A standard trick in integer programming is to replace bounded integer variables with

binary variables, using a bit representation. In a previous paper, we showed that this process

can be used to improve linear programming relaxations of mixed-integer quadratic

programs. In this paper, we show that it can also be used to improve {\em semidefinite}\/

programming relaxations

Galli, Laura

Grainger, Daniel J.

Letchford, Adam N.

Università di Pisa

Universita` di Pisa
Dipartimento di Informatica
Technical Report
Bit Representation Can
Improve SDP Relaxations of
Mixed-Integer Quadratic
Programs
Laura Galli Daniel J. Grainger Adam N. Letchford
July 2018
LICENSE: Creative Commons: Attribution-Noncommercial - No Derivative Works
ADDRESS: Largo B. Pontecorvo 3, 56127 Pisa, Italy. TEL: +39 050 2212700 FAX: +39 050 2212726

Bit Representation Can Improve SDP Relaxations
of Mixed-Integer Quadratic Programs
Laura Galli∗ Daniel J. Grainger† Adam N. Letchford‡
July 2018
Abstract
A standard trick in integer programming is to replace bounded in-
teger variables with binary variables, using a bit representation. In a
previous paper, we showed that this process can be used to improve
linear programming relaxations of mixed-integer quadratic programs.
In this paper, we show that it can also be used to improve semidefinite
programming relaxations.
Keywords: mixed-integer nonlinear programming, global optimisa-
tion, semidefinite programming.
1 Introduction
A folklore result in integer programming is that bounded integer-constrained
variables can be replaced with binary variables. In particular, if xi is known
to lie in [0, ui] ∩ Z, where ui ≥ 2 and integer, then one can replace xi with
its bit representation
blog2 uic∑
s=0
2s x˜is,
where the x˜is are new binary variables [33].
In [4, 25, 26, 29], bit representation was used to derive strong linear pro-
gramming (LP) relaxations of bounded mixed-integer linear programs. It
has also been used to derive strong LP and quadratic programming (QP) re-
laxations of bounded non-convex mixed-integer quadratic programs (MIQPs)
[3, 14, 16]. Our goal in this paper is to show that it can also be used to de-
rive strong semidefinite programming (SDP) relaxations of MIQPs. Perhaps
surprisingly, this is true even in the convex case.
∗Department of Computer Science, University of Pisa, Largo B. Pontecorvo 3, 56124
Pisa, Italy. E-mail: Laura.Galli@di.unipi.it
†Former PhD student at Lancaster University.
‡Department of Management Science, Lancaster University, Lancaster LA1 4YW,
United Kingdom. E-mail: A.N.Letchford@lancaster.ac.uk
1
The paper has a very simple structure. The relevant literature is re-
viewed in Section 2, and the new results are presented in Section 3.
Throughout the paper, we assume that we are given a bounded MIQP
with n variables and m constraints, in the form:
min xTQx + c · x (1)
Ax ≤ b (2)
xi ∈ [0, ui] (i ∈ N) (3)
xi ∈ Z (i ∈ I), (4)
where Q ∈ Qn×n, c ∈ Qn, A ∈ Qm×n, b ∈ Qm, N = {1, . . . , n}, I ⊆ N ,
and ui > 0 for all i ∈ N . We assume without loss of generality that Q
is symmetric and that the ui are positive integers. We also let ri denote
