Error analysis of approximation algorithm for standard bi-quadratic programming

0.43 MB·English

by  Chen Ling

Noname manuscript No.
(will be inserted by the editor)
Error analysis of approximation algorithm for standard bi-quadratic
programming
Chen Ling · Hongjin He · Liqun Qi
Received:date/Accepted:date
5
1
0
2 Abstract We consider the problem of approximately solving a standard bi-quadratic programming (StBQP),

n which is NP-hard. After reformulating the original problem as an equivalent copositive tensor programming,
a
we show how to approximate the optimal solution by approximating the cone of copositive tensors via a serial
J
polyhedralcones.Theestablishedqualityofapproximationshowsthat,apolynomial time approximation scheme
3
1 (PTAS)forsolvingStBQPexistsandcanbeextendedtosolvingstandardmulti-quadraticprogramming.Some

numerical examples are provided to illustrate our approach.
]
C
Keywords Standard bi-quadratic polynomial optimization · copositive tensor · PTAS · quality of approxima-
O
tion · standard multi-quadratic polynomial optimization
.
h
t
a
m
1 Introduction
[

1 We consider a polynomial optimization problem of the form
v
1 n m
(cid:88) (cid:88)
7 pmAin =minpA(x,y):= aijklxixjykyl
8 i,j=1k,l=1 (1.1)
2
s.t.x∈∆n, y∈∆m,
0
.
1 where
50 ∆s :=(cid:40)z∈(cid:60)s+ (cid:12)(cid:12)(cid:12) (cid:88)s zi =1(cid:41)
1
i=1
:
v is the standard simplex and (cid:60)s denotes the non-negative orthant in s-dimensional Euclidean space (cid:60)s. Here,
+
i
X A:=(a ) isareal(2,2)-thordern×n×m×m-dimensionaltensor.Withoutlossofgenerality,
ijkl 1≤i,j≤n,1≤k,l≤m
r
a C.Ling·H.He
DepartmentofMathematics,SchoolofScience,HangzhouDianziUniversity,Hangzhou,310018,China.
E-mail:cling [email protected]
H.He
E-mail:[email protected]
L.Qi
DepartmentofAppliedMathematics,TheHongKongPolytechnicUniversity,HungHom,Kowloon,HongKong.
E-mail:[email protected]
2 ChenLingetal.
we assume that the tensor A satisfies the following symmetry condition:
a =a =a , ∀ i,j =1,2,...,n; k,l=1,2,...,m. (1.2)
ijkl jikl jilk
We call the tensor satisfying (1.2) is partially symmetric. It is easy to see that, in case where all a are
ijkl
independent of the indices k and l, i.e., a = b for every i,j = 1,...,n, the original problem (1.1) reduces
ijkl ij
to the following standard quadratic programming (StQP)
n
(cid:88)
f =minf(x):= b x x ,
min ij i j
(1.3)
i,j=1
s.t.x∈∆n.
Hence,theproblem(1.1)iscalledastandardbi-quadraticprogramming(StBQP).StQPnotonlyoccurfrequently
as subproblem in escape procedures for general quadratic programming, but also have manifold applications,
e.g., in portfolio selection and in maximum weight clique problem for undirected graphs. For details, see, e.g.
[1,5,14,18] and references therein. If we consider portfolio selection problems with two groups of securities
whose investment decisions influence each other, then a generalized mean-variance model can be expressed as
aStBQP,see[4]fordetails.Inthatpaper,someoptimalityconditionsofStBQPwerestudied,andbasedupon
a continuously differentiable penalty function, the original problem was converted into the problem of locating
an unconstrained global minimizer of bi-quartic problem.
In terms of A, it is easy to see that the objective function in (1.1) can be written briefly as
p (x,y)=(Axx(cid:62))•(yy(cid:62))=(yy(cid:62)A)•(xx(cid:62)),
A
where
   
n m
Axx(cid:62) =(cid:88)aijklxixj and yy(cid:62)A=(cid:88) aijklykyl
i,j=1 k,l=1
1≤k,l≤m 1≤i,j≤n
are two m×m and n×n symmetric matrices, respectively, and X•Y stands for usual Frobenius inner product
for matrices, i.e., X•Y =tr(X(cid:62)Y).
The problem of solving (1.1) is NP-hard, even if the objective p is a quadratic function, see [2,16,17].
