SecondOrder Cone Relaxations for Binary Quadratic Polynomial Programs.
ABSTRACT techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our frame work, we rederive previous relaxation schemes and provide new ones. In particular, we present three relax ations for binary quadratic polynomial programs. The rst two relaxations, based on secondorder cone and semidenite programming, represent a signicant improvement over previous practical relaxations for several classes of nonconvex binary quadratic polynomial problems. From a practical point of view, due to the computational cost, semidenitebased relaxations for binary quadratic polynomial problems can be used only to solve small to midsize instances. To improve the computational eciency for solving such problems, we propose a third relaxation based purely on secondorder cone programming. Computational tests on dif ferent classes of nonconvex binary quadratic polynomial problems, including quadratic knapsack problems, show that the secondorder conebased relaxation outperforms the semidenitebased relaxations that are proposed in the literature in terms of computational eciency and is comparable in terms of bounds.
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ABSTRACT: This paper presents a canonical dual method for solving a quadratic discrete value selection problem subjected to inequality constraints. The problem is first transformed into a problem with quadratic objective and 01 integer variables. The dual problem of the 01 programming problem is thus constructed by using the canonical duality theory. Under appropriate conditions, this dual problem is a maximization problem of a concave function over a convex continuous space. Numerical simulation studies, including some large scale problems, are carried out so as to demonstrate the effectiveness and efficiency of the method proposed.05/2012;  SourceAvailable from: Juan C Vera
Conference Paper: An Iterative Scheme for Valid Polynomial Inequality Generation in Binary Polynomial Programming.
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ABSTRACT: Semidefinite programming has been used successfully to build hierarchies of convex relaxations to approximate polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small sizes. We propose an iterative scheme that improves the semidefinite relaxations without incurring exponential growth in their size. The key ingredient is a dynamic scheme for generating valid polynomial inequalities for general polynomial programs. These valid inequalities are then used to construct better approximations of the original problem. As a result, the proposed scheme is in principle scalable to large general combinatorial optimization problems. For binary polynomial programs, we prove that the proposed scheme converges to the global optimal solution for interesting cases of the initial approximation of the problem. We also present examples illustrating the computational behaviour of the scheme and compare it to other methods in the literature.Integer Programming and Combinatoral Optimization  15th International Conference, IPCO 2011, New York, NY, USA, June 1517, 2011. Proceedings; 01/2011  SourceAvailable from: Amir ArdestaniJaafari
Article: Robust Optimization of a Class of BiConvex Functions with Application to Inventory Problems
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ABSTRACT: Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models. Unfortunately, the second property is usually lost when the cost function that needs to be \quoteIt{robustified} is not concave (or linear) with respect to the perturbing parameters. In this work, we propose a new scheme for constructing conservative approximations for the robust optimization of a class of biconvex functions; specifically, those that decompose as the sum of maximum of functions that are biaffine with respect to the decisions and the perturbation. Such functions arise very naturally in a wide range of applications, including newsvendor, inventory, classification problems and multiattribute utility theory. We present conditions under which our approximation scheme is exact, and otherwise show that it can always be used to improve on the approximations obtained using affinely adjustable decision rules. Some empirical results will illustrate the quality of the approximate solutions that are obtained.10/2013;
Page 1
SecondOrder Cone Relaxations for Binary Quadratic
Polynomial Programs
Bissan Ghaddar*, Juan C. Vera, Miguel F. Anjos†
Department of Management Sciences, University of Waterloo, 200 University Av. W., Waterloo, Ontario, N2L 3G1
bghaddar@uwaterloo.ca, jvera@uwaterloo.ca, anjos@stanfordalumni.org
Several types of relaxations for binary quadratic polynomial programs can be obtained using linear, second
order cone, or semidefinite techniques. In this paper, we propose a general framework to construct conic
relaxations for binary quadratic polynomial programs based on polynomial programming. Using our frame
work, we rederive previous relaxation schemes and provide new ones. In particular, we present three relax
ations for binary quadratic polynomial programs. The first two relaxations, based on secondorder cone and
semidefinite programming, represent a significant improvement over previous practical relaxations for several
classes of nonconvex binary quadratic polynomial problems. From a practical point of view, due to the
computational cost, semidefinitebased relaxations for binary quadratic polynomial problems can be used
only to solve small to midsize instances. To improve the computational efficiency for solving such problems,
we propose a third relaxation based purely on secondorder cone programming. Computational tests on dif
ferent classes of nonconvex binary quadratic polynomial problems, including quadratic knapsack problems,
show that the secondorder conebased relaxation outperforms the semidefinitebased relaxations that are
proposed in the literature in terms of computational efficiency and is comparable in terms of bounds.
Key words: binary quadratic polynomial program, polynomial programming, sumofsquares, secondorder
cone.
