Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 5, Article 158, 2006
MAXIMIZATION FOR INNER PRODUCTS UNDER QUASI-MONOTONE
CONSTRAINTS
KENNETH S. BERENHAUT, JOHN D. FOLEY, AND DIPANKAR BANDYOPADHYAY
D EPARTMENT OF M ATHEMATICS
WAKE F OREST U NIVERSITY
W INSTON -S ALEM , NC 27106
berenhks@wfu.edu
URL: http://www.math.wfu.edu/Faculty/berenhaut.html
D EPARTMENT OF M ATHEMATICS
WAKE F OREST U NIVERSITY
W INSTON -S ALEM , NC 27106
folejd4@wfu.edu
D EPARTMENT OF B IOSTATISTICS , B IOINFORMATICS AND E PIDEMIOLOGY
M EDICAL U NIVERSITY OF S OUTH C AROLINA
C HARLESTON , SC 29425
bandyopd@musc.edu
Received 19 August, 2006; accepted 04 September, 2006
Communicated by L. Leindler
A BSTRACT. This paper studies optimization for inner products of real vectors assuming monotonicity properties for the entries in one of the vectors. Resulting inequalities have been useful recently in bounding reciprocals of power series with rapidly decaying coefficients and in
proving that all symmetric Toeplitz matrices generated by monotone convex sequences have offdiagonal decay preserved through triangular decompositions. An example of an application of
the theory to global optimization for inner products is also provided.
Key words and phrases: Inner Products, Recurrence, Monotonicity, Discretization, Global Optimization.
2000 Mathematics Subject Classification. 15A63, 39A10, 26A48.
1. I NTRODUCTION
This paper studies inequalities for inner products of real vectors assuming monotonicity and
boundedness properties for the entries in one of the vectors. In particular, for r ∈ (0, 1], we
consider inner products p·q, for vectors p = (p1 , p2 , . . . , pn ) and q = (q1 , q2 , . . . , qn ), satisfying
p, q ∈ Rn , pi ∈ [A, B] for 1 ≤ i ≤ n, and one of the following properties
(1) (r-quasi-monotonicity) pi+1 ≥ rpi for 1 ≤ i ≤ n − 1.
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
The first author acknowledges financial support from a Sterge Faculty Fellowship.
218-06
2
K ENNETH S. B ERENHAUT , J OHN D. F OLEY , AND D IPANKAR BANDYOPADHYAY
(2) (r-geometric monotonicity) pi+1 ≥ 1r pi for 1 ≤ i ≤ n − 1.
(3) (monotonicity) pi+1 ≥ pi for 1 ≤ i ≤ n − 1.
For discussion of various classes of sequences of monotone type, see for instance, Kijima [12],
and Leindler [15, 14].
Our method involves, for each of the three cases mentioned, obtaining finite sets Pn =
Pn (A, B, r) such that
min{v · q : v ∈ Pn } ≤ p · q ≤ max{v · q : v ∈ Pn },
for all p satisfying the respective monotonicity assumption, above.
The paper proceeds as follows. In Section 2, we consider obtaining the sets Pn corresponding
to Property (1), above. An application to linear recurrences, which has been useful in the recent
