An Explicit Exact SDP Relaxation for Nonlinear
0-1 Programs
Jean B. Lasserre
LAAS-CNRS, 7 Avenue du Colonel Roche,
31077 Toulouse Cédex, France
[email protected]
http://www.laas.fr/˜lasserre
Abstract. We consider the general nonlinear optimization problem in 01 variables and provide an explicit equivalent convex positive semidefinite
program in 2n − 1 variables. The optimal values of both problems are
identical. From every optimal solution of the former one easily find an
optimal solution of the latter and conversely, from every solution of the
latter one may construct an optimal solution of the former.
1
Introduction
This paper is concerned with the general nonlinear problem in 0-1 variables
IP → p∗ :=
min {g0 (x) | gk (x) ≥ 0, k = 1, . . . m}
x∈{0,1}n
(1)
where all the gk (x) : IRn → IR are real-valued polynomials of degree 2vk −1 if odd
or 2vk if even. This general formulation encompasses 0-1 linear and nonlinear
programs, among them the quadratic assignment problem. For the MAX-CUT
problem, the discrete set {0, 1}n is replaced by {−1, 1}n .
In our recent work [4,5,6], and for general optimization problems involving
polynomials, we have provided a sequence {Qi } of positive semidefinite (psd) programs (or SDP relaxations) with the property that inf Qi ↑ p∗ as i → ∞, under a
certain assumption on the semi-algebraic feasible set {gk (x) ≥ 0, k = 1, . . . m}.
The approach was based on recent results in algebraic geometry on the representation of polynomials, positive on a compact semi-algebraic set, a theory
dual to the theory of moments. For general 0 − 1 programs, that is, with the
additional constraint x ∈ {0, 1}n , we have shown in Lasserre [5] that this assumption on the feasible set is automatically satisfied and the SDP relaxations
{Qi } simplify to a specific form with at most 2n − 1 variables, no matter how
large is i. The approach followed in [4] and [5] is different in spirit from the lift
and project iterative procedure of Lovász and Schrijver [7] for 0-1 programs (see
also extensions in Kojima and Tunçel [3]), which requires that a weak separation
oracle is available for the homogeneous cone associated with the constraints. In
the lift and project procedure, the description of the convex hull of the feasible
set is implicit (via successive projections) whereas we provide in this paper an
K. Aardal, B. Gerards (Eds.): IPCO 2001, LNCS 2081, pp. 293–303, 2001.
c Springer-Verlag Berlin Heidelberg 2001
294
J.B. Lasserre
explicit description of the equivalent convex psd program, with a simple interpretation. Although different, our approach is closer in spirit to the successive
LP relaxations in the RLT procedure of Sherali and Adams [10] for 0-1 linear
programs in which each of the linear original constraints is multiplied by suitable
polynomials of the form Πi∈J1 xi Πj∈J2 (1 − xj ) and then linearized in a higher
dimension space via several changes of variables to obtain an LP program. The
last relaxation in the hierarchy of RLT produces the convex hull of the feasible
set. This also extends to a special class of 0-1 polynomial programs (see Sherali
and Adams [10]).
In the present paper, we show that in addition to the asymptotic convergence already proved in [4,5], the sequence of convex SDP relaxations {Qi } is
in fact finite, that is, the optimal value p∗ is also min Qi for all i ≥ n + v with
v := maxk vk . Moreover, every optimal solution y ∗ of Qi is the (finite) vector
of moments of some probability measure supported on optimal solutions of IP.
Therefore, every 0-1 program is in fact equivalent to a continuous convex psd
program in 2n − 1 variables for which an explicit form as well as a simple interpretation are available. The projection of the feasible set defined by the LMI
constraints of this psd program onto the subspace spanned by the first n variables, provides the convex hull of the original feasible set. Note that the result
holds for arbitrary 0-1 constrained programs, that is, with arbitrary polynomial criteria and constraints (no weak separation oracle is needed as in the lift
and project procedure [7], and no special form of IP is assumed as in [10]). This
is because the theory of representation of polynomials positive on a compact
semi-algebraic set and its dual theory of moments, make no assumption on the
semi-algebraic set, except compactness (it can be nonconvex, disconnected). For
illustration purposes, we provide the equivalent psd programs for quadratic 0-1
programs and MAX-CUT problems in IR3 .
For practical computational purposes, the preceding result is of little value
for the number of variables grows exponentially with the size of the problem.
Fortunately, in many cases, the optimal value is also the optimal value of some
Qi for i n. For instance, on a sample of 50 randomly generated MAX-CUT
problems in IR10 , the optimal value p∗ was always obtained at the Q2 relaxation
(in which case Q2 is a psd program with “only” 385 variables to compare with
210 − 1 = 1023).
2
Notation and Definitions
We adopt the notation in Lasserre [4] that for sake of clarity, we reproduce here.
Given any two real-valued symmetric matrices A, B let hA, Bi denote the usual
scalar product trace(AB) and let A B (resp. A B) stand for A − B positive
semidefinite (resp. A − B positive definite). Let
1, x1 , x2 , . . . xn , x21 , x1 x2 , . . . , x1 xn , x2 x3 , . . . , x2n , . . . , xr1 , . . . , xrn ,
(2)
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
295
be a basis for the r-degree real-valued polynomials and let s(r) be its dimension.
Therefore, a r-degree polynomial p(x) : IRn → IR is written
p(x) =
X
pα xα ,
x ∈ IRn ,
α
s(r)
αn
1 α2
the coefficients of the
where xα = xα
1 x2 . . . xn . Denote by p = {pα } ∈ IR
polynomial p(x) in the basis (2). Hence, the respective vectors of coefficients of
the polynomials gi (x), i = 0, 1, . . . m, in (1), are denoted {(gi )α } = gi ∈ IRs(wi ) ,
i = 0, 1, . . . m if wi is the degree of gi . We next define the important notions of
moment matrix and localizing matrix.
2.1
Moment Matrix
Given a s(2r)-sequence (1, y1 , . . . , ), let Mr (y) be the moment matrix of dimension s(r) (denoted M (r) in Curto and Fialkow [1]), with rows and columns
labelled by (2). For instance, for illustration purposes, consider the 2-dimensional
case. The moment matrix Mr (y) is the block matrix {Mi,j (y)}0≤i,j≤2r defined
by