1.Introduction
Binary quadratic polynomial problems (BQPP) can be expressed as optimizing a quadratic poly
nomial objective subject to quadratic polynomial equalities and inequalities. Several types of
relaxations can be obtained using linear, secondorder cone [13; 15], or semidefinite techniques
[3; 7; 14; 25]. In this paper we study relaxations for general BQPPs based on polynomial program
ming.
∗Research supported by a Canada Graduate Scholarship from the Natural Sciences and Engineering Research Council
of Canada.
†Currently on leave at Universit¨ at zu K¨ oln, Institut f¨ ur Informatik, Pohligstrasse 1, 50969 K¨ oln, Germany. Research
supported by the Natural Sciences and Engineering Research Council of Canada, and by a Fellowship from the
Humboldt Foundation.
1
Page 2
2
Polynomial programming includes a broad class of problems and is known to be NPhard.
Polynomial programming problems can be relaxed to tractable problems by using sumofsquares
(SOS) decompositions which lead to semidefinite programming (SDP) relaxations. This technique
was first proposed by Shor [35] to obtain bounds on the optimal value of the unconstrained case.
This idea was then generalized by Parrilo [26; 28] and Lasserre [18] for the constrained case.
In this paper, we use a characterization of nonnegative linear polynomials over the ball to
propose secondorder cone (SOC) relaxations of binary quadratic polynomial problems. We use
the polynomial programming framework to rederive, compare and strengthen existing relaxation
schemes. We present a new secondorder and semidefinitebased construction where we are able to
theoretically show that the resulting relaxations provide bounds stronger than other computation
ally practical semidefinitebased relaxations proposed in the literature. Additionally, our proposed
framework enables us to isolate expensive components of existing relaxations, namely the semidefi
nite terms. By removing the semidefinite terms, we obtain relaxations based purely on secondorder
cones. We present computational tests exploring the performance of these relaxations, comparing
them to existing ones in terms of bounds and computational time on general quadratic constrained
problems, quadratic linear constrained problems, and quadratic knapsack problems. The compu
tational experiments confirm our theoretical results where we obtain that the SOCSDPbased
relaxations give the best bounds. Our experiments also show that the purely SOCbased relaxations
produce bounds that are competitive with the existing SDP bounds but computationally much
more efficient. Furthermore, our approach can be in principle extended to mixed binary polynomial
programs where some of the variables are continuous.
The paper is organized as follows. In Section 2, we present an overview of polynomial pro
gramming and its SOS and SOC relaxations. In Section 3, we describe our solution methodology
and present several relaxations for the binary quadratic polynomial problem including our three
new proposed relaxations. In Section 4, we apply our proposed relaxations to the three classes of
problems mentioned above, and theoretically compare them to other existing relaxations from the
literature. In Section 5, we report computational results for these problems. Finally, conclusions
and future research directions are discussed in Section 6.
2.
2.1.
Background
Preliminaries
Given an ntuple α = (α1,··· ,αn) where αi∈ Z+, the total degree of the monomial xα:=
xα1
n
degree at most d. A polynomial is a finite linear combination of monomials
1xα2
2···xαn
is the nonnegative integer d = α :=?n
iαi. There are N =?n+d
d
?
monomials of
Page 3
3
f(x)=
?
α
cαxα=
?
α
cαxα1
1···xαn
n,
where the vector of coefficients c∈RN. We denote the cone of real polynomials (of degree at most
d) that are SOS by Ψ ⊂ R[x] (resp. Ψd) where R[x] := R[x1,··· ,xn] (resp. Rd[x]) denotes the
set of polynomials in n variables with real coefficients (resp. of degree at most d). Notice that in
particular Ψd= {?N
denote Pd(S) := {p(x) ∈ Rd[x] : p(s) ≥ 0 for all s ∈ S} to be the cone of polynomials of degree at
most d that are nonnegative over S.
i=1pi(x)2: p(x) ∈ R?d
2?[x]} and Ψd= Ψd−1for every odd d. Given S ⊆ Rn, we
2.2. Polynomial Programming
Consider the multivariate polynomials f(x) and gj(x) for 1 ≤ j ≤ m with x ∈ Rn. A polynomial
programming problem has the form:
(PPP) sup f(x)
s.t. gj(x)≥0,
1≤j ≤m.
Equality constraints of the form hj(x)=0 can be included as they can be expressed as the inequal
ity constraints hj(x)≥0 and hj(x)≤0.
