## A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0-1 programming (2001)

Venue: | Mathematics of Operations Research |

Citations: | 88 - 11 self |

### BibTeX

@ARTICLE{Laurent01acomparison,

author = {M. Laurent},

title = {A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0-1 programming},

journal = {Mathematics of Operations Research},

year = {2001},

volume = {28},

pages = {470--496}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

1177 | Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics - Grötschel, Lovász, et al. - 1988 |

971 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming - Goemans, Williamson - 1995 |

345 | Global optimization with polynomials and the problem of moments - Lasserre |

343 | On the Shannon capacity of a graph - Lovász - 1979 |

271 | Cones of matrices and set-functions and 0-1 optimization
- Lovász, Schrijver
- 1991
(Show Context)
Citation Context ...ms iterate is contained in the Lovász-Schrijver iterate which in turn is contained in the Balas-Ceria-Cornuéjols iterate. The latter inclusion is an easy verification and the former was mentioned in (=-=Lovász and Schrijver 1991-=-) as an application of somewhat complicated algebraic manipulations; we present in Section 4 a simple direct proof for this inclusion. The construction of Lasserre is motivated by results about repres... |

255 | Geometry of Cuts and Metrics - Deza, Laurent - 1997 |

221 | A lift-and-project cutting plane algorithm for mixed 0-1 programs - Balas, Ceria, et al. - 1993 |

215 | 2000) Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization - Parrilo |

199 |
A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems
- Sherali, Adams
- 1990
(Show Context)
Citation Context ...near relaxations since the condition (10) can be reformulated as a linear system in y (cf. Lemma 2 below). We present in Section 3.2 the original definition of the Sherali-Adams relaxations given in (=-=Sherali and Adams 1990-=-) and its equivalence with the above definition. 3.1 Preliminary results Let Z denote the square 0 − 1 matrix indexed by P(V ) with entry ZI,J = 1 if and only if I ⊆ J. Its inverse Z −1 has entries Z ... |

169 |
Positive polynomials on compact semi-algebraic sets
- Putinar
- 1993
(Show Context)
Citation Context ... shows the following stronger result which, as we will see in the next subsection, plays a central role in the approach of Lasserre for asymptotically evaluating polynomial programs. i∈I Theorem 22. (=-=Putinar 1993-=-) Let F be a compact semi-algebraic set as in (47). Assume that there exists a polynomial u ∈ Σ 2 + g1Σ 2 + . . . + gmΣ 2 for which the set {x ∈ R n | u(x) ≥ 0} is compact. If p is a polynomial positi... |

149 | Edmonds polytopes and a hierarchy of combinatorial problems - Chvátal - 1973 |

149 | The K-moment problem for compact semi-algebraic sets - Schmüdgen - 1991 |

121 |
On the cut polytope
- Barahona, Mahjoub
- 1986
(Show Context)
Citation Context ...), let CUT(G) and MET(G) denote the projections of CUTn and METn, respectively, on the subspace R E indexed by the edge set of G. Then, CUT(G) ⊆ MET(G) with equality if and only if G has no K5-minor (=-=Barahona and Mahjoub 1986-=-). When applying the Lovász-Schrijver construction to K := MET(G), one finds the relaxation N(MET(G)) of CUT(G). Another possibility is to first apply the LS construction to K := MET(Kn) and then proj... |

109 | Harmonic Analysis on Semigroups. Theory of Positive De¯nite and Related Functions - Berg, Christensen, et al. - 1984 |

102 | functional systems and optimization problems - Nesterov, Squared - 1999 |

101 | reformulation-linearization technique for solving discrete and continuous nonconvex problems, NonconvexOptim - Sherali, Adams - 1999 |

98 | Some concrete aspects of Hilbert’s 17th problem
- Reznick
- 2000
(Show Context)
Citation Context ... For n ≥ 2, not every nongegative polynomial can be expressed as a sum of squares of polynomials. This problem of representing polynomials as sums of squares goes back to Hilbert’s 17th problem; see (=-=Reznick 1998-=-) for a survey. Let us reformulate the result of Schmüdgen in terms of polynomials. Let F be as in (47) and let Σ 2 (g1, . . . , gm) := denote the set of all polynomials of the form ∑ I⊆{1,...,m} i∈I ... |

70 | An explicit exact SDP relaxation for nonlinear 0-1 programs - Lasserre |

54 | Approximation of the stability number of a graph via copositive programming - Klerk, Pasechnik |

52 | The truncated complex K-moment problem - Curto, Fialkow - 2000 |

51 |
Tuncbilek, “A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique
- Sherali, H
- 1992
(Show Context)
Citation Context ... us briefly mention some of the results of (Lasserre 2002). The reformulation-linearization technique of Sherali and Adams (1990,1994) was extended to general polynomial programs as (48) (see, e.g., (=-=Sherali and Tuncbilek 1992-=-,1997)). The basic idea for constructing successive linear approximations is to consider the LP relaxation obtained from the linearization of all possible products (g1(x)) β1 . . .(gm(x)) βm ≥ 0 of th... |