yi+j,0 yi+j−1,1 . . . yi,j
 yi+j−1,1 yi+j−2,2 . . . yi−1,j+1 
.
(3)
Mi,j (y) = 
 ...
... ... ... 
yj,i yi+j−1,1 . . . y0,i+j
To fix ideas, when n = 2 and r = 2, one obtains

1 y10 y01 y20 y11
 y10 y20 y11 y30 y21

 y01 y11 y02 y21 y12
M2 (y) = 
 y20 y30 y21 y40 y31

 y11 y21 y12 y31 y22
y02 y12 y03 y22 y13

y02
y12 

y03 
.
y22 

y13 
y04
Another more intuitive way of constructing Mr (y) is as follows. If Mr (1, i) = yα
and Mr (y)(j, 1) = yβ , then
Mr (y)(i, j) = yα+β ,
with α + β = (α1 + β1 , · · · , αn + βn ).
(4)
Mr (y) defines a bilinear form h., .iy on the space Ar of polynomials of degree at
most r, by
hq(x), v(x)iy := hq, Mr (y)vi, q(x), v(x) ∈ Ar ,
and if y is a sequence of moments of some measure µy , then Mr (y) 0 because
Z
q(x)2 µy (dx) ≥ 0, ∀q ∈ IRs(r) .
(5)
hq, Mr (y)qi =
296
2.2
J.B. Lasserre
Localizing Matrix
If the entry (i, j) of the matrix Mr (y) is yβ , let β(i, j) denote the subscript β of
yβ . Next, given a polynomial θ(x) : IRn → IR with coefficient vector θ, we define
the matrix Mr (θy) by
X
θα y{β(i,j)+α} .
(6)
Mr (θy)(i, j) =
α
For instance, with x →
7 θ(x) = a − x21 − x22 , we obtain