literature is also given. In Section 3, we consider the case of r-geometric monotonicity. The
paper includes examples which provide an application of the theory to global optimization for
inner products, for a specific vector q.
2. T HE C ASE
OF
r- QUASI - MONOTONICITY
In this section we consider the assumption of r-quasi-monotonicity of the entries in p =
(p1 , p2 , . . . , pn ) (as defined in (1), above), i.e.
pi+1 ≥ rpi
(2.1)
for 1 ≤ i ≤ n − 1. The motivation for consideration of such a condition arose in a probability
related context of investigating a monotone sequence {qi } with a geometric bound, i.e.
qi ≤ Ari
where A > 0 and r < 1 (see [2]). In this case the sequence {φi } defined by φi = rqii , satisfies
0 ≤ φi ≤ A, and
qi
qi+1
φi = i ≥ i = φi+1 r.
r
r
For a given vector t = (t0 , t1 , t2 , . . . , tk ) satisfying t0 ≥ 0, ti ≥ 1 for 1 ≤ i ≤ k and
X
(2.2)
ti = N,
i
define the vector v t via
t
z }|0 { 0 1
(2.3)
v t = A 0, 0, . . . , 0; r , r , . . . , rt1 −1 ; r0 , r1 , . . . , rt2 −1 ; r0 , r1 , . . . , rtk −1 .
In addition, define the set of vectors
def
(2.4)
PN = PN (A, 0, r) = {v t : t satisfies (2.2) }.
We have the following result regarding inner products.
Theorem 2.1. Suppose that p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ) are n-vectors where p
satisfies (2.1), for 1 ≤ i ≤ n − 1 and 0 ≤ pi ≤ A for 1 ≤ i ≤ n. We have,
min{w · q : w ∈ Pn } ≤ p · q ≤ max{w · q : w ∈ Pn }
P
where p · q denotes the standard dot product ni=1 pi qi .
(2.5)
The value in Theorem 2.1 lies in the fact that for any given n, Pn is a finite set.
For a vector p = (p1 , p2 , . . . , pn ), we will use the notation pi,j to indicate the vector consisting of the ith through j th entries in p, i.e.
(2.6)
pi,j = (pi , pi+1 , . . . , pj )
J. Inequal. Pure and Appl. Math., 7(5) Art. 158, 2006
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M AXIMIZATION FOR I NNER P RODUCTS
3
Proof of Theorem 2.1. First, suppose p · q > 0, and note that the lower bound in (2.5), for such
vectors, follows from the fact that v t = 0 for t = (n, 0, . . . , 0). We will obtain a sequence of
vectors {pei }n+1
i=1 , satisfying
0 ≤ p · q = pen+1 · q ≤ pen · q ≤ · · · ≤ pe1 · q,
such that pe1 ∈ Pn .
In particular, consider the vectors pei = (pei (1), pei (2), . . . , pei (n)) ∈ Rn , i = 1, . . . , n + 1
defined recursively according to the following scheme.
(1) pen+1 = p.
(2) For 1 ≤ i ≤ n, set
[
Si = {s : i + 1 ≤ s ≤ n and pei+1 (s) = A}, and vi = min Si {n + 1} .
(3) For 1 ≤ i ≤ n, define pei (a function of pei+1 ) via
pei = pei+1 (1), pei+1 (2), . . . , pei+1 (i − 1), ci pei+1 (i), ci pei+1 (i + 1), . . . , ci pei+1 (vi − 1),
A, pei+1 (vi + 1), . . . , pei+1 (n)
(2.7)
(2.8)
= (w1i+1 ; ci w2i+1 ; w3i+1 ),
say, where ci is given by

