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The equivalence of semidefinite relaxations of polynomial 0–1 and ± 1 programs via scaling
Operations Research Letters 36 (2008) 314–316 www.elsevier.com/locate/orl
The equivalence of semidefinite relaxations of polynomial 0–1 and ±1 programs via scaling Kevin K.H. Cheung ∗ School of Mathematics and Statistics, Carleton University, Canada Received 13 March 2007; accepted 12 October 2007 Available online 17 December 2007
Abstract Many combinatorial optimization problems can be modelled as polynomial-programming problems in binary variables that are all 0–1 or ±1. A sufficient condition under which a common method for obtaining semidefinite-programming relaxations of the two models of the same problem gives equivalent relaxations is established. c 2007 Elsevier B.V. All rights reserved.
Keywords: Semidefinite programming; Binary programming; Equivalent relaxation; Scaling
1. Introduction Many combinatorial optimization problems can be modelled as polynomial-programming problems in binary variables of the form min{g0 (x) : gl (x) ≥ 0 for l = 1, . . . , m, x ∈ {0, 1}n }
(1)
where gl is a polynomial in x for l = 0, 1, . . . , m, or of the form min{g00 (u) : gl0 (u) ≥ 0 for l = 1, . . . , m, u ∈ {−1, 1}n }
(2)
where gl0 is a polynomial in u for l = 0, 1, . . . , m. (We may assume that each variable occurs with degree at most one.) One popular method for obtaining a semidefinite-programming (SDP) relaxation of either form replaces each monomial with a variable and requires certain principal submatrices of the moment matrices to be positive semidefinite. (For example, Laurent [3] describes two hierarchies [2,8] of relaxations in this framework.) It is natural to ask under what assumptions, when both forms model the same problem, this method gives SDP ∗ Corresponding address: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada. E-mail address: [email protected].
c 2007 Elsevier B.V. All rights reserved. 0167-6377/$ - see front matter doi:10.1016/j.orl.2007.10.006
relaxations of the two forms that are equivalent. In the quadratic case, equivalence is shown by Helmberg [1] and, for a special case, by Laurent et al. [6]. In this note, we extend their work to higher-order relaxations. The rest of this section is spent on definitions and notation. In this note, vectors are written as columns. For a given integer n > 0, let V := {1, . . . , n}. The collection of all subsets of V is denoted by P(V ). S ⊆ P(V ) is an independence system if for every A ∈ S, B ⊆ A implies that B ∈ S. Given a polynomial g(x1 , . . . , xn ) in which each variable occurs with degree at most one, we use the same symbol g to denote the vector in RP (V ) such that Q for all I ∈ P(V ), the entry indexed by I is the coefficient of i∈I xi in g(x). For a given y ∈ RP (V ) , the entry indexed by I ∈ P(V ) is denoted by y(I ) or y I (with y{i} and y{i, j} abbreviated as yi and yi j , respectively). The set of matrices with rows and columns indexed by the elements of a finite set S is denoted by R S×S . Given S ⊆ P(V ) and y ∈ RP (V ) , the matrix with (I, J )-entry equal to y(I ∪ J ) for all I, J ∈ S is denoted by MS∪ (y); and the matrix with (I, J )-entry equal to y(I 4J ) for all I, J ∈ S is denoted by 4 MS (y) where I 4J := (I ∪ J ) \ (I ∩ J ). The matrices MS∪ (y) 4 and MS (y) are known as moment matrices. In the case when 4 S = P(V ), MS∪ (y) is written simply as M ∪ (y) and MS (y) as M 4 (y). Given a symmetric matrix Y , we write Y 0 to signify that Y is positive semidefinite.