Solving polynomial programming problems is an area being actively studied. For the uncon
strained case, Shor introduced the idea of computing the minimum value λ such that λ−f(x) is a
SOS to obtain an upper bound for the supremum of f [35]. Such a minimum λ can be computed
in polynomialtime using semidefinite programming. This idea was further developed by Parrilo
[26] and Parrilo and Sturmfels [29] for the constrained case using SOS decompositions. Lasserre
[18] proposed a general solution approach for polynomial optimization problems via semidefinite
programming using methods based on moment theory. Refinements of such ideas have been used
in several instances. de Klerk and Pasechnik [8] approximated the copositive cone via a hierarchy
of linear or semidefinite programs of increasing size using decompositions into sumofsquares and
polynomials with nonnegative coefficients. Kojima, Kim, and Waki exploited the sparsity of the
polynomials to reduce the size of the semidefinite problem [16]. Pe˜ na, Vera, and Zuluaga [30] pre
sented solution schemes exploiting the equality constraints. In addition, the idea of approximating
a set of nonnegative polynomials is also present in the work of several authors such as Nesterov
[24], Parrilo [28; 27], Sturmfels, Demmel, and Nie [37], Laurent [20], and Zuluaga, Vera, and Pe˜ na
[40].
Page 4
4
Consider λ to be the optimal value for (PPP), then λ is the smallest value such that λ−f(x)≥0
for all x∈S :={x:gj(x)≥0; 1≤j ≤m}. As a result, we can express problem (PPP) as:
(PPD) inf λ
s.t. λ−f(x)≥0
∀x∈S.
(1)
To obtain computable relaxations (via SDP) of (1), one can use a SOS decomposition with
restricted degree of the (unknown) polynomials. This can be rephrased in terms of a linear system
of equations involving positive semidefinite matrices [40]. Thus, solving a polynomial problem can
be relaxed to solving an easier problem involving SOS which can be recast as a semidefinite
programming problem [35; 39].
The condition λ − f(x) ≥ 0 for all x ∈ S is NPhard in general. Relaxing this condition to
λ−f(x)∈K for a suitable K ⊆Pd(S) and defining
z∗
K= inf λ
s.t. f(x)−λ∈K,
we have z∗
K≥ z∗
PP. Finding a good approximation K of Pd(S) is a key factor in obtaining a good
bound of the original problem. At the same time, having a tractable approximation, i.e., one
that uses linear, secondorder, and semidefinite cones, is essential to be able to solve the resulting
relaxation efficiently using interiorpoint methods.
2.3.SumofSquares and SecondOrder Cone Relaxations
Consider a polynomial p(x) of degree d. A necessary condition for the polynomial p(x) to be non
negative for all x ∈ Rnis that the degree of p is even. A sufficient condition is the existence of a
sumofsquares decomposition, i.e., the existence of polynomials q1(x),··· ,qk(x) such that p(x) =
?k
polynomial for all values of x; however the inverse does not hold. A simple counterexample is the
Motzkin polynomial [23].
i=1qi(x)2, or equivalently, p∈Ψ. If p(x) is a sumofsquares polynomial then it is a nonnegative
SOS conditions can be written as SDP constraints by applying the following theorem:
Theorem 1. [35] A polynomial p(x) of degree d is SOS if and only if p(x) = σ(x)TQσT(x),
where σ is a vector of monomials in the xivariables, σ(x) = [xα] with α ≤d
?n+d/2
2and Q ∈ SN
+, N =
d/2
?=σ.
The size of the matrix Q in the corresponding SDP is?n+d/2
equality constraints. If d is fixed, then this problem is solvable in polynomialtime.
The following results will allow us to use secondorder cone relaxations when working with
d/2
?×?n+d/2
d/2
?. In addition, we have?n+d
d
?
nonnegative polynomials over the ball B :={x:?x?2=n}.
Page 5
5
Lemma 1. f(x)∈P1(B) if and only if f(x)=fT
order cone.
?√n
x
?
with f ∈Ln+1, where Ln+1is the second
Further, by the S−Lemma of Yakubovich (see [33]), the nonnegativity over the ball of a polynomial
f(x) of degree two can be represented using SOS:
Lemma 2. f(x)∈P2(B) if and only if f(x)=s(x)+t(n−?x?2), where s(x) is SOS and t∈R+.
The key feature of semidefinite and secondorder cones is their tractability. As a result, we can use
these techniques to compute global upper bounds for (PPP).
3.Binary Quadratic Polynomial Programming
Binary quadratic polynomial programming problem is a classical combinatorial problem. It is the
problem of minimizing or maximizing a quadratic function of several binary variables, subject to
quadratic and linear constraints. The problem can be formally expressed as:
(BQPP) max xTQx+pTx
s.t. aT
jx=bj
∀j ∈{1,··· ,t}
∀j ∈{1,··· ,u}
∀j ∈{1,··· ,v}
∀j ∈{1,··· ,w}
∀i∈{1,··· ,n}.
(2)
cT
jx≤dj
xTFjx+eT
(3)
jx=kj
(4)
xTGjx+hT
jx≤lj
(5)
xi∈{−1,1}
(6)
Note that constraint (6) can be modified to allow some continuous variables. In this paper we focus
on pure binary quadratic polynomial programs although our solution methodology can be applied
to mixedbinary quadratic polynomial programs with bounded continuous variables.