A


,
if w2i+1 · q i,vi −1 > 0


p

 i
rpi−1
ci =
, if w2i+1 · q i,vi −1 ≤ 0 and i > 1 .


p

i



0,
otherwise
Note that pei+1 = (w1i+1 , w2i+1 , w3i+1 ).
It is not difficult to verify by induction that wji+1 , j = 1, 2, 3, are of the form
(2.9)
w1i+1 = pe1,i−1
i+1 = (p1 , p2 , . . . , pi−1 )
(2.10)
i −1
w2i+1 = pei,v
= (pi , rpi , r2 pi , . . . , rvi −i−1 pi )
i+1
(2.11)
i ,n
w3i+1 = pevi+1
∈ Pn−vi +1 ,
We have that (2.7) and (2.8) imply
pei · q − pei+1 · q = (ci − 1)(w2i+1 · q n−i,vi −1 ) ≥ 0,
and, for 1 ≤ i ≤ n + 1,
(2.12) pei ∈ (p1 , p2 , . . . , pi−1 , rpi−1 , r2 pi−1 , r3 pi−1 , . . . , rvi −i pi−1 ; w3i+1 ),
(p1 , p2 , . . . , pi−1 , A, rA, r2 A, . . . , rvi −i−1 A; w3i+1 ) .
Thus vi−1 ∈ {vi , i}, and in particular, for i = 2, we have
(2.13)
pe2 ∈ (p1 , rp1 , r2 p1 , r3 p1 , . . . , rv2 −2 p1 ; w33 ), (p1 , A, rA, r2 A, . . . , rv2 −3 A; w33 ) .
The vector pe1 then satisfies
(2.14) pe1 ∈ (0, 0, . . . , 0; w33 ), (A, rA, r2 A, r3 A, . . . , rv2 −2 A; w33 ),
(A, A, rA, r2 A, . . . , rv2 −3 A; w33 ), (0, A, rA, r2 A, . . . , rv2 −3 A; w33 ) ⊂ Pn
and the theorem is proven in this case. The proof follows similarly, if p · q ≤ 0, and the proof
of the theorem is complete.
J. Inequal. Pure and Appl. Math., 7(5) Art. 158, 2006
http://jipam.vu.edu.au/
4
K ENNETH S. B ERENHAUT , J OHN D. F OLEY , AND D IPANKAR BANDYOPADHYAY
0
−4
−2
q_i
2
4
q vector
2
4
6
8
10
12
14
i
Figure 2.1: The vector q in (2.15).
The following example provides an application of Theorem 2.1 to global optimization for
inner products, for a specific vector q.
Example 2.1. Consider the vector q ∈ R15 given by
(2.15) q = 0.4361725, 0.6454718, 2.0226176, −4.1395363, 0.9749134,
4.3806500, −4.0035597, 0.6773984, −3.7420053, −2.7051776,
3.8209032, 0.6327872, 1.4719490, 1.2277661, 4.1026365 .
The entries in q are depicted in Figure 2.1. Now, consider optimizing p · q over all p =
(p1 , p2 , . . . , p15 ) ∈ R15 , satisfying 0 ≤ pi ≤ 1 and (2.1) for some 0 < r ≤ 1. Theorem
2.1 implies that we need only compute and compare inner products with q over the finite set
P15 (1, 0, r) as given in (2.4).
The results of the computations for r ∈ {.1, .3, .7, .9}, are given in Figure 2.2.
