This preview shows pages 1–2. Sign up to view the full content.
Complementary pivoting
Carl Lemke created a sensation in the with a computational scheme that can be
thought of as “rewiring” the ideas in George Dantzig’s simplex method.
Lemke’s scheme is called “complementary pivoting.”
It solves linear programs,
quadratic programs (whatever they are) and bimatrix games.
It is easiest to introduce in
the context of a linear program and, in particular, a linear program that has been written
in this format:
LP.
max c x
,
subject to the constraints
(1)
A x
≤
b ,
x
≥
0 .
The dual to this linear program is:
DUAL
.
min y b ,
subject to the constraints
(2)
y A
≥
c ,
y
≥
0 .
Solutions to (1) and (2) are known to be optimal if and only if they satisfy the
condition that is known as
complementary slackness
.
To describe complementary slackness compactly, we insert slack variables in
A x ≤
b ,
surplus variables in
y A
≥
c
and then multiply the latter by minus 1.
This
places (1) and (2) in the equivalent form
(1)
A x
+
s
=
b ,
x
≥
0 ,
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 '10
 ERICDENARDO

Click to edit the document details