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K.K.H. Cheung / Operations Research Letters 36 (2008) 314–316
(S D P0,1 )
min subject to
g0T z z ∅ = 1, MS∪0, j (z) 0 MS∪l, j (M ∪ (z)gl ) 0
for j = 1, . . . , k0 , for all l = 1, . . . , m, jl = 1, . . . , kl
g00T y y∅ = 1, 4 MS0, j (y) 0
for j = 1, . . . , k0 ,
l
and (S D P±1 )
min subject to
4
MSl, j (M l
4
(y)gl0 )
0
for all l = 1, . . . , m, jl = 1, . . . , kl . Box I.
and R −1 is given by R −1 (I, J ) nR |I |is nonsingular |J \I |
2. Main result Observe that (1) can be written as min (P0,1 ) subject to
g0T z glT z ≥ 0 z∅ = 1
for l = 1, . . . , m,
z I ∪J = z I z J for all I, J ∈ P(V ) z ∈ {0, 1}P (V ) , and (2) can be written as T
min g00 y T
(P±1 ) subject to gl0 y ≥ 0 for l = 1, . . . , m, y∅ = 1 y I 4J = y I y J for all I, J ∈ P(V ) y ∈ {−1, 1}P (V ) . Theorem 1 (Corollary 3 in [3]). The convex hull of the set of feasible solutions to (P0,1 ) is given by {z ∈ RP (V ) : z ∅ = 1, M ∪ (z) 0 and M ∪ (M ∪ (z)gl ) 0 for all l = 1, . . . , m}. Using Theorem 1, (P0,1 ) can be rewritten as min g0T z subject to z ∅ = 1 M ∪ (z) 0 M ∪ (M ∪ (z)gl ) 0 for all l = 1, . . . , m. To obtain an SDP relaxation, one chooses certain principal submatrices of M ∪ (z) and M ∪ (M ∪ (z)gl ) to be positive semidefinite. Proving an analogue of Theorem 1 gives that (P±1 ) is equivalent to min g00T y subject to y∅ = 1 M 4 (y) 0 M 4 (M 4 (y)gl0 ) 0 for all l = 1, . . . , m. Let R ∈ RP (V )×P (V ) be the scaled zeta matrix of the lattice P(V ) (cf. [9]; see also [3]) given by 1 I ⊆ J, R(I, J ) = 2|J | 0 otherwise.
=
2 (−1) I ⊆ J, 0 otherwise.
Suppose that gl0 = Rgl for l = 0, 1, . . . , m. Then (P0,1 ) and (P±1 ) are equivalent under the bijection between the feasible solutions given by z = R T y. The following is our main result. Theorem 2. Let gl0 , gl ∈ RP (V ) be such that gl0 = Rgl for i = 0, 1, . . . , m. Let kl be a positive integer for l = 0, 1, . . . , m. Suppose that Sl,1 , . . . , Sl,kl are independence systems for l = 0, 1, . . . , m. Then the SDP problems in Box I are relaxations of (P0,1 ) and (P±1 ), respectively, and are equivalent under the bijection between the feasible solutions given by z = R T y. Using Theorem 2, one can obtain the analogues of the Lasserre hierarchy [2], the Sherali–Adams hierarchy [8], and the Lov´asz–Schrijver [7] hierarchy (defined by N+ ) for (P±1 ) as described and used, for instance, by Laurent [3–5]. Theorem 2 follows easily from the next lemma. Lemma 3. Let g 0 , g, y, z ∈ RP (V ) be such that z = R T y and g 0 = Rg. Let S ⊆ P(V ) be an independence system. Let RS denote the square submatrix of R with rows and columns indexed by the elements of S. Then 4
(i) RST MS (y)RS = MS∪ (z); 4 (ii) RST MS (M 4 (y)g 0 )RS = MS∪ (M ∪ (z)g). 4
Proof. Let Y = RST MS (y)RS . Then (i) follows from ! X X Y (A, B) = RS (C, A) yC4D RS (D, B) C⊆A
= =
= =
D⊆B
1
X X
2|A|+|B|
C⊆A D⊆B
1
X
2|A|+|B|
yC4D X
X
C⊆A\B C 0 ⊆A∩B D⊆B
2|A∩B| X X yC∪D 2|A|+|B| C⊆A\B D⊆B 1
X
2|A∪B|
C⊆A∪B
= z A∪B .