It is possible to apply Theorem 2.1 in sequence to obtain bounds for linear recurrences, as is
shown by the following theorem.
Theorem 2.2. Suppose that {bi } and {αi,j } satisfy
(2.16)
bn =
n−1
X
(−αn,k )bk ,
n ≥ 1,
k=0
where b0 = 1 and
(2.17)
αn,k ∈ [0, A],
for 0 ≤ k ≤ n − 1 and n ≥ 1, and
(2.18)
rαn,k ≤ αn,k+1 .
0
Then, there exist {b0i } and {αi,j
} such that
|bn | ≤ |b0n |
J. Inequal. Pure and Appl. Math., 7(5) Art. 158, 2006
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M AXIMIZATION FOR I NNER P RODUCTS
5
0.0
6
8
10
12
14
2
6
8
10
0.3
minimal
−12.437
0.3
maximal
17.21
p_i
12
14
12
14
12
14
12
14
0.0
0.8
i
6
8
10
12
14
2
4
6
8
10
i
i
0.7
minimal
−7.343
0.7
maximal
12.684
0.0
0.2
p_i
0.8
4
0.8
2
4
6
8
10
12
14
2
4
6
8
10
i
i
0.9
minimal
−2.38
0.9
maximal
11.256
0.0
0.0
p_i
0.8
2
0.8
p_i
4
i
0.8
4
0.0
p_i
2
p_i
0.8
p_i
0.8
0.1
maximal
19.177
0.0
p_i
0.1
minimal
−13.991
2
4
6
8
10
12
14
2
4
i
6
8
10
i
Figure 2.2: Maximal and minimal values for inner products under the constraint in (2.1)
and
(2.19)
b0n
=
n−1
X
0
(−αn,k
)b0k ,
n ≥ 1,
k=0
with each vector
0
0
0
α0i = (αi,0
, αi,1
, . . . , αi,i−1
) ∈ Pi ,
for 1 ≤ i ≤ n, where Pi is as in (2.4).
In fact, there exists a set {α01 , α02 , . . . , α0n }, with α0i ∈ Pi , such that b0i is as large as possible
(with its inherent sign) given b0 , b01 , b02 , . . . , b0i−1 .
Remark 2.3. While Theorem 2.2 looks relatively simple, it has proven indispensable recently
in two quite unrelated interesting contexts. The theorem was crucial, in proving a recent optimal
explicit form of Kendall’s Renewal Theorem (see Berenhaut, Allen and Fraser [2]) stemming
from bounds on reciprocals of power series with rapidly decaying coefficients. In a quite unrelated context, a simpler version of Theorem 2.2 was also employed in Berenhaut and Bandyopadhyay [3] in proving that all symmetric Toeplitz matrices generated by monotone convex
sequences have off-diagonal decay preserved through triangular decompositions.
J. Inequal. Pure and Appl. Math., 7(5) Art. 158, 2006
http://jipam.vu.edu.au/
6
K ENNETH S. B ERENHAUT , J OHN D. F OLEY , AND D IPANKAR BANDYOPADHYAY
Proof of Theorem 2.2. The proof, here, involves applying Theorem 2.1 to successively “scale"
the rows of the coefficient matrix


