arXiv subjects

Alex L. Wang

Publications and source records attributed to Alex L. Wang.

24 records · Page 2Linked to original sources

arXiv2021

Necessary and sufficient conditions for rank-one generated cones

A closed convex conic subset $\mathcal{S}$ of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of $\mathcal{S}$ is closely related to the exactness of SDP relaxations of nonconvex quadratically constrained quadratic programs (QCQPs) related to $\mathcal{S}$. We consider the case where $\mathcal{S}$ is obtained as the intersection of the PSD cone with finitely many homogeneous linear matrix inequalities and conic constraints and identify sufficient conditions that guarantee that $\mathcal{S}$ is ROG. Our general framework allows us to recover a number of well-known results from the literature. In the case of two linear matrix inequalities, we also establish the necessity of our sufficient conditions. This extends one of the few settings from the literature -- the case of one linear matrix inequality and the S-lemma -- where an explicit characterization for the ROG property exists. Finally, we show how our ROG results on cones can be translated into inhomogeneous SDP exactness results and convex hull descriptions in the original space of a QCQP. We close with a few applications of these results; specifically, we recover the well-known perspective reformulation of a simple mixed-binary set via the ROG toolkit.

math.OC↗

arXiv2020

On the tightness of SDP relaxations of QCQPs

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

math.OC↗

arXiv2020

The Generalized Trust Region Subproblem: solution complexity and convex hull results

We consider the Generalized Trust Region Subproblem (GTRS) of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint. A lifting of this problem recasts the GTRS as minimizing a linear objective subject to two nonconvex quadratic constraints. Our first main contribution is structural: we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil. This result may be of interest in building relaxations for nonconvex quadratic programs. Moreover, this result allows us to reformulate the GTRS as the minimization of two convex quadratic functions in the original space. Our next set of contributions is algorithmic: we present an algorithm for solving the GTRS up to an epsilon additive error based on this reformulation. We carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms. Notably, our algorithms run in linear (in the size of the input) time. Furthermore, our algorithm for computing an epsilon-optimal solution has a slightly-improved running time dependence on epsilon over the state-of-the-art algorithm. Our analysis shows that the dominant cost in solving the GTRS lies in solving a generalized eigenvalue problem -- establishing a natural connection between these problems. Finally, generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality-, interval-, and hollow- constrained variants of the GTRS. This gives the first linear-time algorithm in the literature for these variants of the GTRS.

cs.DS↗

arXiv2020

Weighted Cheeger and Buser Inequalities, with Applications to Clustering and Cutting Probability Densities

In this paper, we show how sparse or isoperimetric cuts of a probability density function relate to Cheeger cuts of its principal eigenfunction, for appropriate definitions of `sparse cut' and `principal eigenfunction'. We construct these appropriate definitions of sparse cut and principal eigenfunction in the probability density setting. Then, we prove Cheeger and Buser type inequalities similar to those for the normalized graph Laplacian of Alon-Milman. We demonstrate that no such inequalities hold for most prior definitions of sparse cut and principal eigenfunction. We apply this result to generate novel algorithms for cutting probability densities and clustering data, including a principled variant of spectral clustering.

cs.LG↗

arXiv2020

On convex hulls of epigraphs of QCQPs

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

math.OC↗

arXiv2018

Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues

We present two graph quantities Psi(G,S) and Psi_2(G) which give constant factor estimates to the Dirichlet and Neumann eigenvalues, lambda(G,S) and lambda_2(G), respectively. Our techniques make use of a discrete Hardy-type inequality.

cs.DM↗