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volume 7, issue 5, article 158, 2006.

Received 19 August, 2006;

accepted 04 September, 2006.

Communicated by:L. Leindler

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Journal of Inequalities in Pure and Applied Mathematics

MAXIMIZATION FOR INNER PRODUCTS UNDER QUASI-MONOTONE CONSTRAINTS

KENNETH S. BERENHAUT, JOHN D. FOLEY AND DIPANKAR BANDYOPADHYAY

Department of Mathematics Wake Forest University Winston-Salem, NC 27106 EMail:berenhks@wfu.edu

URL:http://www.math.wfu.edu/Faculty/berenhaut.html Department of Mathematics

Wake Forest University Winston-Salem, NC 27106 EMail:folejd4@wfu.edu

Department of Biostatistics, Bioinformatics and Epidemiology Medical University of South Carolina

Charleston, SC 29425 EMail:bandyopd@musc.edu

c

2000Victoria University ISSN (electronic): 1443-5756 218-06

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Maximization for Inner Products Under Quasi-Monotone

Constraints

Kenneth S. Berenhaut, John D. Foley, and Dipankar Bandyopadhyay

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J. Ineq. Pure and Appl. Math. 7(5) Art. 158, 2006

Abstract

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 off-diagonal decay preserved through triangular decompositions. An example of an application of the theory to global optimization for inner products is also provided.

2000 Mathematics Subject Classification:15A63, 39A10, 26A48.

Key words: Inner Products, Recurrence, Monotonicity, Discretization, Global Opti- mization.

The first author acknowledges financial support from a Sterge Faculty Fellowship.

1. Introduction

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, . . . , p_n)andq = (q₁, q₂, . . . , q_n), satisfyingp,q ∈ Rⁿ,p_i ∈ [A, B]for 1≤i≤n, and one of the following properties

1. (r-quasi-monotonicity)pi+1 ≥rpi for1≤i≤n−1.

2. (r-geometric monotonicity)p_i+1 ≥ ¹_rp_i for1≤i≤n−1.

3. (monotonicity)p_i+1 ≥p_ifor1≤i≤n−1.

For discussion of various classes of sequences of monotone type, see for in- stance, Kijima [12], and Leindler [15,14].

Our method involves, for each of the three cases mentioned, obtaining finite setsP_n =P_n(A, B, r)such that

min{v·q:v ∈ P_n} ≤p·q ≤max{v·q :v ∈ P_n}, for allpsatisfying the respective monotonicity assumption, above.

The paper proceeds as follows. In Section2, we consider obtaining the sets P_ncorresponding to Property (1), above. An application to linear recurrences, which has been useful in the recent literature is also given. In Section3, we consider the case ofr-geometric monotonicity. The paper includes examples which provide an application of the theory to global optimization for inner products, for a specific vectorq.

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2. The Case of r-quasi-monotonicity

In this section we consider the assumption ofr-quasi-monotonicity of the entries inp= (p₁, p₂, . . . , p_n)(as defined in (1), above), i.e.

(2.1) p_i+1 ≥rp_i

for1 ≤i ≤ n−1. The motivation for consideration of such a condition arose in a probability related context of investigating a monotone sequence{q_i}with a geometric bound, i.e.

qi ≤Arⁱ

where A > 0 andr < 1 (see [2]). In this case the sequence {φ_i} defined by φ_i = ^q_rⁱi, satisfies0≤φ_i ≤A, and

φ_i = q_i

rⁱ ≥ q_i+1

rⁱ =φ_i+1r.

For a given vectort = (t₀, t₁, t₂, . . . , t_k)satisfyingt₀ ≥ 0, t_i ≥ 1 for1 ≤ i≤k and

(2.2) X

i

t_i =N, define the vectorvt via

(2.3) vt ^def= A

t0

z }| {

0,0, . . . ,0;r⁰, r¹, . . . , r^t¹⁻¹;

r⁰, r¹, . . . , r^t²⁻¹;r⁰, r¹, . . . , r^t^k⁻¹ .

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In addition, define the set of vectors

(2.4) PN =PN(A,0, r) = {vt :tsatisfies (2.2)}.

We have the following result regarding inner products.

Theorem 2.1. Suppose that p = (p₁, . . . , p_n) and q = (q₁, . . . , q_n) are n- vectors wherepsatisfies (2.1), for1≤i≤n−1and0≤p_i ≤Afor1≤i≤n.