48 | Nondifferentiable Optimization and Polynomial Problems - Shor - 1998 |

41 |
A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-Integer Zero-One
- Sherali, Adams
- 1994
(Show Context)
Citation Context ...ruction of Sherali and Adams (known as the reformulation-linearization technique or RLT) applies more generally to 0 −1 mixed integer polynomial programs which are linear in the continuous variables (=-=Sherali and Adams 1994-=-). However, it is not clear whether the presentation given here in terms of moment matrices extends to the mixed integer case. Strengthenings of the basic RLT method have been proposed for general pol... |

35 |
The multidimensional moment problem
- Fuglede
- 1983
(Show Context)
Citation Context ... characterizing moment sequences. It has been much studied in the literature especially for the semigroup S = Z n +, in which case S ∗ = R n and the moment condition (46) reads as relation (38); see (=-=Fuglede 1983-=-, Berg, Christensen and Ressel 1984) for a survey. Obviously, every F-moment sequence should be positive semidefinite. Much research has been done for characterizing moment sequences for various close... |

33 |
Semidefinite programming relaxation for nonconvex quadratic programs
- Fujie, Kojima
- 1997
(Show Context)
Citation Context ...al semidefinite relaxation of F: ˆF := {x ∈ R n | qℓ Qℓ ) . Then, gℓ(x) = 〈Pℓ, x xx T ) 〉. Therefore, ( ) 1 x = Y e0 for some Y ≽ 0 with 〈Pℓ, Y 〉 ≥ 0 for ℓ = 1, . . . , m} (57) (considered, e.g., in (=-=Fujie and Kojima 1997-=-)). In fact, the set ˆ F coincides with the first Lasserre relaxation Q0(F). Proposition 25. Q0(F) = ˆ F. Proof. By definition, x ∈ Rn belongs to Q0(F) if there exists y = (yα)|α|≤2 satisfying y0 = 1,... |

32 | On cutting-plane proofs in combinatorial optimization. Linear Algebra and its Applications - Chvátal, Cook, et al. - 1989 |

31 | When does the positive semidefiniteness constraint help in lifting procedures - Goemans, Tunçel |

31 | Representing polynomials by positive linear functions on compact convex polyhedra - Handelman - 1988 |

30 | On the matrix-cut rank of polyhedra - Cook, Dash |

28 | On the chvatal rank of polytopes in the 0/1 cube
- Bockmayr, Eisenbrand, et al.
- 1998
(Show Context)
Citation Context ...ot only on the dimension n but also on the coefficients of the inequalities involved. However, when K is assumed to be contained in the cube [0, 1] n then its Chvátal rank is bounded by O(n 2 log n) (=-=Eisenbrand and Schulz 1999-=-). Even if we can optimize a linear objective function over K in polynomial time optimizing a linear objective function over the first Chvátal closure K ′ is a co-NP-hard problem in general (Eisenbran... |

24 | On the membership problem for the elementary closure of a polyhedron, Combinatorica 19 - Eisenbrand - 1999 |

23 | Strengthened semidefinite relaxations via a second lifting for the max-cut problem - Anjos, Wolkowicz |

21 | On a representation of the matching polytope via semidefinite liftings - Stephen, Tunçel - 1999 |

17 | Semidefinite programming vs. LP relaxations for polynomial programming - Lasserre |

16 | Exploiting special structures in constructing a hierarchy of relaxations for 0–1 mixed integer problems - Sherali, Adams, et al. - 1996 |

14 | A remark on the multidimensional moment problem - Berg, Christensen, et al. - 1979 |

13 | On the rank of mixed 0-1 polyhedra - Cornuéjols, Li - 2002 |

13 | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization - Parillo - 2000 |

11 |
On the momentum problem for distributions in more than one dimension
- Haviland
- 1935
(Show Context)
Citation Context ... nonnegative polynomials as sums of squares Let P+(F) denote the set of polynomials p(x) = ∑ α pαxα that are nonnegative on F; that is, p(x) ≥ 0 for all x ∈ F. One of the basic results about moments (=-=Haviland 1935-=-, 1936) is that, given a closed subset F in Rn , y = (yα)α∈Zn + is a F-moment sequence if and only if yTp ≥ 0 for every polynomial p = (pα)α∈Zn in P+(F). + Since a linear functional f on the set R[x1,... |

11 | Hadamard determinants, Möbius functions, and the chromatic number of a graph - Wilf - 1968 |

10 | Optimality conditions and LMI relaxations for 0-1 programs - Lasserre - 2000 |

8 | Tighter linear and semidefinite relaxations for max-cut based on the Lovász–Schrijver lift-and-project procedure - Laurent |

7 | New reformulation-linearization/convexification relaxations for univariate and multivariate polynomial programming problem - Sherali, Tuncbilek - 1997 |

6 | An approach to obtaining global extremums in polynomial mathematical programming problems, Kibernetika 5 - Shor - 1987 |

1 | On the momentum problem for distributions in more than one dimension - K - 1936 |

1 | Semidefinite relaxations for Max-Cut. To appear - Laurent |