blog2 uic for all i ∈ I. Finally, we let n˜ denote n− |I|+
∑
i∈I(ri + 1), which
is the total number of variables obtained when the MIQP is converted into
a mixed 0-1 QP, using bit representation.
2 Literature Review
For brevity, we review only SDP relaxations of quadratic problems in this
section. For surveys of LP relaxations of quadratic problems, see, e.g., [7,
9, 14]. For surveys of SDP relaxations of 0-1 LPs, see, e.g., [20, 21].
2.1 SDP relaxation of 0-1 QPs
We begin with the special case of 0-1 QPs, i.e., problems of the form:
min
{
xTQx : Ax ≤ b, x ∈ {0, 1}n
}
. (5)
The standard SDP relaxation has its roots in early work of Lova´sz [23],
Shor [31] and Ko¨rner [19]. We follow the presentation given in [22, 24, 27].
Introduce the matrix variable X = xxT , along with the augmented matrix
X+ =
(
1
x
)(
1
x
)T
=
(
1 xT
x X
)
.
Note that X+ is psd by definition. Moreover, the condition xi ∈ {0, 1} is
equivalent to the (non-convex) quadratic equation x2i = xi. Accordingly, we
can impose Xii = xi for all i, or, equivalently, diag(X) = x. This leads to
the following SDP relaxation of the 0-1 QP (5):
min
{
Q •X : Ax ≤ b, diag(X) = x, X+ ∈ Sn+1+
}
,
where ‘•’ denotes inner product and Sn+1+ denotes the cone of (real) psd
matrices of order n + 1. This basic relaxation can be strengthened via the
2
addition of various kinds of cutting planes; see, e.g., [1, 17, 18, 24, 32]. As
a simple example, one can add the constraints
Xij ≥ 0, Xij ≤ xi, Xij ≤ xj , Xij ≥ xi + xj − 1 (6)
for 1 ≤ i < j ≤ n (e.g., [1, 18, 24]).
2.2 SDP relaxation of nonconvex QPs
Now consider a non-convex QP of the form:
min
{
xTQx + c · x : Ax ≤ b, x ∈ Rn+
}
.
Constructing the augmented matrix X+, as before, we obtain the following
SDP relaxation [13, 28, 31]:
inf
{
Q •X + c · x : Ax ≤ b, X+ ∈ Sn+1+
}
.
As noted by Anstreicher [2], however, this SDP is unbounded in general. To
make it bounded, one must suppose that x ≤ u for some known u ∈ Qn+.
One can then enforce boundedness by adding the constraints
Xii ≤ uixi (i ∈ N). (7)
Anstreicher also notes one can strengthen the relaxation further by adding
Xij ≥ 0, Xij ≤ ujxi, Xij ≤ uixj , Xij ≥ ujxi + uixj − uiuj (8)
for 1 ≤ i < j ≤ n. Note that these constraints generalise (6). Additional
cutting planes can be found in, e.g., [2, 6, 7, 12, 24, 30].
2.3 SDP relaxation of nonconvex MIQPs
Finally, we consider the case of general bounded MIQPs of the form (1)–
(4). Given the constructions mentioned in the previous two subsections, a
natural SDP relaxation is
inf
{
Q •X + c · x : Ax ≤ b, (7), X+ ∈ Sn+1+
}
.
One can add the constraints (8) to strengthen this relaxation. Other valid
inequalities can be found in, e.g., [5, 7, 8, 9, 15].
Billionnet et al. [3] suggest a rather different approach to bounded MIQPs.
First, they convert the MIQP into a mixed 0-1 QP, via bit representation.
Second, they show that, under certain technical assumptions, the mixed 0-
1 QP can be convexified. Finally, they solve the convexified problem via
branch-and-bound, with convex QP relaxations.
The motivation for applying bit representation in [3] was to convert the
MIQP into a convex problem. We will show that bit representation has a
