Optimizing Range Norm of the Image Set of Matrix … 23
known as matrix interval which corresponds to interval matrix
n n
I A
× ε
∈ R and which is represented with
[ ]
. , A
A A
≈
Definition 1.10. It is defined that
{ [
] }
. ...,
, 2
, 1
; ...,
, ,
2 1
n i
I x
x x
x x
I
i T
n n
= ∈
| =
=
ε ε
R R
Set
n
I
ε
R can be considered as set .
1 ×
ε n
I R
The elements of
n
I
ε
R are called interval vector over
.
ε
R I
Interval vector x corresponds to vector interval
[ ]
, , x
x i.e.,
[ ]
. , x
x x
≈ The concept of interval min-plus algebra is defined in the same way with
interval max-plus algebra concepts. Following are the definitions of complete interval max-plus algebra and complete interval min-plus algebra [11].
Definition 1.11. Complete interval max-plus algebra is set
=
ε
R I
{ }
, ε′
ε
U R
I
[ ]
ε′ ε′
= ε′
,
that is completed with two operations ⊕ and
, ⊗
meanwhile complete interval min-plus algebra is set
{ }
ε =
ε′ ε′
U R
R I
I with the operations
⊕′ and . ⊗′ Furthermore,
ε′ ε
= R R
I I
are written as .
R I
In the same way as in the interval max-plus algebra and interval min-plus algebra, in complete interval max-plus algebra and complete interval min-
plus algebra, can be defined as the set of matrices in the size n
m × are
notated as .
n m
I
×
R
II. Main Results
In this section, the optimizing range norm of the image set of matrix over interval max-plus algebra is presented:
A. Minimizing the range norm of matrix image set over interval max- plus algebra
The following are definitions of the range norm of vector and the image set of matrix over interval max-plus algebra.
Siswanto, Ari Suparwanto and M. Andy Rudhito 24
Definition 2.1. Suppose that
m
I x
R ∈
with
[ ]
, , x
x x
≈ .
,
m
x x
R ∈
The function
[ ]
x x
x x
I δ
δ δ
= δ
, ,
min
is called the range norm of x. In other words, the range norm of x is interval
[ ]
, ,
, min
x x
x δ
δ δ
where minimum
, min
= δ
δ x
x of
x δ or
. x
δ
Definition 2.2. Given that matrix
n m
I A
× ε
∈ R with
[ ]
, , A
A A
≈
, ,
n m
A A
× ε
∈ R
it is then defined that
{ }
. Im
n
I p
p A
A
ε
∈ |
⊗ =
R Based on the above definitions, the problem of minimizing the range
norm of matrix image set can be formulated as follows:
Problem 2.1. Given that matrix
,
n m
I A
× ε
∈ R solve:
minimize b
I δ
subject to .
Im A b
∈
Definition 2.3. Given that matrix
m
I b
ε
∈ R with
[ ]
. , b
b b
≈ The
function b
I δ is said minimum if and only if
b δ and
b δ are minimum.
If the image vector has infinite components, the definition about the range norm and the problem of minimizing the range norm of matrix image
set can be formulated as follows:
Definition 2.4. Suppose that
m
I x
ε
∈ R with
[ ]
. , x
x x
≈
The function
[ ]
min ,
, I
x x
x x
δ =
δ δ
δ is the range norm of x, after only
considering the finite component.
Problem 2.2. Given that matrix
,
n m
I A
× ε
∈ R solve:
minimize I
b δ
subject to .
Im A b
∈
Definition 2.5. Given that matrix
m
I b
ε
∈ R with
[ ]
. , b
b b
≈ The
function I
b δ
is said minimum if and only if b
δ ~
and b
δ ~
are minimum.
Optimizing Range Norm of the Image Set of Matrix … 25
Minimizing the range norm if its image vector is finite
The first we investigate minimizing the range norm if its image vector is finite, i.e., Problem 2.1. The next lemmas, definition and theorems are
needed to answer Problem 2.1:
Lemma 2.6. Suppose that
n m
I A
× ε
∈ R with
[ ]
. , A
A A
≈ If
[ ]
ij
A A
= with
[ ]
ε ε
≠ ,
ij
A for each
, M
i ∈
; N
j ∈
[ ]
b b
b ,
≈ and
[ ]
, ≈
with
⊗′ ⊗
=
∗
A A
b
and
, ⊗′
⊗ =
∗
A A
b
then i
≤ b
and ii
=
i
b
for some .
