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then
r
i
=
a
i
+
b
i
,
a
i
and
d
integers,
n
i
=
1
b
i
|
|≤
/
=
b
i
1
2and
d
will be satisfied.
Theorem 6.
If d, given by (
3.40
) is equal to
0
, then the minimum of (
3.32
) subject
to (
3.33
) is obtained with the values
α
i
=
a
i
for i
=
1
,
2
,...,
n. Moreover, if every
one of the values b
i
given by (
3.39
) satisfies
|
b
i
| <
1
/
2
then there is no other point s
in which the minimum is attained.
Proof.
In fact, for any other point
s
=(
s
1
,
s
2
,...,
s
n
)
,
s
i
integers
s
i
≥
0
,
for
i
=
n
∑
i
=
1
,
2
,...,
n
,
and
s
i
=
s
, we can write
1
n
i
=
1
c
i
(
s
i
−
r
i
)
n
i
=
1
c
i
(
a
i
−
r
i
)
2
2
f
(
s
)
−
f
(
a
)=
−
n
i
=
1
c
i
(
s
i
−
a
i
−
b
i
)
n
i
=
1
c
i
b
i
,
2
=
−
but at least one of the integers
s
i
−
a
i
is different from zero, and for every one of the
terms in which
s
i
−
a
i
=
0,
|
s
i
−
a
i
−
b
i
|≥|
s
i
−
a
i
|−|
b
i
|≥
1
/
2
,
is satisfied, hence
2
2
b
i
(
s
i
−
a
i
−
b
i
)
≥
(
1
/
2
)
≥
and therefore
n
i
=
1
c
i
(
s
i
−
a
i
−
b
i
)
n
i
=
1
c
i
b
i
2
≥
(
)
which proves that
f
a
is the minimum.
Theorem 7.
If d defined by (
3.40
), is different from
0
let A and B be the sets
s
:
s
i
=
A
=
a
i
+
δ
i
,
i
=
1
,
2
,...,
n
,
δ
i
integers,
i
=
1
δ
i
=
d
n
δ
i
=
0
or sgn
δ
i
=
sgn d
,
,
where a
i
,
i
=
1
,
2
,...,
n, are defined by (
3.38
), and
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