MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Outline #9 (Inequality Constraints)
Last modified: December 18, 2007
Reading.
Luenberger, Chapter 8. If you want to understand the term projects,
you should work through these simple examples, then skim the chapter.
This is all optional material, not required for the final exam!
Lecture topics.
1. Positive cones (Luenberger Section 8.2)
Luenberger wants to have several Lagrange multipliers, refer to them as a
single vector
z
*
, and say that
z
*
≥
0.
This can be done by introducing
a positive cone
P
. The simplest example (and the only one we will need
for the moment, is the “positive orthant,” the set of vectors in
E
n
whose
components are all nonnegative. Any vector in
p
is called positive.
How can you now assign a meaning to
x
≥
y
for vectors?
What properties of
≥
for real numbers carry over to vectors?
What important property of
≥
for real numbers does not carry over to
vectors?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Math, Vector Space, Convex function, lagrange multipliers, Convex Optimization, inequality constraint, Luenberger

Click to edit the document details