Therefore,designingsomeefficientalgorithmsforfindingapproximationsolutionsof(1.1)isofinterest.In[13],
some approximation bounds for the standard bi-quadratic optimization problem were presented. Moreover, by
using the variables z2 and w2 to replace x and y respectively, the original problem can be rewritten as
i j i j
n m
(cid:88) (cid:88)
ming(z,w):= a z2z2w2w2
ijkl i j k l
i,j=1k,l=1
s.t.(cid:107)z(cid:107)2 =1, (cid:107)w(cid:107)2 =1, (z,w)∈(cid:60)n×(cid:60)m.
Base on this, a polynomial-time approximation algorithm with relative approximation ratio was studied, the
obtained result is a bi-quadratic version of that presented in [11,24]. It is well-known that StQP does allow a
polynomial time approximation scheme (PTAS), as was shown by Bomze and De Klerk [2]. For the more general
minimizationofpolynomialoffixeddegreeoverthesimplex,DeKlerk,LaurentandParrilo[8]alsoshowedthe
existenceofaPTAS.Recently,byusingBernsteinapproximationandthemultinomialdistribution,anewproof
of PTAS for fixed-degree polynomial optimization over the simplex was presented, see [9] for details. Indeed,
in the case where feasible set is single simplex, the PTAS is particularly simple, and takes the minimum of f
Erroranalysisofapproximationalgorithmforstandardbi-quadraticprogramming 3
on the regular grid ∆n(r)={x∈∆n | (r+2)x∈Nn} for increasing values of r∈N. Denote the minimum over
the grid by
f∆(r) =min{f(x) | x∈∆n(r)}.
It is obvious that the computation of f∆(r) requires |∆n(r)|=(n+r+r+21) evaluations of f.
Moreover, we see that the regular grid mentioned above play an important role in the implement of PTAS.
Severalpropertiesoftheregulargrid∆n(r)havebeenstudiedintheliterature.InBos[6],theLebesgueconstant
of∆n(r)isstudiedinthecontextofLagrangeinterpolationandfiniteelementmethods.Givenapointx∈∆n,
Bomze, Gollowitzer and Yildirim [3] study a scheme to find the closest point to x on ∆n(r) with respect to
certain norms (including (cid:96)q-norms for finite q). Furthermore, for any quadratic polynomial f and r∈N, Sagol
and Yildirim [23] and Yildirim [26] consider the upper bound on fmin defined by minx∈∪rk=0∆n(k)f(x), and
analyze the error bound. The following error bounds are known for the approximation f(r) of f.
∆
Theorem 1.1 (i) [2] For any quadratic polynomial f and r∈N, one has
1
f∆(r)−fmin ≤ r+2(fmax−fmin),
where fmax is the maximum value of the objective in (1.3).
(ii) [8] For any homogeneous polynomial f of degree d≥2 in (1.3) and r∈N\{0}, one has
(cid:32) (cid:33)
f∆(d+r−2)−fmin ≤(1−wr(d)) 2dd−1 dd(fmax−fmin),
where wr(d)= r!((dd++rr))!d.
TheaboveresultsimplytheexistenceofaPTASforthecorrespondingoptimizationproblems.Thisnaturally
raisesthequestionofwhetherthesameholdsforStBQP.Asfarasweknow,thereareveryfewPTASsforsolving
standard bi-quadratic optimization problems. Indeed, the appearance of Cartesian product of two simplices in
(1.1) results in that the designing PTAS becomes a more complex task, which also differs from the problems
considered in [2,8].
In this paper, we focus on approximately solving StBQP, and present a quality of approximation which
shows the existence of a PTAS for solving StBQP. Moreover, we prove that the proposed approach can be
extended to solving standard multi-quadratic optimization problem. Some numerical examples are provided to
illustrate our approach.
Some words about the notation. (cid:60)n denotes the real Euclidean space of column vectors of length n, and
Nn denotes the set of all nonnegative integer vectors of length n. For α = (α1,...,αn)(cid:62) ∈ Nn and d ∈ N,
we define |α| = α1 +α2 +···+αn, α! = α1!α2!···αn! and I(n,d) = {α ∈ Nn | |α| = d}. For two vectors
α,β ∈ (cid:60)n, the inequality α ≤ β is coordinate-wise and means that α ≤ β for every i. Denote by Td,l the
i i n,m
set of all (d,l)-th order n×···×n×m×···×m-dimensional real rectangular tensors, and Sd,l the set of
n,m
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
d l
all partially symmetric tensors in Td,l . Here, the meaning of partially symmetric is similar to that in (1.2).