We have,

(2.5) min{w·q:w∈ P_n} ≤p·q≤max{w·q :w ∈ P_n} wherep·qdenotes the standard dot productPn

i=1piqi.

The value in Theorem2.1lies in the fact that for any givenn, P_n is a finite set.

For a vectorp = (p₁, p₂, . . . , p_n), we will use the notation p^i,j to indicate the vector consisting of thei^ththroughj^thentries inp, i.e.

(2.6) p^i,j = (p_i, p_i+1, . . . , p_j)

Proof of Theorem2.1. First, supposep·q>0, and note that the lower bound in (2.5), for such vectors, follows from the fact thatvt = 0fort = (n,0, . . . ,0).

We will obtain a sequence of vectors{pe_i}ⁿ⁺¹_i=1, satisfying 0≤p·q=pe_n+1·q ≤pe_n·q≤ · · · ≤pe₁·q, such thatpe₁ ∈ P_n.

In particular, consider the vectorspe_i = (pe_i(1),pe_i(2), . . . ,pe_i(n))∈ Rⁿ, i = 1, . . . , n+ 1defined recursively according to the following scheme.

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1. pe_n+1 =p.

2. For1≤i≤n, set

S_i ={s:i+ 1≤s≤n and pe_i+1(s) =A}, and v_i = min

S_i[

{n+ 1}

.

3. For1≤i≤n, definepe_i (a function ofpe_i+1) via pe_i =

pe_i+1(1),pe_i+1(2), . . . ,pe_i+1(i−1), c_ipe_i+1(i), c_ipe_i+1(i+ 1), . . . , c_ipe_i+1(v_i−1), A,pe_i+1(v_i+ 1), . . . ,pe_i+1(n)

= (w¹_i+1;c_iw²_i+1;w³_i+1), (2.7)

say, wherec_i is given by

(2.8) c_i =









 A

p_i, ifw²_i+1·q^i,vⁱ⁻¹ >0 rp_i−1

p_i , ifw²_i+1·q^i,vⁱ⁻¹ ≤0andi >1 0, otherwise

.

Note thatpe_i+1 = (w¹_i+1,w²_i+1,w³_i+1).

It is not difficult to verify by induction thatw^j_i+1,j = 1,2,3, are of the form w¹_i+1 =pe^1,i−1_i+1 = (p₁, p₂, . . . , pi−1)

(2.9)

w²_i+1 =pe^i,v_i+1ⁱ⁻¹ = (p_i, rp_i, r²p_i, . . . , r^vⁱ⁻ⁱ⁻¹p_i) (2.10)

w³_i+1 =pe^v_i+1ⁱ^,n ∈ Pn−v_i+1, (2.11)

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We have that (2.7) and (2.8) imply

pe_i·q−pe_i+1·q = (c_i−1)(w²_i+1·q^n−i,vⁱ⁻¹)≥0, and, for1≤i≤n+ 1,

(2.12) pe_i ∈

(p₁, p₂, . . . , pi−1, rpi−1, r²pi−1, r³pi−1, . . . , r^vⁱ⁻ⁱpi−1;w³_i+1), (p₁, p₂, . . . , pi−1, A, rA, r²A, . . . , r^vⁱ⁻ⁱ⁻¹A;w³_i+1) . Thusvi−1 ∈ {vi, i}, and in particular, fori= 2, we have

(2.13) pe₂ ∈

(p1, rp1, r²p1, r³p1, . . . , r^v²⁻²p1;w³₃),

(p₁, A, rA, r²A, . . . , r^v²⁻³A;w³₃) . The vectorpe₁then satisfies

(2.14) pe₁ ∈

(0,0, . . . ,0;w³₃),(A, rA, r²A, r³A, . . . , r^v²⁻²A;w³₃), (A, A, rA, r²A, . . . , r^v²⁻³A;w³₃),

(0, A, rA, r²A, . . . , r^v²⁻³A;w³₃) ⊂ P_n and the theorem is proven in this case. The proof follows similarly, ifp·q≤0, and the proof of the theorem is complete.

The following example provides an application of Theorem 2.1 to global optimization for inner products, for a specific vectorq.

