
UCSD Center for Computational Mathematics
Slide 3/42, Monday, January 5th, 2009
The problems to be considered
In general we have
“constraint functions”
a
(
x
,
y
,
z
, . . .
)
≥
0
b
(
x
,
y
,
z
, . . .
)
≥
0
.
.
.
.
.
.
For example,
x
and
y
may need to be outside a circle, i.e.,
x
2
+
y
2

1
≥
0
UCSD Center for Computational Mathematics
Slide 4/42, Monday, January 5th, 2009