n,m
Specially, if d=l=2, Tnd,,ml and Snd,,lm are simply written as Tn,m and Sn,m, respectively. For x∈(cid:60)n, denoted
by xd the tensor (x ···x ) , which is a d-th order n-dimensional square tensor. For given X ∈ Td
i1 id 1≤i1,...,id≤n n
(the set of all d-th order n-dimensional square tensors) and Y ∈ Tl, denoted by X ⊗Y the (d,l)-th order
m
n×···×n×m×···×m-dimensional real rectangular tensor with entries X Y for 1 ≤ i ,...,i ≤ n
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) i1···id j1···jl 1 d
d l
4 ChenLingetal.
and 1≤j ,...,j ≤m. Denote by E the tensor of all ones in an appropriate tensor space. We denote by A•B
1 l
the Frobenius inner product for tensors A,B∈Td,l , i.e.,
n,m
(cid:88) (cid:88)
A•B= a b .
i1...idj1...jl i1...idj1...jl
1≤i1,...,id≤n,1≤j1,...,jl≤m
Let Hd,l denote the set of all bi-homogeneous polynomial of degrees d ≥ 2 and l ≥ 2, with respect to the
n,m
variables x∈(cid:60)n and y∈(cid:60)m respectively.
2 Preliminaries
Recall that a set S ⊆(cid:60)n is said to be convex if whenever x,y∈S and t∈[0,1] we have tx+(1−t)y∈S. A set
K ⊆(cid:60)n is said to be convex cone, if K is convex and whenever x∈K and t≥0 we have tx∈K. Let V be a
finite dimensional vector space equipped with a inner product (cid:104)·,·(cid:105), and let K be a convex cone in V. Denote
K∗ ={w∈V | (cid:104)w,k(cid:105)≥0,∀ k∈K},
which is said to be the positive dual cone of K.
For a given cone K and its dual cone K∗, we define the primal and dual pair of conic linear programs:
(P) p∗ :=inf C•X
s.t.X ∈K, A •X =b ,i=1,2,...,m
i i
and
(D) d∗ :=supb(cid:62)y
s.t.y∈(cid:60)m,C−(cid:80)m y A ∈K∗.
i=1 i i
Thefollowingwell-knownconicdualitytheorem,see,e.g.,[22],givesthedualityrelationsbetween(P)and(D).
Theorem 2.1 (Conic duality theorem). If there exists an interior feasible solution X0 ∈ int(K) of (P), and a
feasible solution of (D), then p∗ = d∗ and the supremum in (D) is attained. Similarly, if there exist y0 ∈ (cid:60)m with
C−(cid:80)m y0A ∈int(K∗) and a feasible solution of (P), then p∗ =d∗ and the infimum in (P) is attained.
i=1 i i
We now introduce the concept of copositive rectangular tensers, which is a generalization of the concept of
copositive square tensors presented in [21] and studied in [25].
Definition 2.1 Let G =(g )∈Sd,l . We say that G is a (resp. strictly) copositive tensor, if
i1...idj1...jl n,m
n m
(cid:88) (cid:88)
g x ···x y ···y ≥0 (resp. >0)
i1...idj1...jl i1 id j1 jl
i,j=1k,l=1
for any (x,y)∈(cid:60)n ×(cid:60)m (resp. (x,y)∈((cid:60)n\{0})×((cid:60)m\{0})).
+ + + +
Denote by Cd,l the set of all copositive tensors in Sd,l . It is easy to see that Cd,l is a closed convex cone
n,m n,m n,m
in the vector space Tnd,,ml . The copositive cone in Sn,m is simply defined by Cn,m.
Proposition 2.1 Let A∈Sn,m. Then
(i) A is copositive, if and only if pmin ≥0.
A
(ii) A is strictly copositive, if and only if pmin >0.
A
Erroranalysisofapproximationalgorithmforstandardbi-quadraticprogramming 5
Proof Since x¯:=x/(cid:80)ni=1xi ∈∆n and y¯:=y/(cid:80)mj=1yj ∈∆m for any x∈(cid:60)n+\{0} and y ∈(cid:60)m+\{0}, the desired
results follow from Definition 2.1. (cid:117)(cid:116)
Proposition 2.2 For any given positive integers n,m,d,l∈N with d,l≥2, one has
Cd,l =(Bd,l )∗,
n,m n,m
whereBd,l =conv{xd⊗yl |x∈(cid:60)n, y∈(cid:60)m}whichiscalledtheconeofallpartiallysymmetriccompletelypositive
n,m + +
tensors in Td,l .