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Example 2.1. Consider the vectorq∈R¹⁵given 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 1. Now, consider optimizing p·q over all p= (p₁, p₂, . . . , p₁₅) ∈ R¹⁵, satisfying0 ≤ p_i ≤ 1and (2.1) for some 0< r≤1. Theorem2.1implies that we need only compute and compare inner products withqover the finite setP₁₅(1,0, r)as given in (2.4).

The results of the computations forr∈ {.1, .3, .7, .9}, are given in Figure2.

It is possible to apply Theorem2.1 in sequence to obtain bounds for linear recurrences, as is shown by the following theorem.

Theorem 2.2. Suppose that{b_i}and{α_i,j}satisfy

(2.16) b_n =

n−1

X

k=0

(−α_n,k)b_k, n≥1, whereb₀ = 1and

(2.17) α_n,k ∈[0, A],

for0≤k≤n−1andn ≥1, and

(2.18) rαn,k ≤αn,k+1.

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2 4 6 8 10 12 14

−4−2024

i

q_i

q vector

Figure 1: The vectorqin (2.15).

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2 4 6 8 10 12 14

0.00.8

i

p_i

0.1 minimal

−13.991

2 4 6 8 10 12 14

0.00.8

i

p_i

0.1 maximal

19.177

2 4 6 8 10 12 14

0.00.8

i

p_i

0.3 minimal

−12.437

2 4 6 8 10 12 14

0.00.8

i

p_i

0.3 maximal

17.21

2 4 6 8 10 12 14

0.00.8

i

p_i

0.7 minimal

−7.343

2 4 6 8 10 12 14

0.20.8

i

p_i

0.7 maximal

12.684

2 4 6 8 10 12 14

0.00.8

i

p_i

0.9 minimal

−2.38

2 4 6 8 10 12 14

0.00.8

i

p_i

0.9 maximal

11.256

Figure 2: Maximal and minimal values for inner products under the constraint in (2.1)

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Then, there exist{b⁰_i}and{α⁰_i,j}such that

|b_n| ≤ |b⁰_n| and

(2.19) b⁰_n =

n−1

X

k=0

(−α⁰_n,k)b⁰_k, n≥1, with each vector

α⁰_i = (α⁰_i,0, α⁰_i,1, . . . , α⁰_i,i−1)∈ P_i, for1≤i≤n, whereP_i is as in (2.4).

In fact, there exists a set{α⁰₁,α⁰₂, . . . ,α⁰_n}, withα⁰_i ∈ P_i, such thatb⁰_i is as large as possible (with its inherent sign) givenb₀, b⁰₁, b⁰₂, . . . , b⁰_i−1.

Remark 1. While Theorem2.2looks relatively simple, it has proven indispens- able recently in two quite unrelated interesting contexts. The theorem was cru- cial, 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 con- text, 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 tri- angular decompositions.

Proof of Theorem2.2. The proof, here, involves applying Theorem 2.1to suc-

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cessively “scale" the rows of the coefficient matrix

[−α_i,j] =







−α_1,0 0 · · · 0

−α_2,0 −α_2,1 . .. ... ... ... . .. 0

−α_n,0 −α_n,1 · · · −αn,n−1





 .

while not decreasing the value of|bn|at any step.

First, define the sequences

¯

α_i = (α_i,0, . . . , αi,i−1) and b^k,j = (b_k, . . . , b_j),

for0≤k ≤j ≤n−1and1≤i≤n.

Suppose thatbn>0. Expanding via (2.16),bncan be written as (2.20) b_n=C₁⁰b₀+C₁¹b₁,

whereC₁⁰andC₁¹ are constants, which depend on{α_i,j}. IfC₁¹ >0, then select

¯

α⁰₁ = (α⁰_1,0)∈ P₁ so that−α¯⁰₁·b^0,0is maximal, via Theorem2.1. Similarly, if C₁¹ < 0, select α¯⁰₁ = (α⁰_1,0) ∈ P₁ so that−α¯⁰₁·b^0,0 is minimal. In either case, replacingα_1,0 byα⁰_1,0 in (2.16) will result in a larger (or equal) value forC₁¹b₁, and in turn, referring to (2.20), a larger (or equal) value of|b_n|.

