Results 1 
4 of
4
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
Abstract

Cited by 37 (7 self)
 Add to MetaCart
We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
Theta Bodies for Polynomial Ideals
, 2008
"... Abstract. Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lo ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
Abstract. Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre’s relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals. 1.
A NEW SEMIDEFINITE PROGRAMMING HIERARCHY FOR CYCLES IN BINARY MATROIDS AND CUTS IN GRAPHS
"... Abstract. The theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal. In this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid. Applied to cuts in graphs, this yields a new hi ..."
Abstract
 Add to MetaCart
Abstract. The theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal. In this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid. Applied to cuts in graphs, this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph. If the binary matroid avoids certain minors we can characterize when the first theta body in the hierarchy equals the cycle polytope of the matroid. Specialized to cuts in graphs, this result solves a problem posed by Lovász. 1.
Computing the Grothendieck constant of some graph classes
"... Given a graph G = ([n], E) and w ∈ RE ∑, consider the integer program maxx∈{±1} n ij∈E wijxixj and its canonical semidefinite programming relaxation max ∑ ij∈E wijvT i vj, where the maximum is taken over all unit vectors vi ∈ Rn. The integrality gap of this relaxation is known as the Grothendieck co ..."
Abstract
 Add to MetaCart
Given a graph G = ([n], E) and w ∈ RE ∑, consider the integer program maxx∈{±1} n ij∈E wijxixj and its canonical semidefinite programming relaxation max ∑ ij∈E wijvT i vj, where the maximum is taken over all unit vectors vi ∈ Rn. The integrality gap of this relaxation is known as the Grothendieck constant κ(G) of G. We present a closedform formula for the Grothendieck constant of K5minor free graphs and derive that it is at most 3/2. Moreover, we show that κ(G) ≤ κ(Kk) if the cut polytope of G is defined by inequalities supported by at most k points. Lastly, since the Grothendieck constant of Kn grows as Θ(log n), it is interesting to identify instances with large gap. However this is not the case for the cliqueweb inequalities, a wide class of valid inequalities for the cut polytope, whose integrality ratio is shown to be bounded by 3. Keywords: Grothendieck constant, elliptope, cut polytope, cliqueweb inequality