87 research outputs found
Equivariant semidefinite lifts and sum-of-squares hierarchies
A central question in optimization is to maximize (or minimize) a linear
function over a given polytope P. To solve such a problem in practice one needs
a concise description of the polytope P. In this paper we are interested in
representations of P using the positive semidefinite cone: a positive
semidefinite lift (psd lift) of a polytope P is a representation of P as the
projection of an affine slice of the positive semidefinite cone
. Such a representation allows linear optimization problems
over P to be written as semidefinite programs of size d. Such representations
can be beneficial in practice when d is much smaller than the number of facets
of the polytope P. In this paper we are concerned with so-called equivariant
psd lifts (also known as symmetric psd lifts) which respect the symmetries of
the polytope P. We present a representation-theoretic framework to study
equivariant psd lifts of a certain class of symmetric polytopes known as
orbitopes. Our main result is a structure theorem where we show that any
equivariant psd lift of size d of an orbitope is of sum-of-squares type where
the functions in the sum-of-squares decomposition come from an invariant
subspace of dimension smaller than d^3. We use this framework to study two
well-known families of polytopes, namely the parity polytope and the cut
polytope, and we prove exponential lower bounds for equivariant psd lifts of
these polytopes.Comment: v2: 30 pages, Minor changes in presentation; v3: 29 pages, New
structure theorem for general orbitopes + changes in presentatio
Equivariant semidefinite lifts of regular polygons
Given a polytope P in , we say that P has a positive
semidefinite lift (psd lift) of size d if one can express P as the linear
projection of an affine slice of the positive semidefinite cone
. If a polytope P has symmetry, we can consider equivariant psd
lifts, i.e. those psd lifts that respect the symmetry of P. One of the simplest
families of polytopes with interesting symmetries are regular polygons in the
plane, which have played an important role in the study of linear programming
lifts (or extended formulations). In this paper we study equivariant psd lifts
of regular polygons. We first show that the standard Lasserre/sum-of-squares
hierarchy for the regular N-gon requires exactly ceil(N/4) iterations and thus
yields an equivariant psd lift of size linear in N. In contrast we show that
one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1,
which is exponentially smaller than the psd lift of the sum-of-squares
hierarchy. Our construction relies on finding a sparse sum-of-squares
certificate for the facet-defining inequalities of the regular 2^n-gon, i.e.,
one that only uses a small (logarithmic) number of monomials. Since any
equivariant LP lift of the regular 2^n-gon must have size 2^n, this gives the
first example of a polytope with an exponential gap between sizes of
equivariant LP lifts and equivariant psd lifts. Finally we prove that our
construction is essentially optimal by showing that any equivariant psd lift of
the regular N-gon must have size at least logarithmic in N.Comment: 29 page
Sparse sum-of-squares certificates on finite abelian groups
Let G be a finite abelian group. This paper is concerned with nonnegative
functions on G that are sparse with respect to the Fourier basis. We establish
combinatorial conditions on subsets S and T of Fourier basis elements under
which nonnegative functions with Fourier support S are sums of squares of
functions with Fourier support T. Our combinatorial condition involves
constructing a chordal cover of a graph related to G and S (the Cayley graph
Cay(,S)) with maximal cliques related to T. Our result relies on two
main ingredients: the decomposition of sparse positive semidefinite matrices
with a chordal sparsity pattern, as well as a simple but key observation
exploiting the structure of the Fourier basis elements of G.
We apply our general result to two examples. First, in the case where , by constructing a particular chordal cover of the half-cube
graph, we prove that any nonnegative quadratic form in n binary variables is a
sum of squares of functions of degree at most , establishing
a conjecture of Laurent. Second, we consider nonnegative functions of degree d
on (when d divides N). By constructing a particular chordal
cover of the d'th power of the N-cycle, we prove that any such function is a
sum of squares of functions with at most nonzero Fourier
coefficients. Dually this shows that a certain cyclic polytope in
with N vertices can be expressed as a projection of a section
of the cone of psd matrices of size . Putting gives a
family of polytopes with LP extension complexity
and SDP extension complexity
. To the best of our knowledge, this is the
first explicit family of polytopes in increasing dimensions where
.Comment: 34 page
Semidefinite descriptions of the convex hull of rotation matrices
We study the convex hull of , thought of as the set of
orthogonal matrices with unit determinant, from the point of view of
semidefinite programming. We show that the convex hull of is doubly
spectrahedral, i.e. both it and its polar have a description as the
intersection of a cone of positive semidefinite matrices with an affine
subspace. Our spectrahedral representations are explicit, and are of minimum
size, in the sense that there are no smaller spectrahedral representations of
these convex bodies.Comment: 29 pages, 1 figur
Terracini Convexity
We present a generalization of the notion of neighborliness to non-polyhedral convex cones. Although a definition of neighborliness is available in the non-polyhedral case in the literature, it is fairly restrictive as it requires all the low-dimensional faces to be polyhedral. Our approach is more flexible and includes, for example, the cone of positive-semidefinite matrices as a special case (this cone is not neighborly in general). We term our generalization Terracini convexity due to its conceptual similarity with the conclusion of Terracini's lemma from algebraic geometry. Polyhedral cones are Terracini convex if and only if they are neighborly. More broadly, we derive many families of non-polyhedral Terracini convex cones based on neighborly cones, linear images of cones of positive semidefinite matrices, and derivative relaxations of Terracini convex hyperbolicity cones. As a demonstration of the utility of our framework in the non-polyhedral case, we give a characterization based on Terracini convexity of the tightness of semidefinite relaxations for certain inverse problems
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