Now, suppose that the first through(k−1)^throws of theαmatrix are of the form described in the theorem (i.e. resulting in maximalbi values for1 ≤ i≤

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k −1with respect to the preceedingb_j, 0 ≤ j ≤ i−1), and expressb_nin the form

(2.21) b_n =C_k⁰b₀+C_k¹b₁+· · ·+C_k^kb_k,

via (2.16). IfC_k^k≥ 0, then selectα¯⁰_k ∈ P_kso that−α¯⁰_k·b^0,k−1is maximal, via Theorem2.1. Similarly, ifC_k^k<0, selectα¯⁰_k ∈ P_kso that−α¯⁰_k·b^0,0is minimal.

In either case, referring to (2.21), replacing the values inα¯_k by those inα¯⁰_kin (2.16) will not decrease the value of |b_n|. The result follows by induction for this case. The caseb_n<0is similar and the theorem is proven.

For further results along these lines in the caser= 1andB = 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 Stevi´c [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 ofr-geometric decay and monotonicity mentioned in the introduction.

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3. The Case of r-geometric Monotonicity

In this section we consider the assumption ofr-geometric monotonicity of the entries inp= (p1, p2, . . . , pn), i.e.

p_i+1 ≥ 1 rp_i for1≤i≤n−1.

First, for a given integer0≤t≤n, define the vectorv_tviav₀ =0, and v_t ^def=

n−t

z }| {

0,0, . . . ,0, Ar^t−1, Ar^t−2, . . . , Ar, A . In addition, define the set of vectors

(3.1) P_n² =P_n²(A,0, r) = {vt : 0≤t ≤n}.

Here, we have the following theorem.

Theorem 3.1. Suppose that p = (p₁, . . . , p_n) and q = (q₁, . . . , q_n) are n- vectors wherepsatisfies

(3.2) p_i+1 ≥ 1

rp_i

for1≤i≤n−1, and0≤p_i ≤Afor1≤i≤n. We have,

min{w·q :w∈ P_n} ≤p·q ≤max{w·q:w∈ P_n}.

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Proof. First, supposep·q >0, and note that the lower bound in (2.5) follows from the fact that vt = 0fort = 0. As in the proof of Theorem2.1, we will, again, obtain a sequence of vectors{pe_i}ⁿ⁺¹_i=1, satisfying

0≤p·q=pe_n+1·q ≤pe_n·q≤ · · · ≤pe₁·q, such thatpe₁ ∈ P_n².

In particular, consider the vectorspe_i = (pe_i(1),pe_i(2), . . . ,pe_i(n))∈ Rⁿ, i = 1,2, . . . , n+ 1defined recursively according to the following scheme.

1. pe_n+1 =p.

2. For1 ≤ i ≤ n, setS_i = {s : i+ 1 ≤ s ≤ nandpe_i+1(s) = Ar^n−s}, and v_i = min(S_iS

{n+ 1}).

3. For1≤i≤n, set pe_i =

pe_i+1(1),pe_i+1(2), . . . ,pe_i+1(i−1), cipe_i+1(i), cipe_i+1(i+ 1), . . . , c_ipe_i+1(v_i−1),pe_i+1(v_i),pe_i+1(v_i+ 1), . . . ,pe_i+1(n)

= (w¹_i+1;ciw²_i+1;w³_i+1), (3.3)

wherec_iis given by

(3.4) ci =











Arⁿ⁻ⁱ

p_i , ifw²_i+1·q^i,vⁱ⁻¹ >0

1 rpi−1

p_i , ifw²_i+1·q^i,vⁱ⁻¹ ≤0andi >1 0, otherwise

.

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It is not difficult to verify by induction thatw^j_i+1,j = 1,2,3, are of the form w¹_i+1 =pe^1,i−1_i+1 = (p₁, p₂, . . . , pi−1)

(3.5)

w²_i+1 =pe^1,v_i+1ⁱ⁻¹ =

p_i,1 rp_i, 1

r²p_i, . . . , 1 r^vⁱ⁻ⁱ⁻¹p_i

(3.6)

w³_i+1 =pe^v_i+1ⁱ^,n = (Ar^n−vⁱ, Ar^n−vⁱ⁻¹· · ·, Ar, A)∈ Pn−vi+1. (3.7)

