Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs.

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Second-Order 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 re-derive 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 second-order cone and
semidefinite programming, represent a significant improvement over previous practical relaxations for several
classes of non-convex binary quadratic polynomial problems. From a practical point of view, due to the
computational cost, semidefinite-based relaxations for binary quadratic polynomial problems can be used
only to solve small to mid-size instances. To improve the computational efficiency for solving such problems,
we propose a third relaxation based purely on second-order cone programming. Computational tests on dif-
ferent classes of non-convex binary quadratic polynomial problems, including quadratic knapsack problems,
show that the second-order cone-based relaxation outperforms the semidefinite-based 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, sum-of-squares, second-order
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, second-order 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.
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Polynomial programming includes a broad class of problems and is known to be NP-hard.
Polynomial programming problems can be relaxed to tractable problems by using sum-of-squares
(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 non-negative linear polynomials over the ball to
propose second-order cone (SOC) relaxations of binary quadratic polynomial problems. We use
the polynomial programming framework to re-derive, compare and strengthen existing relaxation
schemes. We present a new second-order and semidefinite-based construction where we are able to
theoretically show that the resulting relaxations provide bounds stronger than other computation-
ally practical semidefinite-based 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 second-order
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 SOC-SDP-based
relaxations give the best bounds. Our experiments also show that the purely SOC-based 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 n-tuple 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 non-negative integer d = |α| :=?n
iαi. There are N =?n+d
d
?
monomials of

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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 non-negative 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:
(PP-P) 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 polynomial-time 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 sum-of-squares and
polynomials with non-negative 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 non-negative 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].

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Consider λ to be the optimal value for (PP-P), 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 (PP-P) as:
(PP-D) 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 re-phrased 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 re-cast as a semidefinite
programming problem [35; 39].
The condition λ − f(x) ≥ 0 for all x ∈ S is NP-hard 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, second-order, and semidefinite cones, is essential to be able to solve the resulting
relaxation efficiently using interior-point methods.
2.3. Sum-of-Squares and Second-Order 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
sum-of-squares 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 counter-example is the
Motzkin polynomial [23].
i=1qi(x)2, or equivalently, p∈Ψ. If p(x) is a sum-of-squares polynomial then it is a non-negative
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 polynomial-time.
The following results will allow us to use second-order cone relaxations when working with
d/2
?×?n+d/2
d/2
?. In addition, we have?n+d
d
?
non-negative polynomials over the ball B :={x:?x?2=n}.

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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 non-negativity 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 second-order cones is their tractability. As a result, we can use
these techniques to compute global upper bounds for (PP-P).
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 mixed-binary quadratic polynomial programs with bounded continuous variables.
There are many well-known problems that can be naturally written as binary quadratic poly-
nomial problems. For instance, folding of proteins in three-dimension 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 single-row facility layout problem [2]
and the quadratic assignment problem [21].