a − y20 − y02 , ay10 − y30 − y12 , ay01 − y21 − y03
M1 (θy) =  ay10 − y30 − y12 , ay20 − y40 − y22 , ay11 − y31 − y13  .
ay01 − y21 − y03 , ay11 − y31 − y13 , ay02 − y22 − y04
If y is a sequence of moments of some measure µy , then
Z
θ(x)q(x)2 µy (dx),
∀q ∈ IRs(r) ,
hq, Mr (θy)qi =
(7)
so that Mr (θy) 0 whenever µy has its support contained in the set {θ(x) ≥ 0}.
In Curto and Fialkow [1], Mr (θy) is called a localizing matrix.
The theory of moments identifies those sequences {yα } with Mr (y) 0,
such that the yα ’s are moments of a representing measure µy . The IK-moment
problem identifies those yα ’s whose representing measure µy has its support
contained in the semi-algebraic set IK. In duality with the theory of moments is
the theory of representation of positive polynomials (and polynomials positive
on a semi-algebraic set IK), which dates back to Hilbert’s 17th problem. For
details and recent results, the interested reader is referred to Curto and Fialkow
[1], Schmüdgen [9] and the many references therein.
Both sides (moments and positive polynomials) of this same theory is reflected in the primal and dual SDP relaxations proposed in Lasserre [4], and
used below in the present context of nonlinear 0-1 programs.
3
Main Result
Consider the 0-1 optimization problem IP in (1). Let the semi-algebraic set
IK := {x ∈ {0, 1}n | gi (x) ≥ 0, i = 1, . . . m}
be the feasible set. For the sake of simplicy in the proofs derived later on, we treat
the 0-1 integrality constraints x2i = xi as two opposite inequalities gm+i (x) =
x2i − xi ≥ 0 and gm+n+i (x) = xi − x2i ≥ 0, and we redefine the set IK to be
IK = {gi (x) ≥ 0, i = 1, . . . m + 2n}.
(8)
However, in view of the special form of the constraints gm+k (x) ≥ 0, k = 1, . . . 2n,
we will provide a simpler form of the LMI relaxations {Qi } below.
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
297
Depending on its parity, let wk := 2vk or wk := 2vk − 1 be the degree of the
polynomial gk (x), k = 1, . . . m + 2n. When needed below, for i ≥ maxk wk , the
vectors gk ∈ IRs(wk ) are extended to vectors of IRs(i) by completing with zeros.
As we minimize g0 we may and will assume that its constant term is zero, that
is, g0 (0) = 0.
For i ≥ maxk vk , consider the following family {Qi } of convex psd programs

X

(g0 )α yα
inf

 y
α
(9)
Qi
Mi (y) 0


 M
(g y) 0, k = 1, . . . m + 2n,
i−vk
k
with respective dual problems

m+2n
X



−X(1,
1)
−
gk (0)Zk (1, 1)
sup

 X,Z 0
k
∗
k=1
Qi
m+2n
X




hZk , Cαk i = (g0 )α , ∀α 6= 0,
hX, Bα i +

(10)
k=1
where we have written
X
X
Bα yα ; Mi−vk (gk y) =
Cαk yα , k = 1, . . . m + 2n,
Mi (y) =
α
α
for appropriate real-valued symmetric matrices Bα , Cαk , k = 1, . . . m + 2n.
Interpretation of Qi . The LMI constraints of Qi state necessary conditions
for y to be the vector of moments up to order 2i, of some probability measure
µy with support contained in IK. This clearly implies that inf Qi ≤ p∗ because
the vector of moments of the Dirac measure at a feasible point of IP, is feasible
for Qi .
Interpretation of Q∗i . Let X, Zk 0 be a feasible solution of Q∗i with value ρ.
Write
X
X
tj t0j and Zk =
tkj t0kj , k = 1, . . . m + 2n.
X =
j
j
Then, from the feasibility of (X, Zk ) in Q∗i , it was shown in Lasserre [4,5] that