Now, note that from (3.2), and the boundp_n≤A, we have that p_i ≤Arⁿ⁻ⁱ,

for 1 ≤ i ≤ n, andpi−1/r ≤ p_i for2 ≤ i ≤ n. Hence, (3.3) and (3.4) imply that

pe_i·q−pe_i−1·q= (c_i−1)(w²_i+1·q^i,vⁱ⁻¹)≥0, and that,

(3.8) pe_i ∈

p₁, p₂, . . . , pi−2, pi−1,1

rpi−1, 1

r²pi−1, . . . , 1

r^vⁱ⁻ⁱ⁻¹pi−1, Ar^n−vⁱ, Ar^n−(vⁱ⁺¹⁾, . . . , Ar, A

, p₁, p₂, . . . , pi−1, Arⁿ⁻ⁱ, Ar^n−(i+1), . . . , Ar^n−(vⁱ⁻¹⁾, Ar^n−vⁱ, Ar^n−(vⁱ⁺¹⁾, . . . , Ar, A

. Thusvi−1 ∈ {v_i, i}, and fori= 2, we have

(3.9) pe₂ ∈

p₁,1 rp₁, 1

r²p₁, . . . , 1

r^v²⁻ⁱ⁻¹pi−1, Ar^n−v², Ar^n−(v²⁺¹⁾,

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. . . , Ar, A

, p₁, Arⁿ⁻², Arⁿ⁻³, . . . , Ar², Ar, A

. The vectorpe₁then satisfies

(3.10) pe₁ ∈

0,0, . . . ,0, Ar^n−v², Ar^n−(v²⁺¹⁾, . . . , Ar, A , (Arⁿ⁻¹, Arⁿ⁻², Arⁿ⁻³, . . . , Ar², Ar, A),

0, Arⁿ⁻², Arⁿ⁻³, . . . , Ar², Ar, A

⊂ P_n², and the theorem is proven in this case. The proof follows similarly, ifp·q≤0, and the proof of the theorem is complete.

Now, for a given integer0≤t ≤n, define the vectorv_tviav₀ =0, and v_t^def=

n−t

z }| { B, B, . . . , B,

t

z }| { A, A, . . . , A

. In addition, define the set of vectors

(3.11) P_n³ =P_n³(A, B,1) ={v_t: 0≤t≤n}.

For the caser= 1in either (2.1) or (3.2), we can similarly prove the following result. ForB = 0the theorem follows directly from either Theorem2.1or Theorem3.1(see also Lemma 2.2 in [4])). For0< B < A, the proof is similar to that for Theorems2.1and3.1, and will be omitted.

Theorem 3.2 (Monotonicity). Suppose thatp= (p₁, . . . , p_n)andq= (q₁, . . . , q_n) aren-vectors wherepsatisfies

pi+1 ≥pi

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for1≤i≤n−1, and0≤B ≤p_i ≤Afor1≤i≤n. We have, min{w·q :w ∈ P_n³} ≤p·q ≤max{w·q:w∈ P_n³}.

We conclude with a return to global optimization for inner products for the vectorqas given in Example2.1.

Example2.1(revisited). Consider the vectorq ∈R¹⁵as given in (2.15).

The entries in q are depicted in Figure 1. Now, consider optimizing p·q over all p = (p₁, p₂, . . . , p₁₅) ∈ R¹⁵, satisfying 0 ≤ p_i ≤ 1 and (3.2) for some 0 < r ≤ 1. Theorem 3.2 implies that we need only check over the finite set P₁₅² (1,0, r) as given in (3.11). The results of the computations for r ∈ {.1, .3, .7, .9}, are given in Figure3.

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2 4 6 8 10 12 14

0.00.8

i

p_i

0.1 minimal

4.102

2 4 6 8 10 12 14

0.00.8

i

p_i

0.1 maximal

4.241

2 4 6 8 10 12 14

0.00.8

i

p_i

0.3 minimal

4.102

2 4 6 8 10 12 14

0.00.8

i

p_i

0.3 maximal

4.651

2 4 6 8 10 12 14

0.00.8

i

p_i

0.7 minimal

4.102

2 4 6 8 10 12 14

0.00.8

i

p_i

0.7 maximal

6.817

2 4 6 8 10 12 14

0.00.8

i

p_i

0.9 minimal

4.102

2 4 6 8 10 12 14

0.00.8

i

p_i

0.9 maximal

9.368

Figure 3: Maximal and minimal values for inner products under the constraint in (3.2).

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References

[1] R.P. AGARWAL, Difference Equations and Inequalities. Theory, Meth- ods and Applications, Second edition (Revised and Expanded), Marcel Dekker, New York, (2000).

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