happy side-effect: it can make the SDP relaxation stronger.
3
3 The Effect of Bit Representation
We now examine the effect of bit representation on the quality of lower
bounds from SDP relaxations. In Subsection 3.1, we show that, if applied
intelligently, it will never make the bounds worse. In Subsection 3.2, we
show that it can make the bounds better, and explain why.
3.1 Bit representation never makes the bound worse
A major issue that could affect our analysis is that there are several different
SDP relaxations of MIQP, which can be obtained by either including or
omitting various families of cutting planes. To get around this, the following
theorem deals with a “generic” SDP relaxation.
Theorem 1 Consider a bounded MIQP of the form (1)–(4), and suppose
that we have constructed an SDP relaxation of it (possibly with various cut-
ting planes added). Let us write the SDP in the form:
inf Q0 •X + c0 · x
s.t. Qj •X + cj · x ≤ bj (j = 1, . . . ,m)
X+ =
(
1 xT
x X
)
 0,
where c0, . . . , cm ∈ Qn, Q0, . . . Qm ∈ Qn×n and b1, . . . , bm ∈ Q. Now suppose
that we apply the following three-step procedure. The first step is to replace:
• xi with
∑ri
s=0 2
sx˜is for i ∈ I,
• Xii with
∑ri
s=0
∑ri
t=0 2
s+t X˜isit for i ∈ I,
• Xij with
∑ri
s=0
∑rj
t=0 2
s+t X˜isjt for {i, j} ⊆ I,
• Xij with
∑ri
s=0 2
sX˜isj for i ∈ I and j ∈ N \ I.
The second step is to add the constraints
X˜isis = x˜is (i = 1, . . . , q; s = 1, . . . , ri).
In the third step, instead of imposing psd-ness on the augmented matrix X+,
we impose it on the expanded augmented matrix
X˜+ =
(
1 x˜T
x˜ X˜
)
,
where x˜ is the vector obtained from x by replacing each component xi, for
i ∈ I, with the components x˜i,0, . . . , x˜i,ri, and X˜ is a matrix variable intended
to represent x˜x˜T . (Note that x˜ ∈ Rn˜ and X˜ ∈ Rn˜×n˜, where n˜ is as defined
at the end of Section 1.)
Then, the lower bound from the new SDP relaxation is no smaller than
the one from the original relaxation.
4
Proof. It suffices to show that, given any feasible solution of the expanded
SDP, there exists a feasible solution of the original SDP of the same value. To
this end, let (x˜∗, X˜∗) ∈ Rn˜+(n˜)2 be a feasible solution to the expanded SDP,
let (X˜+)∗ ∈ R(n˜+1)2 be the corresponding expanded augmented matrix,
let (x∗, X∗) ∈ Rn+n2 be the corresponding pair defined by the mapping
above, and let (X+)∗ ∈ R(n+1)2 be the corresponding augmented matrix.
By construction, (x∗, X∗) satisfies all of the constraints in the original SDP,
and has the same cost as (x˜∗, X˜∗). It therefore only remains to be shown
that (X+)∗ is psd. This is equivalent to showing that vT (X+)∗ v ≥ 0 for all
vectors v ∈ Rn+1. So, let v = (v0, . . . , vn)T be such a vector and construct
an expanded vector v˜ as follows. For i ∈ I, we replace the component vi
with the following ri + 1 components:
vi, 2vi, . . . , 2
rivi.
Now, since (X˜+)∗ is psd by assumption, we have v˜T (X˜+)∗ v˜ ≥ 0. But
vT (X+)∗ v = v˜T (X˜+)∗ v˜ by construction. 
3.2 Bit representation can make the bound better
The following example shows that bit representation can lead to an im-
provement in the lower bound from the SDP relaxation, even when (a) the