m+2n
X
X
X
(11)
tj (x)2 +
gk (x) 
tkj (x)2  ,
g0 (x) − ρ =
j
k=1
j
where the polynomials {tj (x)} and {tkj (x)} have respective coefficient vectors
{tj } and {tkj }, in the basis (2).
As ρ ≤ p∗ , g0 (x) − ρ is nonnegative on IK (strictly positive if ρ < p∗ ).
One recognizes in (11) a decomposition into a weighted sum of squares of the
polynomial g0 (x) − p∗ , strictly positive on IK, as in the theory of representation
298
J.B. Lasserre
of polynomials, strictly positive on a compact semi-algebraic set IK (see e.g.,
Schmüdgen [9], Putinar [8], Jacobi and Prestel [2]). Indeed, when the set IK has
a certain property (satisfied here), the “linear” representation (11) holds (see
Lasserre [4,5]). Hence, both programs Qi and Q∗i illustrate the duality between
the theory of moments and the theory of positive polynomials. Among other
results, it has been shown in Lasserre [5, Th. 3.3] that
inf Qi ↑ p∗ as i → ∞.
(12)
From the construction of the localizing matrices in (6), and the form of the polynomials gi for i > m, the constraints Mi−1 (gm+k y) 0 and Mi−1 (gm+n+k y) 0
for k = 1, . . . n (equivalently Mi−1 (gm+k y) = 0), simply state that the variable
yα with α = (α1 , . . . αn ), can be replaced with the variable yβ with βi := 1
whenever αi ≥ 1. Therefore, a simpler form of Qi is obtained as follows.
Ignore the constraints Mi−vk (gk y) 0 for k = m + 1, . . . m + 2n, and make
the above substitution yα → yβ in the matrices Mi (y) and Mi−vk (gk (y)), k =
1, . . . m. For instance, in IR2 (n = 2), the matrix M2 (y) now reads


1 y10 y01 y10 y11 y01
 y10 y10 y11 y10 y11 y11 


 y01 y11 y01 y11 y11 y01 

,
M2 (y) = 
(13)

 y10 y10 y11 y10 y11 y11 
 y11 y11 y11 y11 y11 y11 
y01 y11 y01 y11 y11 y01
and only the variables y10 , y01 , y11 appear in all the relaxations Qi . Interpreted
in terms of polynomials, the integrality constraints x2i = xi , i = 1, . . . n, imply
αn
1 α2
that a monomial xα
1 x2 · · · xn can be replaced by
0 if αi = 0
(14)
xβ1 1 xβ2 2 · · · xβnn with βi =
1 if αi ≥ 1
Therefore, there are are no more than 2n − 1 variables yβ , the number of monomials as in (14), of degree at most n.
Remark 1. At any relaxation Qi , the matrix Mi (y) can be reduced in size. When
looking at the kth column Mi (y)(., k), if Mi (y)(1, k) = Mi (y)(p) for some p < k,
then the whole column Mi (y)(., k) as well as the corresponding line Mi (y)(k, .)
can be deleted. For instance, with M2 (y) as in (13), the constraint M2 (y) 0
can be replaced by


1 y10 y01 y11


c2 (y) :=  y10 y10 y11 y11  0,
M
 y01 y11 y01 y11 
y11 y11 y11 y11
for the 4th and 6th columns of M2 (y) are identical to the 2nd and 3rd columns,
ci (y) of Mi (y), one retains only the
respectively. Thus, in the simplified form M
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
299
columns (and the rows) corresponding to the monomials in the basis (2) that
are distinct after the simplification x2i = xi . The same simplification occurs for
the matrices of the LMI constraints Mi−vk (gk y) 0, k = 1, . . . m.
Next, we begin with the following crucial result:
Proposition 1. (a) All the relaxations Qi involve at most 2n − 1 variables yα .
(b) For all the relaxations Qi with i > n, one has
rank Mi (y) = rank Mn (y).
(15)
Proof. (a) is just a consequence of the comment preceding Proposition 1.
To get (b) observe that with i > n, one may write