−α1,0
0
···
0


..
 −α2,0 −α2,1 . . .

.


[−αi,j ] = 
.
..
..
..


.
.
.
0


−αn,0 −αn,1 · · ·
−αn,n−1
while not decreasing the value of |bn | at any step.
First, define the sequences
ᾱi = (αi,0 , . . . , αi,i−1 )
and
bk,j = (bk , . . . , bj ),
for 0 ≤ k ≤ j ≤ n − 1 and 1 ≤ i ≤ n.
Suppose that bn > 0. Expanding via (2.16), bn can be written as
bn = C10 b0 + C11 b1 ,
(2.20)
0
)∈
where C10 and C11 are constants, which depend on {αi,j }. If C11 > 0, then select ᾱ01 = (α1,0
0,0
0
0
1
0
P1 so that −ᾱ1 ·b is maximal, via Theorem 2.1. Similarly, if C1 < 0, select ᾱ1 = (α1,0 ) ∈ P1
0
in (2.16) will result in a larger
so that −ᾱ01 ·b0,0 is minimal. In either case, replacing α1,0 by α1,0
1
(or equal) value for C1 b1 , and in turn, referring to (2.20), a larger (or equal) value of |bn |.
Now, suppose that the first through (k − 1)th rows of the α matrix are of the form described in
the theorem (i.e. resulting in maximal bi values for 1 ≤ i ≤ k − 1 with respect to the preceeding
bj , 0 ≤ j ≤ i − 1), and express bn in the form
(2.21)
bn = Ck0 b0 + Ck1 b1 + · · · + Ckk bk ,
via (2.16). If Ckk ≥ 0, then select ᾱ0k ∈ Pk so that −ᾱ0k · b0,k−1 is maximal, via Theorem 2.1.
Similarly, if Ckk < 0, select ᾱ0k ∈ Pk so that −ᾱ0k · b0,0 is minimal. In either case, referring to
(2.21), replacing the values in ᾱk by those in ᾱ0k in (2.16) will not decrease the value of |bn |.
The result follows by induction for this case. The case bn < 0 is similar and the theorem is
proven.
For further results along these lines in the case r = 1 and B = 0, see [4].
Note that, recurrences with varying or random coefficients have been studied by many previous authors. For a partial survey of such literature see Viswanath [22] and [23], Viswanath and
Trefethen [24], Embree and Trefethen [10], Wright and Trefethen [26], Mallik [16], Popenda
[18], Kittapa [13], Odlyzko [17], Berenhaut and Goedhart [6, 7], Berenhaut and Morton [9],
Berenhaut and Foley [5], and Stević [19, 20, 21] (and the references therein). For a comprehensive treatment of difference equations and inequalities, c.f. Agarwal [1].
We now turn to consideration of the remaining cases of r-geometric decay and monotonicity
mentioned in the introduction.
3. T HE C ASE
OF
r- GEOMETRIC M ONOTONICITY
In this section we consider the assumption of r-geometric monotonicity of the entries in
p = (p1 , p2 , . . . , pn ), i.e.
1
pi+1 ≥ pi
r
for 1 ≤ i ≤ n − 1.
J. Inequal. Pure and Appl. Math., 7(5) Art. 158, 2006
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M AXIMIZATION FOR I NNER P RODUCTS
7
First, for a given integer 0 ≤ t ≤ n, define the vector v t via v0 = 0, and
n−t
def
vt =
z }| {
0, 0, . . . , 0, Art−1 , Art−2 , . . . , Ar, A .
In addition, define the set of vectors
(3.1)
Pn2 = Pn2 (A, 0, r) = {v t : 0 ≤ t ≤ n}.
Here, we have the following theorem.
Theorem 3.1. Suppose that p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ) are n-vectors where p
satisfies
1
(3.2)
pi+1 ≥ pi
r
for 1 ≤ i ≤ n − 1, and 0 ≤ pi ≤ A for 1 ≤ i ≤ n. We have,
min{w · q : w ∈ Pn } ≤ p · q ≤ max{w · q : w ∈ Pn }.
Proof. First, suppose p · q > 0, and note that the lower bound in (2.5) follows from the fact that
v t = 0 for t = 0. As in the proof of Theorem 2.1, we will, again, obtain a sequence of vectors
{pei }n+1
i=1 , satisfying
0 ≤ p · q = pen+1 · q ≤ pen · q ≤ · · · ≤ pe1 · q,
2
such that pe1 ∈ Pn .
In particular, consider the vectors pei = (pei (1), pei (2), . . . , pei (n)) ∈ Rn , i = 1, 2, . . . , n + 1
defined recursively according to the following scheme.
(1) pen+1 = p.
(2) For 1 ≤S i ≤ n, set Si = {s : i + 1 ≤ s ≤ n and pei+1 (s) = Arn−s }, and vi =
min(Si {n + 1}).
(3) For 1 ≤ i ≤ n, set
e
pi = pei+1 (1), pei+1 (2), . . . , pei+1 (i − 1), ci pei+1 (i), ci pei+1 (i + 1), . . . , ci pei+1 (vi − 1),
pei+1 (vi ), pei+1 (vi + 1), . . . , pei+1 (n)
(3.3)
(3.4)
= (w1i+1 ; ci w2i+1 ; w3i+1 ),
where ci is given by