objective function is convex, (b) there is only one variable, and (c) there are
no constraints (apart from trivial bounds).
Example: Consider the trivial IQP
min
{
x21 − 3x1 : x1 ∈ Z ∩ [0, 3]
}
.
Following Anstreicher [2], the standard SDP relaxation of this IQP is
min
{
X11 − 3x1 : x1 ∈ [0, 3], X11 ≤ 3x1, X+ =
(
1 x1
x1 X11
)
 0
}
.
One can check (either by hand or with an SDP solver) that the unique
optimal solution to this SDP has x1 = 3/2 and X11 = 9/4, giving a lower
bound of −9/4. On the other hand, if we convert the IQP into a 0-1 QP,
via bit representation, we obtain
min
{
x˜210 + 4x˜10x˜11 + 4x˜
2
11 − 3x˜10 − 6x˜11 : x˜ ∈ {0, 1}2
}
.
The simplest SDP relaxation of this 0-1 QP is
min X˜1010 + 4X˜1011 + 4X˜1111 − 3x˜10 − 6x˜11
s.t. x˜1s = X˜1s1s (s = 0, 1)
X˜+ =
 1 x˜10 x˜11x˜10 X˜1010 X˜1011
x˜11 X˜1110 X˜1111
  0.
5
Xx
0 u
u2
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqq
qqq
Figure 1: The convex set F (u).
One can check with an SDP solver that all optimal solutions to this SDP
have cost −2. So, the transformed SDP yields an improved (in fact optimal)
lower bound for this instance. 
The reason for the improvement in the bound appears to be the con-
straint diag(X˜) = x˜, which has no analog in the general-integer case. The
following lemma and theorem explore this phenomenon in more detail.
Lemma 1 Let u1 be a positive integer. Consider a trivial IQP of the form
min
{
Q11x
2
1 + c1x1 : x1 ∈ Z ∩ [0, u1]
}
, (9)
along with its SDP relaxation
min
{
Q11X11 + c1x1 : x1 ∈ [0, u1], X11 ≤ u1x1, X+  0
}
. (10)
Let F (u) denote the feasible region of this SDP. We have:
F (u) = conv
{
(x1, X11) ∈ R2 : 0 ≤ x1 ≤ u1, X11 = x21
}
.
Proof. This follows from the fact that X+ is psd if and only if its deter-
minant is non-negative, i.e., if and only if X11 ≥ x21 (see Figure 1). 
In other words, the SDP relaxation (10) does not exploit the integrality of
x1 in any way.
Theorem 2 Consider again the trivial IQP (9). Suppose we apply the pro-
6
cedure in Theorem 1 to the SDP relaxation (10), yielding the SDP relaxation
min Q11
r1∑
s=0
r1∑
t=0
2s+t X˜1s1t + c1
r1∑
s=0
2s x˜1s
s.t. 0 ≤
r1∑
s=0
2s x˜1s ≤ u1
r1∑
s=0
r1∑
t=0
2s+t X˜1s1t ≤ u1
r1∑
s=0
2s x˜1s
diag(X˜) = x˜
X˜+ =
(
1 x˜T
x˜ X˜
)
 0.
Now, suppose that we project the feasible region of this SDP onto a two-
dimensional subspace having x1 and X11 as axes, using the linear mappings
x1 =
r1∑
s=0
2sx˜1s
and
X11 =
ri∑
s=0
ri∑
t=0
2s+tX˜1s1t.
Then the projection satisfies the linear inequality
X11 ≥ u¯x1 − (u¯
2 − 1)
4
, (11)
where u¯ =
∑r1
s=0 2
s = 2r1+1 − 1. (See Figure 2 for an illustration for the
case u1 = u¯ = 3.)
Proof. For ease of notation, let us drop the index 1, writing r for r1, x˜s
for x˜1s and X˜st for X˜1s1t. Let v ∈ Zr+1+ be the vector with
• vs = 2s for s = 0, . . . , r − 1;
• vr = 2r − 1.
Also let w = 2r − 1 = bu¯/2c. Since X˜+ is psd, we have:
(−w vT )(1 x˜T
x˜ X˜
)(−w
v
)
≥ 0,
or, equivalently, vT X˜v ≥ (2w)vT x˜− w2. Expanding this gives:
r∑
s=0
r∑
t=0
vsvt X˜st ≥ (u¯− 1)
r∑
s=0
vsx˜s − w2. (12)
7
ﬀX11
x10 1 2 3
9
extra inequality
X11 ≥ 3x1 − 2
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqq
 