There are many wellknown problems that can be naturally written as binary quadratic poly
nomial problems. For instance, folding of proteins in threedimension by Phillips and Rosen [31],
machine scheduling and unconstrained task allocation by Alidaee, Kochenberger, and Ahmadian
[1], capital budgeting and financial analysis such as in Laughhunn [19], as well as other examples
arising in physics and engineering applications such as the spin glass problem and circuit board lay
out design by Gr¨ otschel, J¨ unger, and Reinelt [10]. Furthermore, Boros and Hammer [4] and Boros
and Prekopa [5] formulated many satisfiability problems as BQPPs. In addition, there are several
applications related to combinatorial problems such as the singlerow facility layout problem [2]
and the quadratic assignment problem [21].
Page 6
6
3.1.Polynomial ProgrammingBased Relaxations
Using (1), (BQPP) is equivalent to
min λ
s.t. λ−q(x)∈P2(H ∩S),
where q(x) =?
lj}, and H :={−1,1}n. Note that even checking if a polynomial is in P2(H) is NPhard, therefore
tractable approximations of P2(H ∩S) are needed. A hierarchy of approximations to P2(H ∩S) is
obtained using the cones
?
ij
?
⊆P2(H ∩S),
i,jQijxixj+?
ipixi, S = {x : aT
jx = bj,cT
jx ≥ dj,xTFjx+eT
jx = kj,xTGjx+hT
jx ≥
Kr:=Ψr+2+
?
(1−x2
i)Rr[x]+
?
jx)Rr[x]+
(bj−aT
jx)Rr+1[x]+
?
j
(dj−cT
jx)Ψr+1
?
+
j
(kj−xTFjx−eT
?
j
(lj−xTGjx−hT
jx)Ψr
∩R2[x]
for an integer r ≥0. The result is a hierarchy of relaxations:
(BQPPKr) min λ
s.t. λ−q(x)∈Kr
(7)
whose optimal value converges to the optimal value of (BQPP) due to the fact that at the limit the
cone Krcontains the interior of P2(H ∩S). The following theorem follows by applying Corollary 1
of [30] and Putinar’s theorem [34]
Theorem 2. The sequence of cones Krsatisfies
Kr⊆Kr+1⊆···⊆P2(H ∩S) and int(P2(H ∩S))⊆
∞
?
r=0
Kr⊆P2(H ∩S).
Hence λ∗
BQPPKr↑z∗
BQPP.
The size of the relaxations produced in the previous theorem grows exponentially in r. For this
reason, instead of looking at the hierarchy of relaxations, we will concentrate on the first and
simplest relaxation where r =0,
K0=Ψ2+
?
(kj−xTFjx−eT
i
(1−x2
i)R0+
?
jx)R0+
j
(bj−aT
jx)R1[x]+
?
?
j
(dj−cT
jx)R+
0
+
?
jj
(lj−xTGjx−hT
jx)R+
0.
We study how to improve the approximation of P2(H ∩S) using variations of the cone K0. The
fundamental tool that we use to construct such inner approximations of P2(H ∩S) is Lemma 1,
Page 7
7
a representation theorem for nonnegative linear polynomials over B which results in secondorder
cone conditions. These yield stronger approximations than K0with an insignificant impact on the
computational time.
3.2.New Conic Relaxations of BQPP
In this section, we present three relaxations for the BQPP problem. Two of these relaxations are
based on secondorder cone and semidefinite programming and the final relaxation is solely based
on secondorder cone programming.
3.2.1.SOCSDPbased Relaxations of BQPP
Recall the previous polynomial formulation of the binary quadratic polynomial problem. First,
notice that x∈H implies ?x?2
2=n. Therefore, S ∩H ⊆B and by defining¯K0as
?
jx)P1(B)+
j
¯K0=P2(B)+
?
(dj−cT
i
(1+xi)P1(B)+
i
(1−xi)P1(B)+
(kj−xTFjx−eT
?
i
(1−x2
i)R0+
?
?
j
(bj−aT
jx)R1[x]
+
?
j
?
jx)R0+
j
(lj−xTGjx−hT
jx)R+
0,
we have¯K0⊆P2(S ∩B∩H)=P2(S ∩H).
Using Lemmas 1 and 2, we can write the condition λ−q(x)∈¯K0as
?
+
ηj(x)(dj−cT
λ−q(x)=s(x)+
i
(1+xi)αi(x)+
?
?
i
(1−xi)βi(x)+
θj(kj−xTFjx−eT
?
jx)+
i
γi(1−x2
?
?
i)+
?
j
δj(x)(bj−aT
jx)
?
j
jx)+
jj
ξj(lj−xTGjx−hT
jx),
with s(x) =?1 xT?S
ηT
j
x
We then obtain the following relaxation of (BQPP):
?1
x
?
and S ∈ Sn+1
+
, αi(x) = αT
i
?√n
x
, βi(x) = βT
i
?√n
x
?
, and ηj(x) =
?√n
?
where αi,βi,ηj∈Ln+1, δj(x)∈R1[x], γi,θj∈R, and ξj∈R+.
