Approximate Cone Factorizations
and Lifts of Polytopes
In this paper we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix. This provides a robust generalization of the famous result of Yannakakis that polyhedral lifts of a polytope are controlled by (exact) nonnegative factorizations of its slack matrix. Our approximations behave well under polarity and have efficient representations using second order cones. We establish a direct relationship between the quality of the factorization and the quality of the approximations, and our results extend to generalized slack matrices that arise from a polytope contained in a polyhedron.
A well-known idea in optimization to represent a complicated convex set is to describe it as the linear image of a simpler convex set in a higher dimensional space, called a lift or extended formulation of . The standard way to express such a lift is as an affine slice of some closed convex cone , called a -lift of , and the usual examples of are nonnegative orthants and the cones of real symmetric positive semidefinite matrices . More precisely, has a -lift, where , if there exists an affine subspace and a linear map such that .
Given a nonnegative matrix and a closed convex cone with dual cone , a -factorization of is a collection of elements and such that for all . In particular, a -factorization of , also called a nonnegative factorization of of size , is typically expressed as where has columns and has columns . In [Yannakakis], Yannakakis laid the foundations of polyhedral lifts of polytopes by showing the following.
[Yannakakis] A polytope has a -lift if and only if the slack matrix of has a -factorization.
This theorem was extended in [GPT2012] from -lifts of polytopes to -lifts of convex sets , where is any closed convex cone, via -factorizations of the slack operator of .
The above results rely on exact cone factorizations of the slack matrix or operator of the given convex set, and do not offer any suggestions for constructing lifts of the set in the absence of exact factorizations. In many cases, one only has access to approximate factorizations of the slack matrix, typically via numerical algorithms. In this paper we show how to take an approximate -factorization of the slack matrix of a polytope and construct from it an inner and outer convex approximation of the polytope. Our approximations behave well under polarity and admit efficient representations via second order cones. Further, we show that the quality of our approximations can be bounded by the error in the corresponding approximate factorization of the slack matrix.
Let be a full-dimensional polytope in with the origin in its interior, and vertices . We may assume without loss of generality that each inequality in defines a facet of . If has size , then the slack matrix of is the nonnegative matrix whose -entry is , the slack of the th vertex in the th inequality of . Given an -factorization of , i.e., two nonnegative matrices and such that , an -lift of is obtained as
Notice that this lift is highly non-robust, and small perturbations of make the right hand side empty, since the linear system is in general highly overdetermined. The same sensitivity holds for all -factorizations and lifts. Hence, it becomes important to have a more robust, yet still efficient, way of expressing (at least approximately) from approximate -factorizations of . Also, the quality of the approximations of and their lifts must reflect the quality of the factorization, and specialize to the Yannakakis setting when the factorization is exact. The results in this paper carry out this program and contain several examples, special cases, and connections to the recent literature.
1.1. Organization of the paper
In Section 2 we establish how an approximate -factorization of the slack matrix of a polytope yields a pair of inner and outer convex approximations of which we denote as and where and are the two “factors” in the approximate -factorization. These convex sets arise naturally from two simple inner and outer second order cone approximations of the nonnegative orthant. While the outer approximation is always closed, the inner approximation maybe open if is an arbitrary cone. However, we show that if the polar of is “nice” [Pataki1], then the inner approximation will be closed. All cones of interest to us in this paper such as nonnegative orthants, positive semidefinite cones, and second order cones are nice. Therefore, we will assume that our approximations are closed after a discussion of their closedness.
We prove that our approximations behave well under polarity, in the sense that
where is the polar polytope of . Given and , our approximations admit efficient representations via slices and projections of where is a second order cone of dimension . We show that an -error in the -factorization makes and , thus establishing a simple link between the error in the factorization and the gap between and its approximations. In the presence of an exact -factorization of the slack matrix, our results specialize to the Yannakakis setting.
In Section LABEL:sec:cases we discuss two connections between our approximations and well-known constructions in the literature. In the first part we show that our inner approximation, , always contains the Dikin ellipsoid used in interior point methods. Next we examine the closest rank one approximation of the slack matrix obtained via a singular value decomposition and the approximations of the polytope produced by it.
In Section LABEL:sec:twomatrices we extend our results to the case of generalized slack matrices that arise from a polytope contained in a polyhedron. We also show how an approximation of with a -lift produces an approximate -factorization of the slack matrix of . It was shown in [BraunFioriniPokuttaSteurer] that the max clique problem does not admit polyhedral approximations with small polyhedral lifts. We show that this negative result continues to hold even for the larger class of convex approximations considered in this paper.
2. From approximate factorizations to approximate lifts
In this section we show how to construct inner and outer approximations of a polytope from approximate -factorizations of the slack matrix of , and establish the basic properties of these approximations.
2.1. -factorizations and linear maps
Let be a full-dimensional polytope with the origin in its interior. The vertices of the polytope are , and each inequality for in defines a facet of . The slack matrix of is the matrix with entries . In matrix form, letting and , we have the expression . We assume is a closed convex cone, with dual cone .
([GPT2012]) A -factorization of the slack matrix of the polytope is given by , such that for and . In matrix form, this is the factorization
where and .
It is convenient to interpret a -factorization as a composition of linear maps as follows. Consider as a linear map from , verifying . Similarly, think of as a linear map from verifying . Then, for the adjoint operators, and . Furthermore, we can think of the slack matrix as an affine map from to , and the matrix factorization in Definition 2.1 suggests to define the slack operator, , as , where and .