 
 
 
 
 
  
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqq
qq
Figure 2: Projection of SDP relaxation with bit representation when u1 = 3.
Next, observe that, for s = 0, . . . , r − 1, we have
(−1 1 1)
 1 x˜s x˜rx˜s X˜ss X˜sr
x˜r X˜sr X˜rr
−11
1
 ≥ 0,
which, together with the equation diag(X˜) = x˜, implies 2X˜sr ≥ x˜s + x˜r − 1.
Multiplying these last inequalities by 2s and summing over all s yields:
r−1∑
s=0
2s+1X˜sr ≥
r−1∑
s=0
2sx˜s + wxr − w. (13)
Finally, summing the inequalities (12) and (13), along with the trivial in-
equality u¯ X˜st ≥ 0, we obtain
r∑
s=0
r∑
t=0
2s+t X˜st ≥ u¯
r∑
s=0
2sx˜s − w(w + 1),
which is equivalent to (11). 
We remark that an alternative (but longer) proof of Theorem 2 can be
derived by using Theorem 3.2 of Delorme & Poljak [10] along with the so-
called ‘covariance mapping’ (see Deza & Laurent [11], Section 5.2). We omit
the details for the sake of brevity.
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Bit Representation Can Improve SDP Relaxations of Mixed-Integer Quadratic Programs