(BQPPSS) min λ
s.t. λ−q(x)=?1 xT?S
?1
x
?
+
?
i)+
i
(1+xi)αT
i
?√n
x
?
jx)+
+
?
?
i
(1−xi)βT
i
?√n
x
?
+
?
?
αi,βi,ηj∈Ln+1,
i
γi(1−x2
θj(kj−xTFjx−eT
?
j
δj(x)(bj−aT
j
(dj−cT
jx)ηT
j
?√n
x
?
+
j
jx)+
?
j
ξj(lj−xTGjx−hT
ξj∈R+.
jx),
S ∈Sn+1
+
,γi,θj∈R,
To strengthen this relaxation we can add valid inequalities to the original problem (BQPP) which
is equivalent to adding more variables to the relaxation due to the next lemma.
Page 8
8
Lemma 3. For any S, d, and f ∈Rd[x]
Pd(S ∩{x:f(x)≥0})⊇Pd(S)+f(x)Pd−deg(f)(S).
Notice that products of linear constraints, such as (dk− cT
cT
kx)(1 + xi), (dk− cT
kx)(1 − xi), (dk−
kx)(dl−cT
inequalities and can be added to (BQPPSS) to further strengthen the relaxation. Hence we obtain
lx), (1−xj)(1−xi), (1+xj)(1+xi), and (1−xj)(1+xi) are also considered as valid
(BQPPSS+) min λ
s.t. λ−q(x)=?1 xT?S
?1
x
?
+
?
i)+
i
(1+xi)αT
i
?√n
x
?
jx)+
+
?
?
i
(1−xi)βT
i
?√n
x
?
+
?
?
?
?
?
αi,βi,ηj∈Ln+1,
i
γi(1−x2
θj(kj−xTFjx−eT
σik(dk−cT
νkl(dk−cT
ωij(1+xi)(1+xj)+
?
j
δj(x)(bj−aT
j
(dj−cT
jx)ηT
j
?√n
x
?
+
j
jx)+
?
?
j
ξj(lj−xTGjx−hT
µik(dk−cT
?
φij(1−xi)(1+xj)
γi,θj∈R,
jx)
+
i,k
kx)(1+xi)+
i,k
kx)(1−xi)
+
k≤l
kx)(dl−cT
lx)+
?
i≤j
τij(1−xi)(1−xj)
+
i≤j
i,j
S ∈Sn+1
+
,ξj,σik,µik,νkl,τij,ωij,φij∈R+.
3.2.2. Pure SOCbased Relaxations of BQPP
The relaxation (BQPPSS) can further be relaxed by removing the positive semidefinite variable
leading to the following relaxation:
min λ
s.t. λ−q(x)=
?
+
i
(1+xi)αT
i
?√n
x
?
jx)+
+
?
?
jx),
i
(1−xi)βT
i
?√n
x
?
+
?
?
i
+
γi(1−x2
?
i)
?
?
j
δj(x)(bj−aT
ξj(lj−xTGjx−hT
γi,θj∈R,
j
(dj−cT
jx)ηT
j
?√n
x
j
θj(kj−xTFjx−eT
jx)
+
j
αi,βi,ηj∈Ln+1,ξj∈R+.
One type of valid inequalities that we consider for BQPP is:
−1≤xixj≤1.
(8)
These inequalities are not violated in the presence of the SDP term. However, once the SDP term
is removed these constraints are no longer satisfied and adding them will strengthen the SOC
relaxation.
Page 9
9
Hence, we obtain our proposed SOCbased relaxation:
(BQPPSOC) min λ
s.t. λ−q(x)=
?
+
i
(1+xi)αT
i
?√n
x
?
jx)+
+
?
?
jx)+
i
(1−xi)βT
i
?√n
x
?
+
?
?
i
+
γi(1−x2
?
?
i)
?
?
j
δj(x)(bj−aT
ξj(lj−xTGjx−hT
γi,θj∈R,
j
(dj−cT
?
ξj,h+
jx)ηT
j
?√n
x
j
θj(kj−xTFjx−eT
jx)
+
ji<j
h+
ij(1+xixj)+
i<j
h−
ij(1−xixj)
αi,βi,ηj∈Ln+1,
ij,h−
ij∈R+.
By construction we have the following theorem relating the three presented relaxations:
Theorem 3. Let λ∗
BQPPSOC, λ∗
BQPPSS, and λ∗
BQPPSS+be the optimal solution value of (BQPPSOC),
(BQPPSS), and (BQPPSS+) respectively, then
λ∗
BQPPSOC≥λ∗
BQPPSS≥λ∗
BQPPSS+≥z∗
BQPP.
4.
In this section, we apply our proposed framework to the following classes of constrained BQPPs:
Applications
• General quadratic polynomial problems;
• Quadratic linear constrained problems;
• Quadratic knapsack problems.
