An example of an application of the theory to global optimization for inner products is also provided

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 DEPARTMENT OFMATHEMATICS

WAKEFORESTUNIVERSITY

WINSTON-SALEM, NC 27106 berenhks@wfu.edu

URL:http://www.math.wfu.edu/Faculty/berenhaut.html DEPARTMENT OFMATHEMATICS

WAKEFORESTUNIVERSITY

WINSTON-SALEM, NC 27106 folejd4@wfu.edu

DEPARTMENT OFBIOSTATISTICS, BIOINFORMATICS ANDEPIDEMIOLOGY

MEDICALUNIVERSITY OFSOUTHCAROLINA

CHARLESTON, SC 29425 bandyopd@musc.edu

Received 19 August, 2006; accepted 04 September, 2006 Communicated by L. Leindler

ABSTRACT. This paper studies optimization for inner products of real vectors assuming mono- tonicity properties for the entries in one of the vectors. Resulting inequalities have been use- ful 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.

Key words and phrases: Inner Products, Recurrence, Monotonicity, Discretization, Global Optimization.

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

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 productsp·q, for vectorsp= (p₁, p₂, . . . , p_n)andq= (q₁, q₂, . . . , q_n), satisfying p,q∈Rⁿ,pi ∈[A, B]for1≤i≤n, and one of the following properties

(1) (r-quasi-monotonicity)p_i+1 ≥rp_ifor1≤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) (r-geometric monotonicity)p_i+1 ≥ ¹_rp_ifor1≤i≤n−1.

(3) (monotonicity)p_i+1 ≥p_i for1≤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 P_n = P_n(A, B, r)such that

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

The paper proceeds as follows. In Section 2, we consider obtaining the setsP_ncorresponding 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 vectorq.

2. THECASE OFr-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.

(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.

q_i ≤Arⁱ

whereA > 0andr < 1(see [2]). In this case the sequence {φ_i}defined byφ_i = ^q_rⁱi, satisfies 0≤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 ≥1for1≤i≤kand

(2.2) X

i

ti =N, define the vectorvtvia

(2.3) vt ^def= A

t0

z }| {

0,0, . . . ,0;r⁰, r¹, . . . , r^t1−1;r⁰, r¹, . . . , r^t2−1;r⁰, r¹, . . . , r^tk−1 . In addition, define the set of vectors

(2.4) P_N =P_N(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 where p satisfies (2.1), for1≤i≤n−1and0≤p_i ≤Afor1≤i≤n. We have,

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

i=1p_iq_i.

The value in Theorem 2.1 lies in the fact that for any givenn,P_nis a finite set.

For a vectorp= (p₁, p₂, . . . , p_n), we will use the notationp^i,j to indicate the vector consist- ing of thei^ththroughj^thentries inp, i.e.

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

Proof of Theorem 2.1. 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 ∈ Pn.

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

(1) pe_n+1 =p.

(2) For1≤i≤n, set

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

Si

[{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_iis given by

(2.8) c_i =













 A

p_i, ifw²_i+1·q^i,vi−1 >0 rpi−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, . . . , p_i−1)

(2.9)

w²_i+1 =pe^i,v_i+1ⁱ⁻¹ = (pi, rpi, r²pi, . . . , r^vi−i−1pi) (2.10)

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

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 ∈ {v_i, 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 vectorpe₁ then 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) ⊂ Pn

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

of the theorem is complete.

2 4 6 8 10 12 14

−4−2024

i

q_i

q vector

Figure 2.1: The vectorqin (2.15).

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

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 2.1. Now, consider optimizing p·q over all p = (p₁, p₂, . . . , p₁₅) ∈ R¹⁵, satisfying 0 ≤ p_i ≤ 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 forr∈ {.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{b_i}and{α_i,j}satisfy

(2.16) bn=

n−1

X

k=0

(−αn,k)bk, n ≥1, whereb₀ = 1and

(2.17) α_n,k ∈[0, A],

for0≤k ≤n−1andn≥1, and

(2.18) rα_n,k ≤α_n,k+1.