n,m
Proof Let A∈Cnd,,ml . For any xd⊗yl ∈Bn,m, it is obvious that
n m
A•(xd⊗yl)= (cid:88) (cid:88) a x ···x y ···y ≥0,
i1...idj1...jl i1 id j1 jl
i1,...,id=1j1,...,jl=1
by Definition 2.1. Based upon this, we may know that A•B ≥ 0 for any B ∈ Bd,l , which implies that
n,m
A∈(Bd,l )∗. Hence, Cd,l ⊆(Bd,l )∗.
n,m n,m n,m
Now we prove (Bd,l )∗ ⊆ Cd,l . Let A ∈ (Bd,l )∗. Suppose that A (cid:54)∈ Cd,l . Then there exist x¯ ∈ (cid:60)n and
n,m n,m n,m n,m +
y¯∈(cid:60)m, such that
+
n m
(cid:88) (cid:88)
a x¯ ···x¯ y¯ ···y¯ <0.
i1...idj1...jl i1 id j1 jl
i1,...,id=1j1,...,jl=1
Take Z¯=x¯d⊗y¯l. Then Z¯∈Bd,l . From the above expression, we see that A•Z¯<0, which is a contradiction.
n,m
Therefore, A∈Cd,l , which implies that (Bd,l )∗ ⊆Cd,l . (cid:117)(cid:116)
n,m n,m n,m
We consider the following conic optimization problem
vp =minA•Z
min
s.t.Z ∈Bn,m (2.1)
E•Z =1
with B =B2,2 , whose dual problem is
n,m n,m
vd =maxλ
max
λ∈(cid:60) (2.2)
s.t.A−λE ∈Cn,m.
Itisclearthatforeveryfeasiblepair(x,y)of(1.1),onehasthatxx(cid:62)⊗yy(cid:62) ∈Bn,m andE•(xx(cid:62)⊗yy(cid:62))=1.
Hence, the problem (2.1) is a tensor program relaxation of (1.1), which implies that vp ≤pmin. However, the
min A
followingtheoremshowsthatthisrelaxationisexactlytight,andthesolving(1.1)canbeconvertedequivalently
to the solving (2.1).
Theorem 2.2 The bi-quadratic optimization (1.1) and conic optimization (2.1) are equivalent, that is, (1.1) and
(2.1) have the same optimal value and one optimal solution pair of (1.1) can be obtained from the optimal solution
of (2.1).
Proof LetZ∗beanoptimalsolutionof(2.1)withtheobjectivevaluevmpin.BythedefinitionofBn,m,thereexists
a positive integer t such that Z∗ =(cid:80)t (x(k)(x(k))(cid:62))⊗(y(k)(y(k))(cid:62)) with x(k) ∈(cid:60)n\{0} and y(k) ∈(cid:60)m\{0},
k=1 + +
k = 1,...,t. Let λ = (cid:16)(cid:80)n x(k)(cid:17)2(cid:16)(cid:80)m y(k)(cid:17)2. Then λ > 0 for k = 1,2,...,t, as well as (cid:80)t λ = 1
k i=1 i j=1 j k k=1 k
since E •Z∗ = 1. Moreover, it is easy to see that Z∗ = (cid:80)t λ (x¯(k)(x¯(k))(cid:62))⊗(y¯(k)(y¯(k))(cid:62)), where x¯(k) =
k=1 k
6 ChenLingetal.
x(k)/(cid:80)ni=1x(kk) ∈∆n andy¯(k) =y(k)/(cid:80)mj=1yj(k) ∈∆m.Sincevmpin =A•Z∗,itfollowsthatvmpin =(cid:80)tk=1λkA•
[(x¯(k)(x¯(k))(cid:62))⊗(y¯(k)(y¯(k))(cid:62))], which implies that there must exist an index, say 1, such that
A•[(x¯(1)(x¯(1))(cid:62))⊗(y¯(1)(y¯(1))(cid:62))]≤vp , (2.3)
min
since λk > 0 for k = 1,...,t and (cid:80)tk=1λk = 1. On the other hand, since x¯(1) ∈ ∆n and y¯(1) ∈ ∆m, it is clear
that
pmin ≤p (x¯(1),y¯(1))=A•[(x¯(1)(x¯(1))(cid:62))⊗(y¯(1)(y¯(1))(cid:62))],
A A
whichimplies,togetherwith(2.3)andthefactthatvp ≤pmin,thatpmin =A•[(x¯(1)(x¯(1))(cid:62))⊗(y¯(1)(y¯(1))(cid:62))]=
min A A
vp . We obtain the desired result and complete the proof. (cid:117)(cid:116)
min
For the linear tensor conic optimization problems (2.1) and (2.2), by utilizing Theorem 2.1, we obtain
the following duality result, which means, together with Theorem 2.2, that there exist no polynomial time
algorithms for solving (2.2).
