3,682 research outputs found
Rational Polyhedral Outer-Approximations of the Second-Order Cone
It is well-known that the second-order cone can be outer-approximated to an
arbitrary accuracy by a polyhedral cone of compact size defined by
irrational data. In this paper, we propose two rational polyhedral
outer-approximations of compact size retaining the same guaranteed accuracy
. The first outer-approximation has the same size as the optimal but
irrational outer-approximation from the literature. In this case,we provide a
practical approach to obtain such an approximation defined by the smallest
integer coefficients possible, which requires solving a few, small-size integer
quadratic programs. The second outer-approximation has a size larger than the
optimal irrational outer-approximation by a linear additive factor in the
dimension of the second-order cone. However, in this case, the construction is
explicit, and it is possible to derive an upper bound on the largest
coefficient, which is sublinear in and logarithmic in the dimension.
We also propose a third outer-approximation, which yields the best possible
approximation accuracy given an upper bound on the size of its coefficients.
Finally, we discuss two theoretical applications in which having a rational
polyhedral outer-approximation is crucial, and run some experiments which
explore the benefits of the formulations proposed in this paper from a
computational perspective
Extended Formulations in Mixed-integer Convex Programming
We present a unifying framework for generating extended formulations for the
polyhedral outer approximations used in algorithms for mixed-integer convex
programming (MICP). Extended formulations lead to fewer iterations of outer
approximation algorithms and generally faster solution times. First, we observe
that all MICP instances from the MINLPLIB2 benchmark library are conic
representable with standard symmetric and nonsymmetric cones. Conic
reformulations are shown to be effective extended formulations themselves
because they encode separability structure. For mixed-integer
conic-representable problems, we provide the first outer approximation
algorithm with finite-time convergence guarantees, opening a path for the use
of conic solvers for continuous relaxations. We then connect the popular
modeling framework of disciplined convex programming (DCP) to the existence of
extended formulations independent of conic representability. We present
evidence that our approach can yield significant gains in practice, with the
solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201
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
- …