Mn (y) | B
−
Mi (y) =  −
0
B |C
for appropriate matrices B, C, and we next prove that each column of B is
identical to some column of Mn (y). Indeed, remember from (4) how an element
Mi (y)(k, p) can be obtained. Let yγ = Mi (y)(k, 1) and yα = Mi (y)(1, p). Then
Mi (y)(k, p) = yη with ηi = γi + αi , i = 1, . . . n.
(16)
Now, consider a column Bj of B, that is, some column Mi (y)(., p) of Mi (y),
with first element yα = Bj (1) = Mi (y)(1, p). Therefore, the element B(k) (or,
equivalently, the element Mi (y)(k, p)) is the variable yη in (16). Note that α
correspond to a monomial in the basis (2) of degree larger than n, say α1 · · · αn .
Associate to this column Bj the column v := Mn (y)(., q) of Mn (y) whose element
v(1) = Mn (y)(1, q) = yβ (for some q) with βi = 0 if αi = 0 and 1 otherwise, for
all i = 1, . . . n. Then, the element v(k) = Mn (y)(k, q) is obtained as
v(k) = yδ with δi = γi + βi , i = 1, . . . n.
But then, as for each entry yα of Mj (y), we can make the substitution αi ↔ 1
whenever αi ≥ 1, it follows that the element v(k) is identical to the element
B(k). In other words, each column of B is identical
to some columnof M
n (y).
Mn (y)
B
If we now write Mj (y) = [A|D] with A :=
, and D :=
, then,
B0
C
with exactly same arguments, every column of D is also identical to some column
of A, and consequently, (15) follows.
t
u
For instance, when n = 2 the reader can check that M3 (y) has same rank as
M2 (y) in (13). We now can state our main result:
Theorem 1. Let IP be the problem defined in (1) and let v := maxk=1,...m vk .
Then for every i ≥ n + v:
(a) Qi is solvable with p∗ = min Qi , and to every optimal solution x∗ of IP
corresponds the optimal solution
y ∗ := (x∗1 , · · · , x∗n , · · · , (x∗1 )2i , · · · , (x∗n )2i )
of Qi .
(17)
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J.B. Lasserre
(b) Every optimal solution y ∗ of Qi is the (finite) vector of moments of a
probability measure finitely supported on s optimal solutions of IP, with s =
rank Mi (y) = rank Mn (y).
Proof. Let y be an admissible solution of Qn+v . From Proposition 1, we have
that rank Mi (y) = rank Mn (y) for all i > n (in particular for i = n + v). From
a deep result of Curto and Fialkow [1, Th 1.6, p. 6], it follows that y is the
vector of moments of some rank Mn (y) atomic measure µy . Mn+1 (y) is called a
flat positive extension of Mn (y) and it follows that Mn+1 (y) admits unique flat
extension moment matrices Mn+2 (y), Mn+3 (y), etc. (see Curto and Fialkow [1, p.
3]). Moreover, from the constraints Mn+v−vk (gk y) 0, for all k = 1, . . . m + 2n,
it also follows that µy is supported on IK. Theorem 1.6 in Curto and Fialkow [1]
is stated in dimension 2 for the complex plane, but is valid for n real or complex
variables (see comments page 2 in [1]). In the notation of Theorem 1.6 in Curto
and Fialkow [1], we have M (n) 0 and M (n + 1) has a flat positive extension
M (n + 1) (hence unique flat positive extensions M (n + k) for all k ≥ 1), with
Mgk (n + vk ) 0, k = 1, . . . m + 2n.
But then, as µy is supported on IK, it also follows that
Z
X
(g0 )α yα =
g0 (x) µy (dx) ≥ p∗ .
IK
α
From this and inf Qi ≤ p∗ , the first statement (a) in Theorem 1 follows for
i = n + v. For i > n + v, the result follows from p∗ ≥ inf Qi+1 ≥ inf Qi , for all i.
Finally, y ∗ in (17) is obviously admissible for Qi with value p∗ , and therefore, is
an optimal solution of Qi .
(b) follows from same arguments as in (a). First observe that from (a), Qi is
solvable for all i ≥ n+v. Now, let y ∗ be an optimal solution of Qi . From (a), y ∗ is
the vector of moments of an atomic measure µy∗ supported on s (= rank Mn (y))
points x1 , · · · xs ∈ IK, that is, with δ{.} the Dirac measure at 00 .00 ,
µ
y∗
=
s
X
γj δxj with γj ≥ 0,
j=1
s
X
γj = 1.
j=1
Therefore, from g0 (xj ) ≥ p∗ for all j = 1, . . . s, and
p∗ = min Qi =
Z
=
IK
X
α
(g0 )α yα∗
g0 (x) µy∗ (dx) =
s
X
γj g0 (xj ),
j=1
it follows that each point xj must be an optimal solution of IP.
t
u
Hence, Theorem 1 shows that every 0-1 program is equivalent to a convex psd
program with 2n − 1 variables. The projection of the feasible set of Qn+v onto
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
301
the space spanned by the n variables y10...0 , . . . , y0...01 is the convex hull of IK in
(8).
Interpretation . The interpretation of the relaxation Qn+v is as follows. The
unknown variables {yα } should be interpreted as the moments of some probability measure µ, up to order n + v. The LMI constraints Mn+v−vk (gk y) 0 state
that
Z
for all polynomials q of degree at most n,
gk (x)q(x)2 dµ ≥ 0,
(as opposed to only gk (x) ≥ 0 in the original description). As the integral constraints x2i = xi are included in the constraints, and due to the result of Curto
and Fialkow, the above constraint (for k = 1, . . . m + 2n) will ensure that the
support of µ is concentrated on {0, 1}n ∩ [∩m
k=1 {gk (x) ≥ 0}].
For illustration purposes, we provide below the explicit description of the
equivalent convex psd programs for quadratic 0-1 programs and MAX-CUT
programs in IR3 , respectively.
3.1
Quadratic 0-1 Programs
Consider the quadratic program min{x0 Ax | x ∈ {0, 1}n } for some real-valued
symmetric matrix A ∈ IRn×n . As the only constraints are the integral constraints
x ∈ {0, 1}n , with n = 3, it is equivalent to the convex psd program
inf A11 y100 + A22 y010 + A33 y001 + 2A12 y110 + 2A13 y101 + 2A23 y011