Theorem 2.3 For the conic optimization problems (2.1) and (2.2), one has vp =vd .
min max
Proof InordertoinvokeTheorem2.1,wehavetoshowthatthereisaλ∈(cid:60)withA−λE ∈int(Bn∗,m)=int(Cn,m),
and that there is a feasible solution of (2.1).
Take λ¯ ∈ (cid:60) such that A¯:= A−λ¯E is positive tensor, i.e., all entries of A−λ¯E are positive, which implies
that A¯is strictly copositive. Hence A−λ¯E ∈int(Cn,m)=int(Bn∗,m).
On the other hand, by taking Z¯= n21m2e(n)(e(n))(cid:62)⊗e(m)(e(m))(cid:62) ∈Bn,m, where e(n) and e(m) are the two
vectorsofallonesin(cid:60)n and(cid:60)m respectively,wemayverifythatE•Z¯=1,whichmeansZ¯isafeasiblesolution
of (2.1). By Theorem 2.1, we obtain the desired result. (cid:117)(cid:116)
3 Approximation of copositive tensor cones
Copositive programming is a useful tool in dealing with all sorts of optimization problems. However, it is well-
known that the problem of checking whether a symmetric matrix belongs to the cone of copositive matrices
or not is co-NP-complete [15]. The appearance of copositive tensor in (2.2) results in the considered problem
becomes a more complex task, since copositive tensor hides more complex structures than matrix in terms of
computational solvability. In this section, we focus attention on studying how to approximate the copositive
tensor cone C .
n,m
By virtue of pG(x,y) in (1.1) with G ∈Sn,m, we define
(cid:32) n (cid:33)s m r
PG(s,r)(x,y):=pG(x,y) (cid:88)xi (cid:88)yj , (3.1)
i=1 j=1
wheresandr areanygivennon-negativeintegers.WeconsiderwhenP(s,r)(x,y)hasnonegativecoefficients.It
G
is clear that the set of all partially symmetric tensors G satisfying this condition forms a convex cone in Sn,m.
Definition 3.1 The convex cone Cns,,mr consists of the tensors in Sn,m for which PG(s,r)(x,y) in (3.1) has no
negative coefficients.
Obviously, these cones are contained in each other: Cs,r ⊆ Cs+1,r ⊆ Cs+1,r+1 for all non-negative integers s
n,m n,m n,m
and r. The approximation of Cns,,mr to Cn,m is essentially by examination of the proof of the following theorem,
which is a bi-quadratic version of the famous theorem of P´olya [10,19] (see also Powers and Reznick [20]).
Erroranalysisofapproximationalgorithmforstandardbi-quadraticprogramming 7
Theorem 3.1 LetG ∈Sn,m andpG(x,y)beabi-quadraticformdefinedin (1.1).SupposethatpG(x,y)ispositiveon
theCartesianproductoftwosimplices∆n×∆m,i.e.,pmGin >0.ThenthepolynomialpG(x,y)((cid:80)ni=1xi)r((cid:80)mj=1yj)s
has non-negative coefficients for all sufficiently large integers r and s.
Proof In terms of I(n,2) and I(m,2), we rewrite p (x,y) in (1.1) as
G
(cid:88) xαyβ
p (x,y)= p¯ , (3.2)
G αβ α!β!
α∈I(n,2),β∈I(m,2)
that is, p¯ =p α!β!. Define
αβ αβ
φ(x,y,t,s)= (cid:88) p¯ (x1)αt1···(xn)αtn(y1)βs1···(ym)βsm,
αβ α!β!
α∈I(n,2),β∈I(m,2)
where (a)q :=a(a−t)···(a−(q−1)t) with (a,t,q)∈(cid:60)×(cid:60) ×N. It is clear that φ(x,y,0,0) =p(x,y), and φ
t +
is continuous in ∆n×∆m×[0,1]×[0,1]. Consequently, by the given condition, there exist positive numbers µ
and ε¯∈[0,1], such that
φ(x,y,t,s)>φ(x,y,0,0)−(1/2)µ≥(1/2)µ>0 (3.3)
for (x,y,t,s)∈∆n×∆m×[0,ε¯]×[0,ε¯].