First, we start with the most general class of binary quadratic polynomial problems (BQPP)
where we have quadratic and linear constraints. Then we consider the special case with only linear
constraints and finally we consider problems with a single linear constraint. We rederive existing
relaxations that have been proposed in the literature for each of these problems and theoretically
compare our proposed two SOCSDPbased relaxations to them. We show theoretically that we
obtain stronger relaxations based on applying the methodology of Section 3. In addition, in Section
5 we compare the relaxations computationally for each of these three classes of binary quadratic
problems. Our computational results show that more efficient relaxations are obtained if the SDP
term is omitted.
4.1.General Quadratic Polynomial Problems
In this section, we consider the general binary quadratic problem (BQPP). Lasserre [17] introduced
SDP relaxations for binary polynomial programs by approximating P2(H ∩S) using the cone
?
ii
?
for even r ≥0.
Γr:=Ψr+2+
?
(1−x2
i)Ψr+
?
?
(x2
i−1)Ψr+
?
ix−ki)Ψr+
i
(bi−aT
ix)Ψr+
?
i
(aT
ix−bi)Ψr+
?
?
i
(di−cT
ix)Ψr
+
i
(ki−xTFix−eT
ix)Ψr+
i
(xTFix+eT
?
i
(li−xTGix−hT
ix)Ψr
∩R2[x],
Page 10
10
Lemma 4. Γr⊆Kr.
Notice that from the definition of Ψr, Γris equal to Γr−1for all odd r. Taking r =0, we obtain the
following relaxation for the BQPP problem:
(BQPPLas) min λ
s.t. λ−q(x)∈Γ0.
Theorem 4. Let λ∗
BQPPLas, λ∗
BQPPSS, and λ∗
BQPPSS+be the optimal solution value of (BQPPLas),
(BQPPSS), and (BQPPSS+) respectively, then
λ∗
BQPPLas≥λ∗
BQPPSS≥λ∗
BQPPSS+≥z∗
BQPP.
Proof: Define
H1=Ψ2+
?
(ki−xTFix−eT
i
(1−x2
i)R0+
?
ix)R0+
?
i
(bi−aT
ix)R1[x]+
?
(1+xi)P1(B)+
?
?
?
i
(di−cT
ix)P1(B)
+
?
ii
(li−xTGix−hT
ix)R+
0.
H2=H1+(di−cT
H3=H2+
+
ix)P1(B)+
Ψ0(1+xi)(dk−cT
Ψ0(1+xi)(1+xj)+
i
?
i
(1−xi)P1(B)=¯K0.
?
i,k
kx)+
i,k
Ψ0(1−xi)(dk−cT
Ψ0(1−xi)(1−xj)+
kx)+
?
?
k≤l
Ψ0(dk−cT
kx)(dl−cT
lx)
?
i≤ji≤j
i,j
Ψ0(1+xi)(1−xj).
We have
Ψ1=R+
0⊆P1(B)⇒K0⊆H1.
In addition, from Lemma 4, by setting r to zero we have Γ0⊆K0and therefore,
Γ0⊆K0⊆H1⊆H2⊆H3.
?
From Theorem 4, we obtain that (BQPPSS+) provides the best bound for the BQPP problem
while (BQPPSS) has a better bound than Lasserre’s relaxation. Further, as presented in Table 1
the computational complexity of the three problems is similar. Table 1 summarizes the number of
variables (and for SDPs, the dimension) for each of the resulting optimization problems. Recall
that the (BQPP) problem has t linear equalities, u linear inequalities, v quadratic equalities, w
quadratic inequalities, and n binary variables.
Page 11
11
Table 1
Variables
SDP
SOC
Linear Nonnegative 2n+2t+2v +u+w
Linear Free
Problem dimension for various BQPP relaxations.
(BQPPLas)
1, (n+1)×(n+1) 1, (n+1)×(n+1)
(BQPPSS) (BQPPSS+)
1, (n+1)×(n+1)
(2n+u) , (n+1)
?+2?n+1
 (2n+u) , (n+1)
w w+2tn+?t+1
22
?+n2

n+(n+1)t+vn+(n+1)t+v
4.2.Quadratic Linear Constrained Problems
Without loss of generality, we formulate the binary quadratic linear constrained problem as:
(QLCP) max xTQx+pTx
s.t. aT
jx≤bj
x∈{−1,1}n.
∀j ∈{1,··· ,m}
Specializing the results of Section 3.2 to (QLCP), we obtain the following polynomial programming
relaxations:
(QLCPSS) min λ
s.t. λ−q(x)=?1 x?S
?1
x
?
+
m
?
?√n
j=1
(bj−aT
?
S ∈Sn+1
jx)dT
j
?√n
x
?
+
n
?
i=1
(1+xi)fi
T
?√n
x
?
+
n
?
fi,gi,dj∈Ln+1,
i=1
(1−xi)giT
x
+
n
?
i=1
ci(1−x2
;
i)
ci∈R,
+
(QLCPSS+) min λ
s.t. λ−q(x)=?1 x?S
?1
x
?
