Math 414 Lecture 3
THEOREM. For linear functions on nonempty closed bounded
convex sets:
(a) Absolute maximas and minimas always exist.
(b) Local maximas and minimas are absolute maximas and
minimas.
CUTTING PLANE METHOD
Convert the problem to standard form with integer
coefficients. Solve.
If a primal variable is not integral, cut it off with add a new
integer-coefficient constraint.
Cutting it o
Math 414 Lecture 15
One step of the Simplex method makes feasible
(constant-column is > 0) but nonoptimal solutions (objective row
has a negative coefficient) more optimal.
The dual method makes super
Math 414 Lecture 14
Exam 2 covers Lectures 8-14
Find an optimal solution for the primal and dual.
PRIMAL
min z = x y2u
r: x u = 0
s: -x y > -3
t: x + y > 1
x, y, u > 0
Write the canonical version with
Math 414 Lecture 13
FROM LAST LECTURE.
DUAL
PRIMAL PROBLEM CANONICAL+EXTRAS
max z = C:X
max z = C:X
min z = B:W
ATW > C
AX ? B
AX + . = B
X >0
X, ., > 0
W?0
Where ? is some combination of <, >, =.
The
Math 414 Lecture 21.31
Exam 3 covers Lectures 15 - 21.
In a transportation problem, suppose the total supply s
produced exceeds the total demand d, d < s.
Solution: cut back on the supply produced.
Sh
Math 414 Lecture 17
Integer Programming
In many problems the variables must range over integers:
the number of people you must hire, the number of chairs
you must rent, a price in cents, . .
DEFINITIO
Math 414 Lecture 26
LEMMA. The amount I can expect to win by consistently
playing the strategy for a given row is the minimum payoff
for that row.
THEOREM. The strategy for I with the maximum expected
Math 414 Lecture 24
LEMMA.
v The distance from the origin to another node = the
minimum of the distances to the nodes edges.
v If the estimated distance of a cut edge is < the estimated
distances of o
Math 414 Lecture 22
Not included in Exam 3.
ASSIGNMENT ALGORITHM
Input: an nxn cost matrix [cij ].
Output: a minimum cost assignment [xij ].
Row step. From each row, subtract the row minimum.
teach ro
Math 414 Lecture 23
Maximum Flows
`The amount of water flowing into node x increases by 5.
How much can the flow into adjacent nodes increase? The
upper number = the capacity; the lower = the flow.
4
Math 414 Lecture 16
Constant column sensitivity
Let B(brp) be the original constant column B with the
constant br replaced by a new variable p.
Let B' be the constant column (and hence solution) of th
Math 414 Lecture 19
All linear inequalities can be rewritten in standard form:
ax + by < c.
`y-3 < 2x
0 < x+y < 5
is equivalent to 2x y 3.
can be written as 2 linear inequalities.
In the problems belo
Math 414 Lecture 4
THEOREM. Every general linear programming problem can be
rewritten in standard and in canonical form.
PROOF.
`Rewriting min in terms of max.
x minimizes f (x) iff x maximizes -f (x)
Math 414 Lecture 1
Homework assignments are always due at the beginning of the next lecture.
Operations Research studies linear programming and associated
algorithms. Businesses use it to find resourc
Math 414 Lecture 6
CONTINUING ASSUMPTION.
AX = b, X > 0 is a canonical problem with
n variables (columns),
m independent equations (rows) and
k= n m = the number of parameters
LEMMA.
Every basic solut
Math 414 Lecture 5
Word problems Write as a general linear programming problem.
`Golf carts are made in Detroit and Newark and shipped to
dealers in Miami, Houston and LA.
Production:
v Detroit can ma
Math 414 Lecture 2 Everyone have a laptop?
THEOREM. Let v1, .,vk be k vectors in an n-dimensional space
and A = [v1; . ; vk]
v1, ., v k independent
k<n
v1, ., v k span the space
n<k
v1, ., v k a basis
Math 414 Lecture 8
`Solve by geometry.
Max z = x + y
with r: 2x + y < 6
s: x +2y < 6,
6
(0,3)
s
x
(0,0)
x, y > 0.
(2,2)
r
y (3,0)
6
Run the simplex algorithm by looking at the picture.
At each stage w
Math 414 Lecture 7
Exam 1 a week from today. Practice Exam 1 also due then.
For today, assume all BASIC variables are positive. Later we
consider the degenerate case where a basic variable is 0.
CONTI
Math 414 Lecture 10
RECALL:
Upward constraints dual variables > 0,
Downward constraints dual variables < 0,
equalities unrestricted dual variables.
The original problem is called the primal problem.
F
Math 414 Lecture 9
TWO PHASE METHOD
If our matrix has a set of m independent identity columns
and the constant column b is > 0, then the matrix is in tableau
form and we can start the simplex algorith
Math 414 Lecture 11
DOUBLE DUAL THEOREM. The dual of a dual is the original
primal problem.
PROOF FOR STANDARD CASE.
DUAL
DUAL OF DUAL
PRIMAL
max z = C:X
min z = B:W
max z = C:X
with
with
with
W: ATTX
Math 414 Lecture 12
`You have 100 lbs. of flour and 60 lbs. of sugar.
1 lb. of cookies = .6 lbs. of flour + .4 lbs. of sugar.
1 lb. of candy
=
1 lb. of sugar.
1 lb. of crackers = .8 lbs. of flour + .2
Math 414 Lecture 25
Job scheduling
A project requires completing several jobs.
Some are independent, some must precede others.
Find the minimum time required to complete the project.
We model this pro