1 y100
 y100 y100

 y010 y110


c3 (y) =  y001 y101
M
 y110 y110

 y101 y101

 y011 y111
y111 y111
y010
y110
y010
y011
y110
y111
y011
y111
y001
y101
y011
y001
y111
y101
y011
y111
y110
y110
y110
y111
y110
y111
y111
y111
y101
y101
y111
y101
y111
y101
y111
y111
y011
y111
y011
y011
y111
y111
y011
y111

y111
y111 

y111 

y111 

y111 

y111 

y111 
y111
0,
c3 (y) is the simplified form of M3 (y) (see Remark 1).
where M
3.2
MAX-CUT Problem
In this case, g0 (x) := xAx for some real-valued symmetric matrix A = {Aij } ∈
IRn×n with null diagonal and the constraint set is {−1, 1}n . Now, we look for
solutions in {−1, 1}n , which yields the following obvious modifications in the
relaxations {Qi }. In the matrix Mi (y), replace any entry yα by yβ with βi = 1
whenever αi is odd, and βi = 0 otherwise. As for 0-1 programs, the matrix Mi (y)
302
J.B. Lasserre
ci (y) (by eliminating identical rows
can be reduced in size to a simplified form M
and columns). Hence, MAX-CUT in IR3 is equivalent to the psd program
inf A12 y110 + A13 y101 + A23 y011