+
m
?
?√n
j=1
(bj−aT
?
jx)dT
j
?√n
αik(1+xi)(bk−aT
n
?
?
lx)+
i=1
+,fi,gi,dj∈Ln+1,
x
?
+
n
?
i=1
(1+xi)fi
T
?√n
x
?
+
n
?
n
?
n
?
m
?
αik,βik,γij,δij,ζij,ηkl∈Rn
i=1
(1−xi)giT
m
?
n
?
m
?
x
+
n
?
kx)+
i=1
m
?
k=1
kx)
+
i=1
k=1
βik(1−xi)(bk−aT
i=1
?
?
n
?
ζij(1+xi)(1−xj)
n
ci(1−x2
j=i
γij(1+xi)(1+xj)
+
i=1j=i
δij(1−xi)(1−xj)+
n
i=1
n
j=1
+
k=1l=k
ηkl(bk−aT
kx)(bl−aT
i)
ci∈R,S ∈Sn+1
+
;
Page 12
12
(QLCPSOC) min λ
s.t. λ−q(x)=
m
?
+
j=1
?
h+
(bj−aT
jx)dT
j
?√n
?
fi,gi,dj∈Ln+1.
x
?
+
n
?
i=1
(1+xi)fi
T
?√n
x
?
+
n
?
i=1
(1−xi)giT
?√n
x
?
n
i=1
ci(1−x2
i)+
i<j
h+
ij(1+xixj)+
n
?
i<j
h−
ij(1−xixj)
ci∈R,
ij,h−
ij∈R+
4.2.1. The Relaxation of Burer and Lov´ aszSchrijver
Burer [6] presented an SDPbased relaxation for the QLCP where the variables are 01. We
introduce the following relaxation that is at least as strong as the relaxation presented by Burer
[38]:
(QLCPBurer’) min λ
s.t. λ−q(x)=?1 x s t?(M +N)
?
ci∈R,
1
x
s
t
+
n
?
i=1
cixi(1−xi)+
n
?
i=1
(1−xi−si)li(x)
+
m
j=1
li,ki∈R1[x,s,t],
(bj−aT
jx−tj)kj(x)
M ∈S2n+m+1
+
,N ∈R2n+m+1
+
,
where m is the number of linear constraints. Further (QLCPBurer’) is equivalent to:
min λ
s.t. λ−q(x)=?1 x 1−x b−aTx?(M +N)
ci∈R,
1
x
1−x
b−aTx
,
+
n
?
i=1
cixi(1−xi)
M ∈S2n+m+1
+
,N ∈R2n+m+1
+
Page 13
13
which can be written as
min λ
s.t. λ−q(x)=?1 x?M?
?1
αikxi(bk−aT
x
?
+
n
?
i=1
cixi(1−xi)
+
n
?
n
?
n
?
αik,βik,γij,δij,ζij,ηkl∈R+,
i=1
m
?
n
?
n
?
k=1
kx)+
n
?
δij(1−xi)(1−xj)
m
?
i=1
m
?
k=1
βik(1−xi)(bk−aT
kx)
+
i=1j=i
γijxixj+
n
?
i=1
n
?
j=i
+
i=1
j=1
ζijxi(1−xj)+
k=1
m
?
l=k
ηkl(bk−aT
M?∈Sn+1
kx)(bl−aT
lx)
ci∈R,
+
.
Notice that (QLCPBurer’) reduces to the N+relaxation of Lov´ asz and Schrijver [22] by setting the
variables γij,δij,ζij, and ηklto zero. That is, N+is equivalent to the following relaxation:
(QLCPN+) min λ
s.t. λ−q(x)=?1 x?S
?1
βik(1−xi)(bk−aT
αik,βik∈R+,
x
?
+
n
?
i=1
cixi(1−xi)+
n
?
i=1
m
?
k=1
αikxi(bk−aT
kx)
+
n
?
i=1
m
?
k=1
kx)
ci∈R,S ∈Sn+1
+
.
4.2.2.Comparing Relaxations for QLCP
We can prove the following result:
Theorem 5. Let λ∗
QLCPN+, λ∗
QLCPBurer’, and λ∗
QLCPSS+
be the optimal solution value of
(QLCPN+),(QLCPBurer’), and (QLCPSS+) respectively, then
λ∗
QLCPN+≥λ∗
QLCPBurer’≥λ∗
QLCPSS+≥z∗
QLCP.
Proof: Define
H4=Ψ2+
?
?
i,k
Ψ0(1+xi)(bk−aT
Ψ0(1+xi)(1+xj)+
?
?
kx)+
?
Ψ0(1−xi)(1−xj)+
i,k
Ψ0(1−xi)(bk−aT
kx)+
?
?
Ψ0(1+xi)(1−xj)
i
(1−x2
i)R0
H5=H4+
i≤j
?
i≤j
i,j
+Ψ0
k≤l
(bj−aT
(bk−aT
kx)(bl−aT
lx)
?
