Ch8convexsets - CHAPTER VIII CONVEX SETS We present some...

CHAPTER VIII CONVEX SETS
We present some geometric aspects of linear programming.
Definition. A subset S of ℜ
n
is said to be convex, if given x and y in S, then the line segment [ x,y ] connecting x and
y lies in S. The line segment [ x,y ] connecting x and y is the set of all points
[ x,y ] = { λ x + (1 - λ ) y | 0 ≤ λ ≤ 1}.
Picture of a Convex Set.
Example (1). Let A be an m x n matrix and let b be in ℜ
n
. Then the set
S = { x ∈ ℜ
n
| A x = b }
is a convex set. In fact, we have that A x = b and A y = b implies that
A( λ x + (1 - λ ) y ) = λ A x + (1 - λ )A y = t b + (1 - λ ) b = b,
for 0 ≤ λ ≤ 1. Thus, [ x, y ] ⊂ S.
Example (2). Let A be an m x n matrix and let b be in ℜ
n
. Then the set
S = { x ∈ ℜ
n
| A x = b, x ≥ 0 }
is a convex set. As in Example (1), we have that A x = b and A y = b implies that
A( λ x + (1 - λ ) y ) = λ A x + (1 - λ )A y = t b + (1 - λ ) b = b.
In addition we have that
λ x + (1 - λ ) y ≥ 0,
for 0 ≤ λ ≤ 1. Thus, [ x, y ] ⊂ S.
Example (3). Let A be an m × n matrix and let b be in ℜ
n
. Then the set
S = { x ∈ ℜ
n
| A x ≤ b }
is a convex set since
A( λ x + (1 - λ ) y ) = λ A x + (1 - λ )A y ≤ λ b + (1 - λ ) b = b,
for 0 ≤ λ ≤ 1.
Definition. A real-valued function f on a convex set S is said to be convex if
f( λ x + (1 - λ ) y ) ≤ λ f( x ) + (1 - λ )f( y ),
for all x and y in S.
Graph of a Real Valued Convex Function.
Chapter VIII Convex Sets page 1

Example of a Convex Function. Let c ∈
ℜ
n
and let f( x ) = c ⋅ x (dot product).
Then f is a convex function. In fact, the function f is linear and every linear function is convex.
Proposition. Let f be a convex function on a convex set S; then every local minimum for f is a global minimum.
Proof.Let f( x
o
) be a local minimum for f. By the definition of local minimum there is a d > 0 such that f( x
o
) ≤ f( x ) for
every x in S with || x -x
o
|| < δ . Now, for arbitrary y in S, we need to show that f( x
o
) ≤ f( y ) to show that f( x
o
)
is a global minimum. We use the convexity of S for this. In fact, we have that
|| x
o
- ( λ y + (1 - λ ) ) x
o
)|| ≤ | λ | || x
o
- y ||
and thus that
|| x
o
- (t y + (1-t) x
o
)|| < d
whenever 0 < λ < δ /(|| x
o
|| + || y || + 1). Thus, we have that
f( x
o
) ≤ f( λ y + (1 - λ ) x
o
) ≤ λ f( y ) +(1 - λ )f( x
o
),
or equivalently,
tf( x
o
) ≤ tf( y )
whenever 0 < t < d/(|| x
o
|| + || y || + 1). Thus, f( x
o
) ≤ f( y ). Q.E.D.
The Proposition shows that every local minimum in a Linear Programming Problem is a global minimum.
Definition. Let S be a convex set in ℜ
n
; then a point x in S is said to be an extreme point of S if given y and z in S
with x ∈ [ y, z ], then x = y = z.
Note that x is an extreme point of S if it is not properly inside any line segment in S.
Examples of Extreme Points.
Chapter VIII Convex Sets page 2

The extreme points of the Triangle The extreme points of the circle
are the vertices of the triangle. is the boundary of the circle
.
Definition. A set of the form
H = { x ∈ ℜ
n
| c ⋅ x = a}
is said to be a hyperplane in ℜ
n
. Here c is a nonzero vector in ℜ
n
and c ⟩ x represents the dot product of c and x .