70 research outputs found
Lift & Project Systems Performing on the Partial-Vertex-Cover Polytope
We study integrality gap (IG) lower bounds on strong LP and SDP relaxations
derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and
Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the
t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover
problem in which only t edges need to be covered. t-PVC admits a
2-approximation using various algorithmic techniques, all relying on a natural
LP relaxation. Starting from this LP relaxation, our main results assert that
for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P
systems that have been used for positive algorithmic results (but the Lasserre
hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices
of the input graph. Our lower bounds are nearly tight.
Our results show that restricted yet powerful models of computation derived
by many L&P systems fail to witness c-approximate solutions to t-PVC for any
constant c, and for t = O(n). This is one of the very few known examples of an
intractable combinatorial optimization problem for which LP-based algorithms
induce a constant approximation ratio, still lift-and-project LP and SDP
tightenings of the same LP have unbounded IGs.
We also show that the SDP that has given the best algorithm known for t-PVC
has integrality gap n/t on instances that can be solved by the level-1 LP
relaxation derived by the LS system. This constitutes another rare phenomenon
where (even in specific instances) a static LP outperforms an SDP that has been
used for the best approximation guarantee for the problem at hand. Finally, one
of our main contributions is that we make explicit of a new and simple
methodology of constructing solutions to LP relaxations that almost trivially
satisfy constraints derived by all SDP L&P systems known to be useful for
algorithmic positive results (except the La system).Comment: 26 page
Approximation Limits of Linear Programs (Beyond Hierarchies)
We develop a framework for approximation limits of polynomial-size linear
programs from lower bounds on the nonnegative ranks of suitably defined
matrices. This framework yields unconditional impossibility results that are
applicable to any linear program as opposed to only programs generated by
hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations
for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound
applies to linear programs using a certain encoding of CLIQUE as a linear
optimization problem.) Moreover, we establish a similar result for
approximations of semidefinite programs by linear programs. Our main ingredient
is a quantitative improvement of Razborov's rectangle corruption lemma for the
high error regime, which gives strong lower bounds on the nonnegative rank of
certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure
Dualities in Convex Algebraic Geometry
Convex algebraic geometry concerns the interplay between optimization theory
and real algebraic geometry. Its objects of study include convex semialgebraic
sets that arise in semidefinite programming and from sums of squares. This
article compares three notions of duality that are relevant in these contexts:
duality of convex bodies, duality of projective varieties, and the
Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the
optimal value of a polynomial program is an algebraic function whose minimal
polynomial is expressed by the hypersurface projectively dual to the constraint
set. We give an exposition of recent results on the boundary structure of the
convex hull of a compact variety, we contrast this to Lasserre's representation
as a spectrahedral shadow, and we explore the geometric underpinnings of
semidefinite programming duality.Comment: 48 pages, 11 figure
Lifts of convex sets and cone factorizations
In this paper we address the basic geometric question of when a given convex
set is the image under a linear map of an affine slice of a given closed convex
cone. Such a representation or 'lift' of the convex set is especially useful if
the cone admits an efficient algorithm for linear optimization over its affine
slices. We show that the existence of a lift of a convex set to a cone is
equivalent to the existence of a factorization of an operator associated to the
set and its polar via elements in the cone and its dual. This generalizes a
theorem of Yannakakis that established a connection between polyhedral lifts of
a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts
of convex sets can also be characterized similarly. When the cones live in a
family, our results lead to the definition of the rank of a convex set with
respect to this family. We present results about this rank in the context of
cones of positive semidefinite matrices. Our methods provide new tools for
understanding cone lifts of convex sets.Comment: 20 pages, 2 figure
Lower bounds on matrix factorization ranks via noncommutative polynomial optimization
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the completely positive rank, and their symmetric analogues: the positive semidefinite rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples
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