yC
yC∪(C 0 4D)
316
K.K.H. Cheung / Operations Research Letters 36 (2008) 314–316
For (ii), note that MS∪ (M ∪ (z)g) = MS∪ (M ∪ (z)R −1 g 0 ) = MS∪ (R T M 4 (y)R R −1 g 0 ) by (i) 4
= RST MS (R −T (R T M 4 (y)g 0 ))RS 4 = RST MS ((M 4 (y)g 0 ))RS .
by (i)
Acknowledgements The research for this paper was supported by a grant of NSERC of Canada. The author would like to thank the anonymous referee for helpful comments. References [1] C. Helmberg, Fixing variables in semidefinite relaxations, SIAM J. Matrix Anal. Appl. 21 (2000) 952–969.
[2] J.B. Lasserre, An explicit equivalent positive semidefinite program for nonlinear 0–1 programs, SIAM J. Optim. 12 (2002) 756–769. [3] M. Laurent, A comparison of the Sherali–Adams, Lov´asz–Schrijver, and Lasserre relaxations for 0–1 programming, Math. Oper. Res. 28 (2003) 470–496. [4] M. Laurent, Lower bound for the number of iterations in semidefinite hierarchies for the cut polytope, Math. Oper. Res. 28 (2003) 871–883. [5] M. Laurent, Tighter linear and semidefinite relaxations for max-cut based on the Lov´asz–Schrijver lift-and-project procedure, SIAM. J. Optim. 12 (2001/02) 345–375. [6] M. Laurent, S. Poljak, F. Rendl, Connections between semidefnite relaxations of the max-cut and stable set problems, Math. Program. Ser. B 77 (1997) 225–246. [7] L. Lov´asz, A. Schrijver, Cones of matrices and set-functions and 0–1 optimization, SIAM J. Optim. 1 (1991) 166–190. [8] H.D. Sherali, W.P. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM J. Discrete Math. 3 (1990) 411–430. [9] H.S. Wilf, Hadamard determinants, M¨obius functions, and the chromatic number of a graph, Bull. Amer. Math. Soc. 74 (1968) 960–964.
The equivalence of semidefinite relaxations of polynomial 0–1 and ±1 programs via scaling Kevin K.H. Cheung ∗ School of Mathematics and Statistics, Carleton University, Canada Received 13 March 2007; accepted 12 October 2007 Available online 17 December 2007
Abstract Many combinatorial optimization problems can be modelled as polynomial-programming problems in binary variables that are all 0–1 or ±1. A sufficient condition under which a common method for obtaining semidefinite-programming relaxations of the two models of the same problem gives equivalent relaxations is established. c 2007 Elsevier B.V. All rights reserved.
Keywords: Semidefinite programming; Binary programming; Equivalent relaxation; Scaling
1. Introduction Many combinatorial optimization problems can be modelled as polynomial-programming problems in binary variables of the form min{g0 (x) : gl (x) ≥ 0 for l = 1, . . . , m, x ∈ {0, 1}n }
(1)
where gl is a polynomial in x for l = 0, 1, . . . , m, or of the form min{g00 (u) : gl0 (u) ≥ 0 for l = 1, . . . , m, u ∈ {−1, 1}n }
(2)
where gl0 is a polynomial in u for l = 0, 1, . . . , m. (We may assume that each variable occurs with degree at most one.) One popular method for obtaining a semidefinite-programming (SDP) relaxation of either form replaces each monomial with a variable and requires certain principal submatrices of the moment matrices to be positive semidefinite. (For example, Laurent [3] describes two hierarchies [2,8] of relaxations in this framework.) It is natural to ask under what assumptions, when both forms model the same problem, this method gives SDP ∗ Corresponding address: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada. E-mail address: [email protected].