H6=H5+
j
jx)P1(B)+
i
(1+xi)P1(B)+
?
i
(1−xi)P1(B).
Hence,
H4⊆H5⊆H6.
Page 14
14
After a simple change of variables from {−1,1} to {0,1}, H4 and H5 correspond to the repre
sentations (QLCPN+) and (QLCPBurer’) respectively, while H6corresponds to the representation
(QLCPSS+).
Table 2 lists the number of variables required to formulate the various relaxations for the QLCP
?
problem of Theorem 5 where we have m linear constraints and n binary variables. While both
relaxations have the same computational complexity, (QLCPSS+) provides the best bounds as shown
in Theorem 5 and confirmed by the computational results of Section 5.
Table 2 Problem dimension for various QLCP relaxations.
Variables
SDP
SOC
Linear Nonnegative
Linear Free
(QLCPN+)
1, (n+1)×(n+1)
(QLCPBurer’)
1, (n+1)×(n+1)
(QLCPSS+)
1, (n+1)×(n+1)
(2n+m),(n+1)
2nm+n2+2?n+1

?
2nm 2nm+n2+2?n+1
2
?+?m+1
2
2
?+?m+1
2
?
nnn
Remark 1. We are unable to compare theoretically (QLCPSS) with (QLCPN+) and
(QLCPBurer’). However in our computational experiments in Section 5.2, (QLCPSS) always provides
a strictly better bound than (QLCPN+) while (QLCPBurer’) provides a strictly better bound than
(QLCPSS).
4.3. Quadratic Knapsack Problem
In this section, we consider the quadratic knapsack problem (QKP) which is the particular case
of QLCP where m=1. The QKP was introduced by Gallo, Hammer, and Simeone [9] and is NP
hard. The QKP can be interpreted as follows: we are given n items with a nonnegative weight
wiassigned to item i, and a (n+1)×(n+1) symmetric matrix Q with real entries. The QKP is
the problem of selecting a subset of items so as to maximize the overall profit such that the total
weight of the selected items does not exceed a given capacity c. Introducing the binary variable xi
such that
?
−1
the problem may be formulated as:
xi=
1if item i is selected
otherwise,
(QKPP) max q(x)=?1 x?Q
s.t. wTx≤c
?1
x
?
x∈{−1,1}n.
The QKP is a generalization of the linear knapsack problem (where the objective function is
linear). As in the case of the linear knapsack problem, the QKP often appears as a subproblem to
Page 15
15
other complex problems such as the graph partitioning problem described in Johnson, Mehrotra,
and Nemhauser [12]. Since the QKP is a constrained version of the binary quadratic problem, all
valid inequalities for the unconstrained BQPP problem are also valid for the QKP and hence they
can be used to tighten bounds for this problem. Using the same approach as in Section 3.2, we
obtain the following relaxations of (QKPP):
(QKPSS) min λ
s.t. λ−q(x)=?1 x?S
?1
x
?
+(c−wTx)dT
?√n
?√n
ci(1−x2
;
x
?
+
n
?
i)
i=1
(1+xi)fi
T
?√n
x
?
+
n
?
fi,gi,d∈Ln+1,
i=1
(1−xi)giT
x
?
S ∈Sn+1
+
n
?
i=1
ci∈R,
+
(QKPSS+) min λ
s.t. λ−q(x)=?1 x?S
?1
x
?
+(c−wTx)dT
?√n
?√n
αi(1+xi)(c−wTx)+
?
x
?
+
n
?
i=1
(1+xi)fi
T
?√n
n
?
?
x
?
+
n
?
?
n
?
αi,βi,γij,δij,ζij∈R+,
i=1
(1−xi)giT
γij(1+xj)(1+xi)+
x
?
+
n
?
i=1i=1
βi(1−xi)(c−wTx)
ζij(1−xj)(1+xi)
+
i≤ji≤j
δij(1−xj)(1−xi)+
i,j
+
i=1
ci(1−x2
i)
ci∈R,fi,gi,d∈Ln+1,S ∈Sn+1
+
;
(QKPSOC) min λ
s.t. λ−q(x)=(c−wTx)dT
?√n
x
?
+
n
?
h+
i=1
(1+xi)fi
T
?√n
?
x
?
+
n
?
i=1
(1−xi)giT
?√n
x
?
+
n
?
h+
i=1
ci(1−x2
i)+
?
fi,gi,d∈Ln+1.
i<j
ij(1+xixj)+
i<j
h−
ij(1−xixj)
ci∈R,
ij,h−
ij∈R+
4.3.1.HelmbergRendlWeismantel QKP Relaxation
Helmberg et al. [11] presented four SDPbased relaxations for the QKP where the discrete set
is {0,1}n. These relaxations are obtained by considering the semidefinite matrix X = xxT. In
particular they studied the relaxation