On the other hand, we also have that for any s,r≥2,
(cid:32)(cid:88)n (cid:33)r−2(cid:88)m s−2 (cid:88) xγyλ
xi  yj =(r−2)!(s−2)! γ!λ! . (3.4)
i=1 j=1 γ∈I(n,r−2),λ∈I(m,s−2)
By multiplying (3.2) and (3.4), we obtain
(cid:32)(cid:88)n (cid:33)r−2(cid:88)m s−2 (cid:88) (cid:88) xα+γyβ+λ
pG(x,y) xi  yj =(r−2)!(s−2)! p¯αβ γ!λ!α!β! .
i=1 j=1 α∈I(n,2) γ∈I(n,r−2)
β∈I(m,2) λ∈I(m,s−2)
Denote α+γ =ξ and β+λ=ζ. Then, it holds that ξ∈I(n,r) and ζ ∈I(m,s). Moreover, it is not difficult to
see that
(cid:32)(cid:88)n (cid:33)r−2(cid:88)m s−2 (cid:88) xξyζ (cid:88) (cid:32)ξ(cid:33)(cid:32)ζ(cid:33)
pG(x,y) xi  yj =(r−2)!(s−2)! ξ!ζ! p¯αβ α β , (3.5)
i=1 j=1 ξ∈I(n,r) α∈I(n,2)
ζ∈I(m,s) β∈I(m,2)
α≤ξ,β≤ζ
where (ξ)= ξ! and (ζ)= ζ! . Since (b)=0 for any a,b∈N with b<a, by (3.5), we have
α α!(ξ−α)! β β!(ζ−β)! a
(cid:32) n (cid:33)r−2 m s−2
(cid:88) (cid:88)
pG(x,y) xi  yj
i=1 j=1
(cid:32) (cid:33)(cid:32) (cid:33)
(cid:88) xξyζ (cid:88) ξ ζ
=(r−2)!(s−2)! p¯
ξ!ζ! αβ α β
ξ∈I(n,r) α∈I(n,2)
ζ∈I(m,s) β∈I(m,2)
(cid:88) xξyζ
=(r−2)!(s−2)!r2s2 φ(ξ/r,ζ/s,1/r,1/s) .
ξ!ζ!
ξ∈I(n,r)
ζ∈I(m,s)
Since, φ here is positive for sufficiently large r and s by (3.3), we obtain the desired result and complete the
proof. (cid:117)(cid:116)
8 ChenLingetal.
For every M ∈ Cn,m, we claim that M ∈ Cns,,mr for sufficiently large s and r. In fact, it is clear that
M¯t :=M+tE is strictly copositive for any t>0, which implies pmM¯int >0 by Proposition 2.1. Consequently, by
Theorem 3.1, we know that M¯t lies in some cone Cns,,mr for s,r sufficiently large. Since Cns,,mr is closed for any
fixed s and r, by letting t→0, we know that M∈Cs,r for sufficiently large s and r.
n,m
For given α∈(cid:60)n, define

|α|!
 ,if α∈Nn,
c(α)= α! (3.6)
 0, otherwise.
Moreover, for given ξ∈(cid:60)n, ζ ∈(cid:60)m and G ∈Sn,m, define
n m
Q¯ (G)= (cid:88) (cid:88) g c(ξ(n)(i,j))c(ζ(m)(k,l)), (3.7)
ξζ ijkl
i,j=1k,l=1
where ξ(n)(i,j)=ξ−e(n)−e(n) for e(n),e(n) ∈(cid:60)n, and ζ(m)(k,l)=ζ−e(m)−e(m) for e(m),e(m) ∈(cid:60)m. Here,
i j i j k l k l
e(n) is the i-th column vector of the identity matrix in (cid:60)n×n.
i
By the multinomial law, it holds that
P(s,r)(x,y)
G
 n m (cid:32) n (cid:33)s(cid:32) m (cid:33)r
(cid:88) (cid:88) (cid:88) (cid:88)
= gijklxixjykyl xp yq
i,j=1k,l=1 p=1 q=1
n m
= (cid:88) (cid:88) (cid:88) (cid:88) g s!r!x x y y xαyβ
ijklα!β! i j k l
α∈I(n,s)β∈I(m,r)i,j=1k,l=1
n m
= (cid:88) (cid:88) (cid:88) (cid:88) gijklαs!!rβ!!xα+e(in)+e(jn)yβ+e(km)+e(lm) (3.8)
α∈I(n,s)β∈I(m,r)i,j=1k,l=1
 
n m
= (cid:88) (cid:88) (cid:88) (cid:88) gijklξ(n)(i,j)s!!ζr(!m)(k,l)!xξyζ
ξ∈I(n,s+2),ξ(n)(i,j)≥0 ζ∈I(m,r+2),ζ(m)(k,l)≥0 i,j=1k,l=1
 
n m
= (cid:88) (cid:88) (cid:88) (cid:88) gijklc(ξ(n)(i,j))c(ζ(m)(k,l))xξyζ
ξ∈I(n,s+2)ζ∈I(m,r+2) i,j=1k,l=1
= (cid:88) (cid:88) Q¯ (G)xξyζ,
ξζ
ξ∈I(n,s+2)ζ∈I(m,r+2)
where the last second equality is due to (3.6), and the last equality comes from (3.7). From (3.8), we see that
Q¯ (G) as given by (3.7), are exactly the coefficients of P(s,r).