c 2007 Elsevier B.V. All rights reserved. 0167-6377/$ - see front matter doi:10.1016/j.orl.2007.10.006
relaxations of the two forms that are equivalent. In the quadratic case, equivalence is shown by Helmberg [1] and, for a special case, by Laurent et al. [6]. In this note, we extend their work to higher-order relaxations. The rest of this section is spent on definitions and notation. In this note, vectors are written as columns. For a given integer n > 0, let V := {1, . . . , n}. The collection of all subsets of V is denoted by P(V ). S ⊆ P(V ) is an independence system if for every A ∈ S, B ⊆ A implies that B ∈ S. Given a polynomial g(x1 , . . . , xn ) in which each variable occurs with degree at most one, we use the same symbol g to denote the vector in RP (V ) such that Q for all I ∈ P(V ), the entry indexed by I is the coefficient of i∈I xi in g(x). For a given y ∈ RP (V ) , the entry indexed by I ∈ P(V ) is denoted by y(I ) or y I (with y{i} and y{i, j} abbreviated as yi and yi j , respectively). The set of matrices with rows and columns indexed by the elements of a finite set S is denoted by R S×S . Given S ⊆ P(V ) and y ∈ RP (V ) , the matrix with (I, J )-entry equal to y(I ∪ J ) for all I, J ∈ S is denoted by MS∪ (y); and the matrix with (I, J )-entry equal to y(I 4J ) for all I, J ∈ S is denoted by 4 MS (y) where I 4J := (I ∪ J ) \ (I ∩ J ). The matrices MS∪ (y) 4 and MS (y) are known as moment matrices. In the case when 4 S = P(V ), MS∪ (y) is written simply as M ∪ (y) and MS (y) as M 4 (y). Given a symmetric matrix Y , we write Y 0 to signify that Y is positive semidefinite.
315
K.K.H. Cheung / Operations Research Letters 36 (2008) 314–316
(S D P0,1 )
min subject to
g0T z z ∅ = 1, MS∪0, j (z) 0 MS∪l, j (M ∪ (z)gl ) 0
for j = 1, . . . , k0 , for all l = 1, . . . , m, jl = 1, . . . , kl
g00T y y∅ = 1, 4 MS0, j (y) 0
for j = 1, . . . , k0 ,
l
and (S D P±1 )
min subject to
4
MSl, j (M l
4
(y)gl0 )
0
for all l = 1, . . . , m, jl = 1, . . . , kl . Box I.
and R −1 is given by R −1 (I, J ) nR |I |is nonsingular |J \I |
2. Main result Observe that (1) can be written as min (P0,1 ) subject to
g0T z glT z ≥ 0 z∅ = 1
for l = 1, . . . , m,
z I ∪J = z I z J for all I, J ∈ P(V ) z ∈ {0, 1}P (V ) , and (2) can be written as T
min g00 y T
(P±1 ) subject to gl0 y ≥ 0 for l = 1, . . . , m, y∅ = 1 y I 4J = y I y J for all I, J ∈ P(V ) y ∈ {−1, 1}P (V ) . Theorem 1 (Corollary 3 in [3]). The convex hull of the set of feasible solutions to (P0,1 ) is given by {z ∈ RP (V ) : z ∅ = 1, M ∪ (z) 0 and M ∪ (M ∪ (z)gl ) 0 for all l = 1, . . . , m}. Using Theorem 1, (P0,1 ) can be rewritten as min g0T z subject to z ∅ = 1 M ∪ (z) 0 M ∪ (M ∪ (z)gl ) 0 for all l = 1, . . . , m. To obtain an SDP relaxation, one chooses certain principal submatrices of M ∪ (z) and M ∪ (M ∪ (z)gl ) to be positive semidefinite. Proving an analogue of Theorem 1 gives that (P±1 ) is equivalent to min g00T y subject to y∅ = 1 M 4 (y) 0 M 4 (M 4 (y)gl0 ) 0 for all l = 1, . . . , m. Let R ∈ RP (V )×P (V ) be the scaled zeta matrix of the lattice P(V ) (cf. [9]; see also [3]) given by 1 I ⊆ J, R(I, J ) = 2|J | 0 otherwise.
=
2 (−1) I ⊆ J, 0 otherwise.