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

|b_n| ≤ |b⁰_n|

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

and

(2.19) b⁰_n=

n−1

X

k=0

(−α⁰_n,k)b⁰_k, n ≥1, with each vector

α⁰_i = (α_i,0⁰ , α_i,1⁰ , . . . , α⁰_i,i−1)∈ P_i, for1≤i≤n, wherePiis 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 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 un- related context, a simpler version of Theorem 2.2 was also employed in Berenhaut and Bandy- opadhyay [3] in proving that all symmetric Toeplitz matrices generated by monotone convex sequences have off-diagonal decay preserved through triangular decompositions.

Proof of Theorem 2.2. The proof, here, involves applying Theorem 2.1 to successively “scale"

the rows of the coefficient matrix

[−α_i,j] =















−α_1,0 0 · · · 0

−α_2,0 −α_2,1 . .. ... ... ... . .. 0

−αn,0 −αn,1 · · · −αn,n−1













 .

while not decreasing the value of|b_n|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 thatb_n>0. Expanding via (2.16),b_ncan be written as

(2.20) b_n =C₁⁰b₀+C₁¹b₁,

whereC₁⁰ andC₁¹are constants, which depend on{α_i,j}. IfC₁¹ >0, then selectα¯⁰₁ = (α⁰_1,0)∈ P₁so that−α¯⁰₁·b^0,0is maximal, via Theorem 2.1. Similarly, ifC₁¹ <0, selectα¯⁰₁ = (α⁰_1,0)∈ P₁ so that−α¯⁰₁·b^0,0is minimal. In either case, replacingα1,0byα⁰_1,0in (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 maximalb_ivalues for1≤i≤k−1with respect to the preceeding b_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 select α¯⁰_k ∈ P_k so that−α¯⁰_k·b^0,k−1 is maximal, via Theorem 2.1.

Similarly, ifC_k^k < 0, selectα¯⁰_k ∈ P_k so that−α¯⁰_k·b^0,0 is minimal. In either case, referring to (2.21), replacing the values in α¯_k by those in α¯⁰_k in (2.16) will not decrease the value of|b_n|.

The result follows by induction for this case. The case b_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 previ- ous 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 comprehen- sive 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.

3. THECASE OFr-GEOMETRIC MONOTONICITY

In this section we consider the assumption of r-geometric monotonicity of the entries in p= (p₁, p₂, . . . , p_n), 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 where p satisfies

(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}.

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 Theorem 2.1, 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+ 1 defined recursively according to the following scheme.

(1) pe_n+1 =p.

(2) For 1 ≤ i ≤ n, set S_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), c_ipe_i+1(i), c_ipe_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;c_iw²_i+1;w³_i+1), (3.3)

wherec_i is given by

(3.4) c_i =















Arⁿ⁻ⁱ

p_i , ifw²_i+1·q^i,vi−1 >0

1 rpi−1

pi

, ifw²_i+1·q^i,vi−1 ≤0andi >1

0, otherwise

.

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

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

for1≤i≤n, andpi−1/r≤p_ifor2≤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 ∈ {vi, 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), . . . , Ar, A

,

p₁, Arⁿ⁻², Arⁿ⁻³, . . . , Ar², Ar, A

. The vectorpe₁ then 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 vectorvtviav0 =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 = 1 in either (2.1) or (3.2), we can similarly prove the following result. For B = 0the theorem follows directly from either Theorem 2.1 or Theorem 3.1 (see also Lemma 2.2 in [4])). For0< 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, . . . , p_n) and q = (q₁, . . . , q_n) are n- vectors wherepsatisfies

p_i+1 ≥p_i

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 Example 2.1.

Example 2.1 (revisited). Consider the vectorq∈R¹⁵as 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 ≤ p_i ≤ 1 and (3.2) for some 0 < r ≤ 1. Theorem 3.2 implies that we need only check over the finite setP₁₅² (1,0, r)as given in (3.11). The results of the computations forr∈ {.1, .3, .7, .9}, are given in Figure 3.1.

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

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