ξζ G
For G ∈ Sn,m, denote by A(i) (i = 1,...,n) the m×m symmetric matrix with entries being giikl (k,l =
1,2,...,m), B(k) (k =1,...,m) the n×n symmetric matrix with entries being g (i,j =1,2,...,n), and C
ijkk
then×mmatrixwithentriesbeingg (i=1,...,n, k=1,2,...,m).Thefollowingauxiliaryresultsimplifies
iikk
the expressions Q¯ (G) considerably.
ξζ
Lemma 3.1 Let G ∈Sn,m, ξ∈I(n,s+2) and ζ ∈I(m,r+2). Let Q¯ξζ(G) be defined in (3.7). Then,
(cid:32) n
Q¯ (G)= c(ξ)c(ζ) (Gξξ(cid:62))•(ζζ(cid:62))−(cid:88)ξ (ζ(cid:62)A(i)ζ)
ξζ (s+2)(s+1)(r+2)(r+1) i
m i=1 (cid:33) (3.9)
−(cid:88)ζ (ξ(cid:62)B(k)ξ)+ξ(cid:62)Cζ ,
k
k=1
where c(·) is defined in (3.6).
Erroranalysisofapproximationalgorithmforstandardbi-quadraticprogramming 9
Proof It is easy to verify that, if ξ(n)(i,j)∈(cid:60)n\Nn, then c(ξ(n)(i,j))=0; otherwise, we have
 ξ (ξ −1)
c(ξ(n)(i,j))=c(ξ)(s+i 2i)(s+1),if i=j,
ξ ξ
c(ξ) i j ,otherwise.
(s+2)(s+1)
Similar result on c(ζ(m)(k,l)) also holds. Consequently, by (3.7), it holds that
Q¯ (G)= (cid:88) (cid:88) ξi(ξi−1)ζk(ζk−1) g c(ξ)c(ζ)
ξζ (s+2)(s+1)(r+2)(r+1) iikk
1≤i≤n1≤k≤m
+ (cid:88) (cid:88) ξi(ξi−1)ζkζl g c(ξ)c(ζ)
(s+2)(s+1)(r+2)(r+1) iikl
1≤i≤n1≤k(cid:54)=l≤m
+ (cid:88) (cid:88) ξiξjζk(ζk−1) g c(ξ)c(ζ)
(s+2)(s+1)(r+2)(r+1) ijkk
1≤i(cid:54)=j≤n1≤k≤m
+ (cid:88) (cid:88) ξiξjζkζl g c(ξ)c(ζ)
(s+2)(s+1)(r+2)(r+1) ijkl
1≤i(cid:54)=j≤n1≤k(cid:54)=l≤m

c(ξ)c(ζ)  (cid:88) (cid:88) (cid:88) (cid:88)
= g ξ ξ ζ ζ − g ξ ζ ζ
(s+2)(s+1)(r+2)(r+1) ijkl i j k l iikl i k l

1≤i,j≤n1≤k,l≤m 1≤i≤n1≤k,l≤m

(cid:88) (cid:88) (cid:88) (cid:88) 
− g ξ ξ ζ + g ξ ζ ,
ijkk i j k iikk i k

1≤i,j≤n1≤k≤m 1≤i≤n1≤k≤m
which exactly corresponds to (3.9). (cid:117)(cid:116)
It is not difficult to see that, when G =E ∈Sn,m, for any ξ∈I(n,s+2) and ζ ∈I(m,r+2), one has
Q¯ (E)=c(ξ)c(ζ), (3.10)
ξζ
since in this case, by a simple computation, we have
(cid:88) (cid:88) (cid:88) (cid:88)
g ξ ξ ζ ζ =(s+2)2(r+2)2, g ξ ζ ζ =(s+2)(r+2)2,
ijkl i j k l iikl i k l
1≤i,j≤n1≤k,l≤m 1≤i≤n1≤k,l≤m
and
(cid:88) (cid:88) (cid:88) (cid:88)
g ξ ξ ζ =(s+2)2(r+2), g ξ ζ =(s+2)(r+2).
