The understanding of the solution methods requires the proof of a theorem. To follow the
proof of the theorem we need to go through some definitions. These are introduced
below:
Definition 1.1
. A
convex combination
of
n
dimensional points
m
U
U
U
,...
,
2
1
is a point
m
m
U
U
U
U
...
2
2
1
1
where
i
are scalars,
m
i
i
i
1
.
1
and
0
1
Let us consider points
U
1
= (1, 1) and
U
2
= (4, 5) in 2dimensional case. Then the convex
combination of
2
1
and
U
U
is the line segment joining these points as shown by the bold
portion of the line in the following figure.
Y
(4.5)
(1, 1)
X
Figure 1.8 Graphical representation of convex combination of 2 points in a plane
The equation of the line passing through (1, 1) and (4,5) is 3y = 4x 1. Let us check it for
3
2
,
3
1
2
1
. Then
3
11
,
3
)
5
,
4
(
3
2
)
1
,
1
(
3
1
2
2
1
1
U
U
U
, which satisfy the
equation of the line. Check that it belongs to the line segment between (1,1) and (4,5).
Definition1.2
: A subset C of
R
n
is called a
convex set
if, for all pair of points
2
1
and
U
U
in C, any convex combination
2
1
)
1
(
U
U
U
is also in C. That is, if
the line segment joining any two points of a set wholly lies in the set, then this set is
called a convex set. Thus if we consider the set of all points on a triangle including the
boundary points, then this set is a convex set, because the line segment joining any two
points of the set also lies in the set. But the set of all boundary points of the triangle is not
a convex set (Why?). Examples of convex sets are the whole space
R
n
, a circle, a
rectangle, a square and a cube. Some convex and nonconvex sets are shown in the
following figure:
A portion of segments of the lines is outside of the set
Nonconvex
Figure1.9
: Examples of convex and nonconvex set
11
Conv
exx
Conv
ex
N
on

co
nv
ex
Definition 1.3:
A convex set C is called
unbounded
if for every point
U in C, a point
0
T
can be found such that the points
T
U
belongs to C for all
.
0
Otherwise C is
called
bounded
. Thus C is
bounded
if there is a point
U in C such that, for every
,
0
T
there exists a
for which
T
U
does not belongs to C. In the following figures
bounded and unbounded convex sets are shown by the shaded region and arrow head
lines respectively.
Y
Y
X
X
(a) Bounded convex set
(b) Unbounded convex set
Figure1.10
: Bounded and unbounded convex sets
Definition 1.4.
A point U in a convex set C is called an
extreme point
of the convex set
C, if it cannot be expressed as a convex combination of any other two distinct points
in C.
That is, if U does not lie on a line segment joining any other two distinct points of C.
Thus the vertices of a triangle are extreme points. The points on the circumference of a
circle are extreme points.
Definition1.5
: Now let us define the general
Linear Programming Problem
(
LPP
) for
minimization (maximization) problem. The purpose of an LPP is to find out a vector
X
)
,...
,...,
,
(
2
1
n
j
x
x
x
x
which minimizes (maximizes) the objective function
f
(
X
)
n
n
j
j
x
c
x
c
x
c
x
c
...
....