Suppose that gl0 = Rgl for l = 0, 1, . . . , m. Then (P0,1 ) and (P±1 ) are equivalent under the bijection between the feasible solutions given by z = R T y. The following is our main result. Theorem 2. Let gl0 , gl ∈ RP (V ) be such that gl0 = Rgl for i = 0, 1, . . . , m. Let kl be a positive integer for l = 0, 1, . . . , m. Suppose that Sl,1 , . . . , Sl,kl are independence systems for l = 0, 1, . . . , m. Then the SDP problems in Box I are relaxations of (P0,1 ) and (P±1 ), respectively, and are equivalent under the bijection between the feasible solutions given by z = R T y. Using Theorem 2, one can obtain the analogues of the Lasserre hierarchy [2], the Sherali–Adams hierarchy [8], and the Lov´asz–Schrijver [7] hierarchy (defined by N+ ) for (P±1 ) as described and used, for instance, by Laurent [3–5]. Theorem 2 follows easily from the next lemma. Lemma 3. Let g 0 , g, y, z ∈ RP (V ) be such that z = R T y and g 0 = Rg. Let S ⊆ P(V ) be an independence system. Let RS denote the square submatrix of R with rows and columns indexed by the elements of S. Then 4
(i) RST MS (y)RS = MS∪ (z); 4 (ii) RST MS (M 4 (y)g 0 )RS = MS∪ (M ∪ (z)g). 4
Proof. Let Y = RST MS (y)RS . Then (i) follows from ! X X Y (A, B) = RS (C, A) yC4D RS (D, B) C⊆A
= =
= =
D⊆B
1
X X
2|A|+|B|
C⊆A D⊆B
1
X
2|A|+|B|
yC4D X
X
C⊆A\B C 0 ⊆A∩B D⊆B
2|A∩B| X X yC∪D 2|A|+|B| C⊆A\B D⊆B 1
X
2|A∪B|
C⊆A∪B
= z A∪B .
yC
yC∪(C 0 4D)
316
K.K.H. Cheung / Operations Research Letters 36 (2008) 314–316
For (ii), note that MS∪ (M ∪ (z)g) = MS∪ (M ∪ (z)R −1 g 0 ) = MS∪ (R T M 4 (y)R R −1 g 0 ) by (i) 4
= RST MS (R −T (R T M 4 (y)g 0 ))RS 4 = RST MS ((M 4 (y)g 0 ))RS .
by (i)
Acknowledgements The research for this paper was supported by a grant of NSERC of Canada. The author would like to thank the anonymous referee for helpful comments. References [1] C. Helmberg, Fixing variables in semidefinite relaxations, SIAM J. Matrix Anal. Appl. 21 (2000) 952–969.
[2] J.B. Lasserre, An explicit equivalent positive semidefinite program for nonlinear 0–1 programs, SIAM J. Optim. 12 (2002) 756–769. [3] M. Laurent, A comparison of the Sherali–Adams, Lov´asz–Schrijver, and Lasserre relaxations for 0–1 programming, Math. Oper. Res. 28 (2003) 470–496. [4] M. Laurent, Lower bound for the number of iterations in semidefinite hierarchies for the cut polytope, Math. Oper. Res. 28 (2003) 871–883. [5] M. Laurent, Tighter linear and semidefinite relaxations for max-cut based on the Lov´asz–Schrijver lift-and-project procedure, SIAM. J. Optim. 12 (2001/02) 345–375. [6] M. Laurent, S. Poljak, F. Rendl, Connections between semidefnite relaxations of the max-cut and stable set problems, Math. Program. Ser. B 77 (1997) 225–246. [7] L. Lov´asz, A. Schrijver, Cones of matrices and set-functions and 0–1 optimization, SIAM J. Optim. 1 (1991) 166–190. [8] H.D. Sherali, W.P. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM J. Discrete Math. 3 (1990) 411–430. [9] H.S. Wilf, Hadamard determinants, M¨obius functions, and the chromatic number of a graph, Bull. Amer. Math. Soc. 74 (1968) 960–964.