ijkk i j k iikk i k
1≤i,j≤n1≤k≤m 1≤i≤n1≤k≤m
Based upon Lemma 3.1, we can immediately derive a polyhedral representation of the cones Cs,r .
n,m
Theorem 3.2 For all n,m,s,r∈N, one has
(cid:40) n
Cns,,mr = G ∈Sn,m | (Gξξ(cid:62))•(ζζ(cid:62))−(cid:88)ξi(ζ(cid:62)A(i)ζ)
m i=1 (cid:41)
−(cid:88)ζ (ξ(cid:62)B(k)ξ)+ξ(cid:62)Cζ ≥0, for all ξ∈I(n,s+2),ζ ∈I(m,r+2) ,
k
k=1
where A(i) (i=1,...,n), B(k) (k=1,...,m) and C are defined before Lemma 3.1.
Proof It follows from (3.8) and (3.9). The proof is completed. (cid:117)(cid:116)
10 ChenLingetal.
From Theorem 3.2, we know that
Cns,,mr ={G ∈Sn,m | G•[(ξξ(cid:62))⊗(ζζ(cid:62))−Diag(ξ)⊗(ζζ(cid:62))−(ξξ(cid:62))⊗Diag(ζ)
+Diag(ξ)⊗Diag(ζ)]≥0, for all ξ∈I(n,s+2),ζ ∈I(m,r+2)},
and hence
(Cs,r )∗ ⊃{(ξξ(cid:62))⊗(ζζ(cid:62))−Diag(ξ)⊗(ζζ(cid:62))−(ξξ(cid:62))⊗Diag(ζ)
n,m
+Diag(ξ)⊗Diag(ζ) | ξ∈I(n,s+2),ζ ∈I(m,r+2)}(cid:54)=∅.
4 Quality of approximation
For any given non-negative integers s,r∈N, define
min(cid:8)A•X | E•X =1,X ∈(Cs,r )∗(cid:9) (4.1)
n,m
which has dual problem
max(cid:8)λ | A−λE ∈Cs,r ,λ∈(cid:60)(cid:9). (4.2)
n,m
By taking ξ¯=(s+2,0,...,0)(cid:62) ∈I(n,s+2) and ζ¯=(r+2,0,...,0)(cid:62) ∈I(m,r+2), we may find a feasible
point of (4.1). On the other hand, it is obvious that there exists a λ¯ ∈(cid:60) such that all elements of A−λ¯E are
positive,whichimpliesA−λ¯E ∈int(Cs,r ).ByTheorem2.1,weknowthattheoptimalvalue,denotedbyp(s,r),
n,m C
of (4.1) equals to one of (4.2). Moreover, we know that problem (4.2) is a relaxation of problem (2.2), in which
the copositive cone Cn,m is approximated by Cns,,mr in the sense that every M ∈ Cn,m must lie in some cone
Cs,r for sufficiently large s and r. It therefore follows that p(s,r) ≤ pmin for all sufficiently large s,r. We now
n,m C A
provide an alternative representation of p(s,r). This representation uses the following two rational grids which
C
approximate the standard simplices ∆n and ∆m in (1.1):
∆n(s):=(cid:8)x∈∆n | (s+2)x∈Nn(cid:9) and ∆m(r):=(cid:8)y∈∆m | (r+2)y∈Nm(cid:9).
Consequently, a natural approximation of problem (1.1) would be
p(∆s,r) :=min{pA(x,y) | x∈∆n(s),y∈∆m(r)}, (4.3)
which satisfies p(s,r) ≥pmin.
∆ A
We now establish the connection between p(s,r) and p(s,r).
C ∆
Theorem 4.1 For any given s,r ∈ N. We consider the rational discretization ∆n(s)×∆m(r) of ∆n ×∆m. If
A∈Sn,m, then
p(Cs,r) = ((ss++21))((rr++21)) min{p¯A(x,y) | x∈∆n(s),y∈∆m(r)}, (4.4)
where
n m
p¯ (x,y):=p (x,y)−(cid:88)x y(cid:62)A˜(i)y−(cid:88)y x(cid:62)Aˆ(k)x+x(cid:62)Aˇy
A A i k
i=1 k=1
with
 1
 A˜(i) = s+12(aiikl)1≤k,l≤m, (i=1,...,n),
Aˆ(k) = (a ) , (k=1,...,m),
r+2 ijkk 1≤i,j≤n
 Aˇ= (s+2)1(r+2)(aiikk)1≤i≤n,l≤k≤m.