Arn−i


, if w2i+1 · q i,vi −1 > 0


p

i

1
p
ci =
.
r i−1

, if w2i+1 · q i,vi −1 ≤ 0 and i > 1


pi



0,
otherwise
It is not difficult to verify by induction that wji+1 , j = 1, 2, 3, are of the form
(3.5)
(3.6)
(3.7)
w1i+1 = pe1,i−1
i+1 = (p1 , p2 , . . . , pi−1 )
1
1
1
1,vi −1
2
wi+1 = pei+1 = pi , pi , 2 pi , . . . , vi −i−1 pi
r r
r
vi ,n
3
n−vi
n−vi −1
wi+1 = pei+1 = (Ar
, Ar
· · · , Ar, A) ∈ Pn−vi +1 .
Now, note that from (3.2), and the bound pn ≤ A, we have that
pi ≤ Arn−i ,
J. Inequal. Pure and Appl. Math., 7(5) Art. 158, 2006
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8
K ENNETH S. B ERENHAUT , J OHN D. F OLEY , AND D IPANKAR BANDYOPADHYAY
for 1 ≤ i ≤ n, and pi−1 /r ≤ pi for 2 ≤ i ≤ n. Hence, (3.3) and (3.4) imply that
pei · q − pei−1 · q = (ci − 1)(w2i+1 · q i,vi −1 ) ≥ 0,
and that,
1
1
1
p1 , p2 , . . . , pi−2 , pi−1 , pi−1 , 2 pi−1 , . . . , vi −i−1 pi−1 ,
r
r
r
Arn−vi , Arn−(vi +1) , . . . , Ar, A , p1 , p2 , . . . , pi−1 , Arn−i , Arn−(i+1) , . . . ,
(3.8) pei ∈
Ar
n−(vi −1)
, Ar
n−vi
, Ar
n−(vi +1)
, . . . , Ar, A .
Thus vi−1 ∈ {vi , i}, and for i = 2, we have
1
1
1
n−v2
n−(v2 +1)
(3.9) pe2 ∈
p1 , p1 , 2 p1 , . . . , v2 −i−1 pi−1 , Ar
, Ar
, . . . , Ar, A ,
r
r
r
p1 , Ar
n−2
, Ar
n−3
, . . . , Ar , Ar, A .
2
The vector pe1 then satisfies
(3.10) pe1 ∈ 0, 0, . . . , 0, Arn−v2 , Arn−(v2 +1) , . . . , Ar, A , (Arn−1 , Arn−2 , Arn−3 ,
. . . , Ar2 , Ar, A), 0, Arn−2 , Arn−3 , . . . , Ar2 , Ar, A ⊂ Pn2 ,
and the theorem is proven in this case. The proof follows similarly, if p · q ≤ 0, and the proof
of the theorem is complete.
Now, for a given integer 0 ≤ t ≤ n, define the vector v t via v0 = 0, and
n−t
def
vt =
t
z
}|
{ z
}|
{
B, B, . . . , B, A, A, . . . , A .
In addition, define the set of vectors
(3.11)
Pn3 = Pn3 (A, B, 1) = {v t : 0 ≤ t ≤ n}.
For the case r = 1 in either (2.1) or (3.2), we can similarly prove the following result. For
B = 0 the theorem follows directly from either Theorem 2.1 or Theorem 3.1 (see also Lemma
2.2 in [4])). For 0 < B < A, the proof is similar to that for Theorems 2.1 and 3.1, and will be
omitted.
Theorem 3.2 (Monotonicity). Suppose that p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ) are nvectors where p satisfies
pi+1 ≥ pi
for 1 ≤ i ≤ n − 1, and 0 ≤ B ≤ pi ≤ A for 1 ≤ i ≤ n. We have,
min{w · q : w ∈ Pn3 } ≤ p · q ≤ max{w · q : w ∈ Pn3 }.
We conclude with a return to global optimization for inner products for the vector q as given
in Example 2.1.
Example 2.1 (revisited). Consider the vector q ∈ R15 as given in (2.15).
The entries in q are depicted in Figure 2.1. Now, consider optimizing p · q over all p =
(p1 , p2 , . . . , p15 ) ∈ R15 , satisfying 0 ≤ pi ≤ 1 and (3.2) for some 0 < r ≤ 1. Theorem 3.2
2
implies that we need only check over the finite set P15
(1, 0, r) as given in (3.11). The results of
the computations for r ∈ {.1, .3, .7, .9}, are given in Figure 3.1.
J. Inequal. Pure and Appl. Math., 7(5) Art. 158, 2006
http://jipam.vu.edu.au/
M AXIMIZATION FOR I NNER P RODUCTS
9
6
8
10
12
14
2
6
8
10
0.3
minimal
4.102
0.3
maximal
4.651
14
12
14
12
14
12
14
0.0
p_i
12
0.8
i
6
8
10
12
14
2
4
6
8
10
i
i
0.7
minimal
4.102
0.7
maximal
6.817
0.0
p_i
0.0
0.8
4
0.8
2
4
6
8
10
12
14
2
4
6
8
10
i
i
0.9
minimal
4.102
0.9
maximal
9.368
0.0
0.0
p_i
0.8
2
0.8
p_i
4
i
0.8
4
0.0
p_i
2
p_i
0.8
0.0
p_i
0.8
0.1
maximal
4.241
0.0
p_i
0.1
minimal
4.102
2
4
6
8
10
12
14
i
2
4
6
8
10
i
Figure 3.1: Maximal and minimal values for inner products under the constraint in (3.2).
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K ENNETH S. B ERENHAUT , J OHN D. F OLEY ,
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D IPANKAR BANDYOPADHYAY
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