UnipiEprints

Universita` di PisaDipartimento di InformaticaTechnical ReportBit Representation CanImprove SDP Relaxations ofMixed-Integer QuadraticProgramsLaura Galli Daniel J. Grainger Adam N. LetchfordJuly 2018LICENSE: Creative Commons: Attribution-Noncommercial - No Derivative WorksADDRESS: Largo B. Pontecorvo 3, 56127 Pisa, Italy. TEL: +39 050 2212700 FAX: +39 050 2212726Bit Representation Can Improve SDP Relaxationsof Mixed-Integer Quadratic ProgramsLaura Galli∗ Daniel J. Grainger† Adam N. Letchford‡July 2018AbstractA standard trick in integer programming is to replace bounded in-teger variables with binary variables, using a bit representation. In aprevious paper, we showed that this process can be used to improvelinear programming relaxations of mixed-integer quadratic programs.In this paper, we show that it can also be used to improve semidefiniteprogramming relaxations.Keywords: mixed-integer nonlinear programming, global optimisa-tion, semidefinite programming.1 IntroductionA folklore result in integer programming is that bounded integer-constrainedvariables can be replaced with binary variables. In particular, if xi is knownto lie in [0, ui] ∩ Z, where ui ≥ 2 and integer, then one can replace xi withits bit representationblog2 uic∑s=02s x˜is,where the x˜is are new binary variables [33].In [4, 25, 26, 29], bit representation was used to derive strong linear pro-gramming (LP) relaxations of bounded mixed-integer linear programs. Ithas also been used to derive strong LP and quadratic programming (QP) re-laxations of bounded non-convex mixed-integer quadratic programs (MIQPs)[3, 14, 16]. Our goal in this paper is to show that it can also be used to de-rive strong semidefinite programming (SDP) relaxations of MIQPs. Perhapssurprisingly, this is true even in the convex case.∗Department of Computer Science, University of Pisa, Largo B. Pontecorvo 3, 56124Pisa, Italy. E-mail: Laura.Galli@di.unipi.it†Former PhD student at Lancaster University.‡Department of Management Science, Lancaster University, Lancaster LA1 4YW,United Kingdom. E-mail: A.N.Letchford@lancaster.ac.uk1The paper has a very simple structure. The relevant literature is re-viewed in Section 2, and the new results are presented in Section 3.Throughout the paper, we assume that we are given a bounded MIQPwith n variables and m constraints, in the form:min xTQx + c · x (1)Ax ≤ b (2)xi ∈ [0, ui] (i ∈ N) (3)xi ∈ Z (i ∈ I), (4)where Q ∈ Qn×n, c ∈ Qn, A ∈ Qm×n, b ∈ Qm, N = {1, . . . , n}, I ⊆ N ,and ui > 0 for all i ∈ N . We assume without loss of generality that Qis symmetric and that the ui are positive integers. We also let ri denoteblog2 uic for all i ∈ I. Finally, we let n˜ denote n− |I|+∑i∈I(ri + 1), whichis the total number of variables obtained when the MIQP is converted intoa mixed 0-1 QP, using bit representation.2 Literature ReviewFor brevity, we review only SDP relaxations of quadratic problems in thissection. For surveys of LP relaxations of quadratic problems, see, e.g., [7,9, 14]. For surveys of SDP relaxations of 0-1 LPs, see, e.g., [20, 21].2.1 SDP relaxation of 0-1 QPsWe begin with the special case of 0-1 QPs, i.e., problems of the form:min{xTQx : Ax ≤ b, x ∈ {0, 1}n}. (5)The standard SDP relaxation has its roots in early work of Lova´sz [23],Shor [31] and Ko¨rner [19]. We follow the presentation given in [22, 24, 27].Introduce the matrix variable X = xxT , along with the augmented matrixX+ =(1x)(1x)T=(1 xTx X).Note that X+ is psd by definition. Moreover, the condition xi ∈ {0, 1} isequivalent to the (non-convex) quadratic equation x2i = xi. Accordingly, wecan impose Xii = xi for all i, or, equivalently, diag(X) = x. This leads tothe following SDP relaxation of the 0-1 QP (5):min{Q •X : Ax ≤ b, diag(X) = x, X+ ∈ Sn+1+},where ‘•’ denotes inner product and Sn+1+ denotes the cone of (real) psdmatrices of order n + 1. This basic relaxation can be strengthened via the2addition of various kinds of cutting planes; see, e.g., [1, 17, 18, 24, 32]. Asa simple example, one can add the constraintsXij ≥ 0, Xij ≤ xi, Xij ≤ xj , Xij ≥ xi + xj − 1 (6)for 1 ≤ i < j ≤ n (e.g., [1, 18, 24]).2.2 SDP relaxation of nonconvex QPsNow consider a non-convex QP of the