M i
∈
Proof. Suppose that
[ ]
b b
b ,
≈ with
⊗′ ⊗
=
∗
A A
b
and ⊗
= A b
. ⊗′
∗
A Since
[ ]
ij
A A
= with
[ ]
ε ε
≠ ,
ij
A for each
M i
∈ and ,
N j
∈
[ ]
ij
A A
= with
ε ≠
ij
A for each
M i
∈
and N
j ∈ and
[ ]
ij
A A
= with
ε ≠
ij
A for each
M i
∈ and .
N j
∈ According to concept in max-plus
algebra, i
≤ b
and
, ≤
b
ii =
i
b and
=
i
b for some
. M
i ∈
Since
[ ]
b b
b ,
≈ and
[ ]
, ,
≈ i
, ≤
b ii 0
=
i
b for some
. M
i ∈
Lemma 2.7. If
m
I y
x R
∈ ,
and R
I a
∈ with
[ ]
, , x
x x
≈
[ ]
y y
y ,
≈ and
[ ]
, , a
a a
≈ then
i ,
y I
x I
y x
I δ
⊕ δ
≤ ⊕
δ ii
. x
a I
x I
⊗ δ
= δ
Siswanto, Ari Suparwanto and M. Andy Rudhito 26
Proof.
Since
[ ]
, , x
x x
≈
[ ]
y y
y ,
≈ and
[ ]
, , a
a a
≈
[ ]
y x
y x
y x
⊕ ⊕
≈ ⊕
, and
[ ]
. ,
x a
x a
x a
⊗ ⊗
≈ ⊗
Next, according to concept in max-plus algebra, i
y x
y x
δ ⊕
δ ≤
⊕ δ
and
, y
x y
x δ
⊕ δ
≤ ⊕
δ
ii
x a
x ⊗
δ =
δ
and
. x
a x
⊗ δ
= δ
Therefore, i
[ ]
y x
y x
y x
y x
I ⊕
δ ⊕
δ ⊕
δ =
⊕ δ
, ,
min
[ ]
y x
y x
y x
δ ⊕
δ δ
⊕ δ
δ ⊕
δ ≤
, ,
min
[ ] [ ]
y y
y x
x x
δ δ
δ ⊕
δ δ
δ =
, ,
min ,
, min
, y
I x
I δ
⊕ δ
= ii
[ ]
x x
x x
I δ
δ δ
= δ
, ,
min
[ ]
. ,
, min
x a
I x
a x
a x
a ⊗
δ =
⊗ δ
⊗ δ
⊗ δ
=
Definition 2.8. The interval matrix
,
n m
I A
×
∈ R
[ ]
A A
A ,
≈ is
considered as double
R I
-astic if A is double R -astic for each
[ ]
. , A
A A
∈
Theorem 2.9. The matrix
n m
I A
×
∈ R with
[ ]
A A
A ,
≈ is double
R I
-astic if and only if A is double R -astic.
Proof. It is known that
n m
I A
×
∈ R
with
[ ]
A A
A ,
≈ is double
R I
-astic. Based on the definition, A is double R -astic for each
[ ]
. , A
A A
∈ If
, A
A = then A is double R -astic. It is known that A is
double R -astic. Since A is the lower bound matrix for the matrix interval
[ ]
, , A
A
. A
A ≤
Therefore, since it is a matrix A double R -astic, matrix A
Optimizing Range Norm of the Image Set of Matrix … 27
double R -astic for each
[ ]
, , A
A A
∈ thus
n m
I A
×
∈ R double
R I
-astic is obtained.
~ The following theorem presents solution to Problem 2.1:
Theorem 2.10. Suppose that
n m
I A
× ε
∈ R is double
R I
-astic and
m a
I v
R ∈
is a vector whose each component is equal to .