1 y100
 y100 1

 y010 y110

 y001 y101
c
M3 (y) = 
 y110 y010

 y101 y001

 y011 y111
y111 y011
y010
y110
1
y011
y100
y111
y001
y101
y001
y101
y011
1
y111
y100
y010
y110
y110
y010
y100
y111
1
y011
y101
y001
y101
y001
y111
y100
y011
1
y110
y010
y011
y111
y001
y010
y101
y110
1
y100

y111
y011 

y101 

y110 
 0.
y001 

y010 

y100 
1
Despite its theoretical interest, this result is of little value for computational
purposes for the number of variables is exponential in the problem size. However,
in many cases, low order SDP relaxations Qi , that is, with i n, will provide the
optimal value p∗ . To illustrate the power of the SDP relaxations Qi , we have run
a sample of 50 MAX-CUT problems in IR10 , with non diagonal entries randomly
generated (uniformly) between 0 and 1 (in some examples, zeros and negative
entries were allowed). In all cases, min Q2 provided the optimal value. The SDP
ci (y) 0) of dimension 56 × 56 and 385
relaxation Q2 has one LMI constraint (M
variables yβ .
Remark 2. Consider the MAX-CUT problem with equal weights, that is,
X
min n
xi xj .
p∗ =
x∈{−1,1}
From
i<j


" n
" n
#2
#
X
X
X
1
1
n

xi xj  +
xi +
(1 − xi )2 ,
=
2
2
2
i<j
i=1
i=1
we can prove that min Q1 = −n/2. Indeed, with en ∈ IRn being a vector of ones,
let X ∈ IR(n+1)×(n+1) , Zk ∈ IR, be defined as
1 0
1
× [0, en ],
(18)
Zk = , k = 1, . . . n; X :=
2
2 en
Pn
so that X 0 and −X(1, 1) − k=1 Zk (1, 1) = −n/2. Thus, (X, {Zk }) is admissible for Q∗1 with value −n/2.
Let n = 2p, and let x ∈ IRn be defined as xi = 1, i = 1, . . . p; xi = −1, i =
p + 1, . . . n. Let y be the vector of first and second moments of the Dirac at x
(which, from M1 (y) 0, implies that y is admissible for Q1 ). Its value, i.e., the
sum of all yα ’s corresponding to xi xj , ∀i < j, is −n/2. Moreover,
" n
#2
X
1 0
0
xi
= 0,
,X
i =
hM1 (y), Xi = h
en
2 en
i=1
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
303
so that y and (X, {Zk }) are optimal solutions of Q1 and Q∗1 respectively. It also
follows that −n/2 = p∗ , as x is feasible for IP.
Next, consider the case where n is odd. Let y be defined with the correspondances
xi → yα := 0; i = 1, 2, . . . n; xi xj → yα := −1/(n − 1), ∀i < j.
With X ∈ IR(n+1)×(n+1) , Zk ∈ IR as in (18), one may check that M1 (y) 0 and
hM1 (y), Xi = 0. The sum of all yα corresponding to xi xj , i < j is −n(n − 1)/2 ×
(n − 1)−1 = −n/2. Hence, again, y and (X, {Z}k ) are optimal solutions of Q1
and Q∗1 respectively. But y is not a moment vector and −n/2 is only a (strict)
lower bound on p∗ .
4
Conclusion
We have provided an explicit equivalent continuous convex psd program for
arbitrary constrained nonlinear 0-1 programs, the last in a finite hierarchy of
SDP relaxations. For practical computation, in many cases, an SDP relaxation
of low order is exact, i.e., provides the optimal value p∗ , as confirmed on the
MAX-CUT problem as well as on a (small) sample of nonconvex continuous
problems in Lasserre [4]. However, for large or even moderate size problems, the
resulting SDP relaxations Qi might still be too large for the present status of
SDP software packages. Therefore, we hope that this work will help stimulate
further effort in designing efficient solving procedures for large SDP programs.
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