form:min{xTQx + c · x : Ax ≤ b, x ∈ Rn+}.Constructing the augmented matrix X+, as before, we obtain the followingSDP relaxation [13, 28, 31]:inf{Q •X + c · x : Ax ≤ b, X+ ∈ Sn+1+}.As noted by Anstreicher [2], however, this SDP is unbounded in general. Tomake it bounded, one must suppose that x ≤ u for some known u ∈ Qn+.One can then enforce boundedness by adding the constraintsXii ≤ uixi (i ∈ N). (7)Anstreicher also notes one can strengthen the relaxation further by addingXij ≥ 0, Xij ≤ ujxi, Xij ≤ uixj , Xij ≥ ujxi + uixj − uiuj (8)for 1 ≤ i < j ≤ n. Note that these constraints generalise (6). Additionalcutting planes can be found in, e.g., [2, 6, 7, 12, 24, 30].2.3 SDP relaxation of nonconvex MIQPsFinally, we consider the case of general bounded MIQPs of the form (1)–(4). Given the constructions mentioned in the previous two subsections, anatural SDP relaxation isinf{Q •X + c · x : Ax ≤ b, (7), X+ ∈ Sn+1+}.One can add the constraints (8) to strengthen this relaxation. Other validinequalities can be found in, e.g., [5, 7, 8, 9, 15].Billionnet et al. [3] suggest a rather different approach to bounded MIQPs.First, they convert the MIQP into a mixed 0-1 QP, via bit representation.Second, they show that, under certain technical assumptions, the mixed 0-1 QP can be convexified. Finally, they solve the convexified problem viabranch-and-bound, with convex QP relaxations.The motivation for applying bit representation in [3] was to convert theMIQP into a convex problem. We will show that bit representation has ahappy side-effect: it can make the SDP relaxation stronger.33 The Effect of Bit RepresentationWe now examine the effect of bit representation on the quality of lowerbounds from SDP relaxations. In Subsection 3.1, we show that, if appliedintelligently, it will never make the bounds worse. In Subsection 3.2, weshow that it can make the bounds better, and explain why.3.1 Bit representation never makes the bound worseA major issue that could affect our analysis is that there are several differentSDP relaxations of MIQP, which can be obtained by either including oromitting various families of cutting planes. To get around this, the followingtheorem deals with a “generic” SDP relaxation.Theorem 1 Consider a bounded MIQP of the form (1)–(4), and supposethat we have constructed an SDP relaxation of it (possibly with various cut-ting planes added). Let us write the SDP in the form:inf Q0 •X + c0 · xs.t. Qj •X + cj · x ≤ bj (j = 1, . . . ,m)X+ =(1 xTx X) 0,where c0, . . . , cm ∈ Qn, Q0, . . . Qm ∈ Qn×n and b1, . . . , bm ∈ Q. Now supposethat we apply the following three-step procedure. The first step is to replace:• xi with∑ris=0 2sx˜is for i ∈ I,• Xii with∑ris=0∑rit=0 2s+t X˜isit for i ∈ I,• Xij with∑ris=0∑rjt=0 2s+t X˜isjt for {i, j} ⊆ I,• Xij with∑ris=0 2sX˜isj for i ∈ I and j ∈ N \ I.The second step is to add the constraintsX˜isis = x˜is (i = 1, . . . , q; s = 1, . . . , ri).In the third step, instead of imposing psd-ness on the augmented matrix X+,we impose it on the expanded augmented matrixX˜+ =(1 x˜Tx˜ X˜),where x˜ is the vector obtained from x by replacing each component xi, fori ∈ I, with the components x˜i,0, . . . , x˜i,ri, and X˜ is a matrix variable intendedto represent x˜x˜T . (Note that x˜ ∈ Rn˜ and X˜ ∈ Rn˜×n˜, where n˜ is as definedat the end of Section 1.)Then, the lower bound from the new SDP relaxation is no smaller thanthe one from the original relaxation.4Proof. It suffices to show that, given any feasible solution of the expandedSDP, there exists a feasible solution of the original SDP of the same value. Tothis end, let (x˜∗, X˜∗) ∈ Rn˜+(n˜)2 be a feasible solution to the expanded SDP,let (X˜+)∗ ∈ R(n˜+1)2 be the corresponding expanded augmented matrix,let (x∗, X∗) ∈ Rn+n2 be the corresponding pair defined by the mappingabove, and let (X+)∗ ∈ R(n+1)2 be the corresponding augmented matrix.By construction, (x∗, X∗) satisfies all of the constraints in the original SDP,and has the same cost as (x˜∗, X˜∗). It therefore only remains to be shownthat (X+)∗ is psd. This is equivalent to showing that vT (X+)∗ v ≥ 0 for allvectors v ∈ Rn+1. So, let v = (v0, . . . , vn)T be such a vector and constructan expanded vector v˜ as follows. For i ∈ I, we replace the component viwith the following ri + 1 components:vi, 2vi, . . . , 2rivi.Now, since (X˜+)∗ is psd by assumption, we have v˜T (X˜+)∗ v˜ ≥ 0. ButvT (X+)∗ v = v˜T (X˜+)∗ v˜ by construction. 