R I
a ∈
Then
[ ]
b b
b ,
≈ is a solution for Problem 2.1 such that
a
v A
A b
⊗′ ⊗
≤
∗
and
.
a
v A
A b
⊗′ ⊗
=
∗
Proof. Suppose that
[ ]
, ,
A A
A ≈
[ ]
a a
a
v v
v ,
≈
and
[ ]
. , a
a a
≈
Since
n m
I A
× ε
∈ R is double
R
I
-astic,
m a
I v
R
∈
and
a
v
is a vector whose each component is equal to
, R
I a
∈
which means that: i Matrix
n m
A
× ε
∈ R
is double R -astic and
a
v
is a vector whose component is equal to
. R
∈ a
Based on concept in max-plus algebra,
⊗ A
a
v A
⊗′
∗
is a solution for Problem 1.1. ii Matrix
n m
A
× ε
∈
R is double
R -astic and
a
v
is a vector whose component is equal to
.
R ∈
a
Based on concept in max-plus algebra,
⊗ A
a
v A
⊗′
∗
is a solution for Problem 1.1. Therefore,
[ ]
b b
, with
a
v A
A b
⊗′ ⊗
≤
∗
and
a
v A
A b
⊗′ ⊗
=
∗
is a solution for Problem 2.1.
~
Minimizing the range norm if its image vector is not finite
To solve Problem 2.2, the following theorem is used:
Theorem 2.11.
Suppose that
,
n m
I A
× ε
∈ R
where
[ ]
ij
A A
= for each
M i
∈ and
N j
∈ with
[ ]
, ,
ε ε
≠
ij
A and
m a
I v
R ∈
is a vector whose each component is equal to constant interval
a
.
If
n
I x
ε
∈ R
such that ≤
′ b
Siswanto, Ari Suparwanto and M. Andy Rudhito 28
x A
⊗ is a solution for Problem
2.2,
then N
j ∈
∀ satisfies
: a.
[ ]
ε ε
=
,
j
x or
b.
[ ]
. ,
, min
j j
a j
a j
a j
v A
v A
v A
x ⊗′
⊗′ ⊗′
=
∗ ∗
∗
Proof.
Suppose that
[ ]
, , A
A A
≈
[ ]
a a
a
v v
v ,
≈ and
[ ]
, , a
a a
≈ Since
[ ]
ij
A A
= for each
M i
∈ and N
j ∈ with
[ ]
, ,
ε ε
≠
ij
A and
m a
I v
R ∈
is a vector whose each component is equal to constant interval a, it means that:
i Matrix
[ ]
ij
A A
= for each
M i
∈ and N
j ∈ with
ε ≠
ij
A and
m a
v R
∈ is a vector whose each component is equal to constant
. a
ii Matrix
[ ]
ij
A A
= for each
M i
∈ and N
j ∈ with
ε ≠
ij
A and
m a
v R
∈ is a vector whose each component is equal to constant .
a Then, suppose that
[ ]
, , x
x x
≈
[ ]
b b
b ′
′ ≈
′ ,
and
[ ]
. ,
j j
j
x x
x ≈
It is known that
n
I x
ε
∈ R
such that x
A b
⊗ =
′ is a solution for Problem 2.2,
which according to concept in max-plus algebra means: iii
n
x
ε
∈ R with x
A b
⊗ ≤
′ is the solution for Problem 1.2, so
N j
∈ ∀
satisfies: a
′. ε
=
j
x or
b ′.
. ,
min
j a
j a
j
v A
v A
x ⊗′
⊗′ =
∗ ∗
iv
n
x
ε
∈ R with x
A b
⊗ =
′ is the solution for Problem 1.2, so
N j
∈ ∀
satisfies: a
′. ε
=
j
x or
b ′.
.
j a
j
v A
x ⊗′
=
∗
Optimizing Range Norm of the Image Set of Matrix … 29
Thus, from i and iii and also from ii and iv, we have: If
n
I x
ε
∈ R such that
x A
b ⊗
= ′
is the solution for Problem 2.2, then N
j ∈
∀ satisfies:
a.
[ ]
ε ε
= ,
j
x or
b.
[ ]
. ,
, min
j j
a j
a j
a j
v A
v A
v A
x ⊗′
⊗′ ⊗′
=
∗ ∗
∗
B. Maximizing the range norm of matrix image set over interval max- plus algebra