3.2 Bit representation can make the bound betterThe following example shows that bit representation can lead to an im-provement in the lower bound from the SDP relaxation, even when (a) theobjective function is convex, (b) there is only one variable, and (c) there areno constraints (apart from trivial bounds).Example: Consider the trivial IQPmin{x21 − 3x1 : x1 ∈ Z ∩ [0, 3]}.Following Anstreicher [2], the standard SDP relaxation of this IQP ismin{X11 − 3x1 : x1 ∈ [0, 3], X11 ≤ 3x1, X+ =(1 x1x1 X11) 0}.One can check (either by hand or with an SDP solver) that the uniqueoptimal solution to this SDP has x1 = 3/2 and X11 = 9/4, giving a lowerbound of −9/4. On the other hand, if we convert the IQP into a 0-1 QP,via bit representation, we obtainmin{x˜210 + 4x˜10x˜11 + 4x˜211 − 3x˜10 − 6x˜11 : x˜ ∈ {0, 1}2}.The simplest SDP relaxation of this 0-1 QP ismin X˜1010 + 4X˜1011 + 4X˜1111 − 3x˜10 − 6x˜11s.t. x˜1s = X˜1s1s (s = 0, 1)X˜+ = 1 x˜10 x˜11x˜10 X˜1010 X˜1011x˜11 X˜1110 X˜1111  0.5Xx0 uu2qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqFigure 1: The convex set F (u).One can check with an SDP solver that all optimal solutions to this SDPhave cost −2. So, the transformed SDP yields an improved (in fact optimal)lower bound for this instance. The reason for the improvement in the bound appears to be the con-straint diag(X˜) = x˜, which has no analog in the general-integer case. Thefollowing lemma and theorem explore this phenomenon in more detail.Lemma 1 Let u1 be a positive integer. Consider a trivial IQP of the formmin{Q11x21 + c1x1 : x1 ∈ Z ∩ [0, u1]}, (9)along with its SDP relaxationmin{Q11X11 + c1x1 : x1 ∈ [0, u1], X11 ≤ u1x1, X+  0}. (10)Let F (u) denote the feasible region of this SDP. We have:F (u) = conv{(x1, X11) ∈ R2 : 0 ≤ x1 ≤ u1, X11 = x21}.Proof. This follows from the fact that X+ is psd if and only if its deter-minant is non-negative, i.e., if and only if X11 ≥ x21 (see Figure 1). In other words, the SDP relaxation (10) does not exploit the integrality ofx1 in any way.Theorem 2 Consider again the trivial IQP (9). Suppose we apply the pro-6cedure in Theorem 1 to the SDP relaxation (10), yielding the SDP relaxationmin Q11r1∑s=0r1∑t=02s+t X˜1s1t + c1r1∑s=02s x˜1ss.t. 0 ≤r1∑s=02s x˜1s ≤ u1r1∑s=0r1∑t=02s+t X˜1s1t ≤ u1r1∑s=02s x˜1sdiag(X˜) = x˜X˜+ =(1 x˜Tx˜ X˜) 0.Now, suppose that we project the feasible region of this SDP onto a two-dimensional subspace having x1 and X11 as axes, using the linear mappingsx1 =r1∑s=02sx˜1sandX11 =ri∑s=0ri∑t=02s+tX˜1s1t.Then the projection satisfies the linear inequalityX11 ≥ u¯x1 − (u¯2 − 1)4, (11)where u¯ =∑r1s=0 2s = 2r1+1 − 1. (See Figure 2 for an illustration for thecase u1 = u¯ = 3.)Proof. For ease of notation, let us drop the index 1, writing r for r1, x˜sfor x˜1s and X˜st for X˜1s1t. Let v ∈ Zr+1+ be the vector with• vs = 2s for s = 0, . . . , r − 1;• vr = 2r − 1.Also let w = 2r − 1 = bu¯/2c. Since X˜+ is psd, we have:(−w vT )(1 x˜Tx˜ X˜)(−wv)≥ 0,or, equivalently, vT X˜v ≥ (2w)vT x˜− w2. Expanding this gives:r∑s=0r∑t=0vsvt X˜st ≥ (u¯− 1)r∑s=0vsx˜s − w2. (12)7ﬀX11x10 1 2 39extra inequalityX11 ≥ 3x1 − 2qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq        qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqFigure 2: Projection of SDP relaxation with bit representation when u1 = 3.Next, observe that, for s = 0, . . . , r − 1, we have(−1 1 1) 1 x˜s x˜rx˜s X˜ss X˜srx˜r X˜sr X˜rr−111 ≥ 0,which, together with the equation diag(X˜) = x˜, implies 2X˜sr ≥ x˜s + x˜r − 1.Multiplying these last inequalities by 2s and summing over all s yields:r−1∑s=02s+1X˜sr ≥r−1∑s=02sx˜s + wxr − w. (13)Finally, summing the inequalities (12) and (13), along with the trivial in-equality u¯ X˜st ≥ 0, we obtainr∑s=0r∑t=02s+t X˜st ≥ u¯r∑s=02sx˜s − w(w + 1),which is equivalent to (11). 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Bit Representation Can Improve SDP Relaxations of Mixed-Integer Quadratic Programs

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