arXiv subjects

Juan C. Vera

Publications and source records attributed to Juan C. Vera.

26 records · Page 2Linked to original sources

arXiv2018

Positive semidefinite approximations to the cone of copositive kernels

It has been shown that the maximum stable set problem in some infinite graphs, and the kissing number problem in particular, reduces to a minimization problem over the cone of copositive kernels. Optimizing over this infinite dimensional cone is not tractable, and approximations of this cone have been hardly considered in literature. We propose two convergent hierarchies of subsets of copositive kernels, in terms of non-negative and positive definite kernels. We use these hierarchies and representation theorems for invariant positive definite kernels on the sphere to construct new SDP-based bounds on the kissing number. This results in fast-to-compute upper bounds on the kissing number that lie between the currently existing LP and SDP bounds.

math.OC↗

arXiv2018

Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations

In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robust feasibility problems with an empty first stage. This is equivalent to deciding whether the uncertainty set is contained within the projection of the feasible region onto the uncertainty-space. Moreover, the considered sets are assumed to be described by polynomials. For answering this question, two very general approaches using methods from polynomial optimization are presented - one for showing feasibility and one for showing infeasibility. The developed methods are approximated through sum of squares polynomials and solved using semidefinite programs. Deciding robust feasibility and infeasibility is important for gas network operations, which is a nonconvex feasibility problem where the feasible set is described by a composition of polynomials with the absolute value function. Concerning the gas network problem, different topologies are considered. It is shown that a tree structured network can be decided exactly using linear programming. Furthermore, a method is presented to reduce a tree network with one additional arc to a single cycle network. In this case, the problem can be decided by eliminating the absolute value functions and solving the resulting linearly many polynomial optimization problems. Lastly, the effectivity of the methods is tested on a variety of small cyclic networks. It turns out that for instances where robust feasibility or infeasibility can be decided successfully, level 2 or level 3 of the Lasserre relaxation hierarchy typically is sufficient.

math.OC↗

arXiv2017

Positive polynomials on unbounded domains

Certificates of non-negativity such as Putinar's Positivstellensatz have been used to obtain powerful numerical techniques to solve polynomial optimization (PO) problems. Putinar's certificate uses sum-of-squares (sos) polynomials to certify the non-negativity of a given polynomial over a domain defined by polynomial inequalities. This certificate assumes the Archimedean property of the associated quadratic module, which in particular implies compactness of the domain. In this paper we characterize the existence of a certificate of non-negativity for polynomials over a possibly unbounded domain, without the use of the associated quadratic module. Next, we show that the certificate can be used to convergent linear matrix inequality (LMI) hierarchies for PO problems with unbounded feasible sets. Furthermore, by using copositive polynomials to certify non-negativity, instead of sos polynomials, the certificate allows the use of a very rich class of convergent LMI hierarchies to approximate the solution of general PO problems. Throughout the article we illustrate our results with various examples certifying the non-negativity of polynomials over possibly unbounded sets defined by polynomial equalities or inequalities.

math.OC↗

arXiv2016

On the convergence rate of grid search for polynomial optimization over the simplex

We consider the approximate minimization of a given polynomial on the standard simplex, obtained by taking the minimum value over all rational grid points with given denominator ${r} \in \mathbb{N}$. It was shown in [De Klerk, E., Laurent, M., Sun, Z.: An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution. {\em SIAM J. Optim.} 25(3) 1498--1514 (2015)] that the relative accuracy of this approximation depends on $r$ as $O(1/r^2)$ if there exists a rational global minimizer. In this note we show that the rational minimizer condition is not necessary to obtain the $O(1/r^2)$ bound.

math.OC↗

arXiv2014

Improved Bounds on the Phase Transition for the Hard-Core Model in 2-Dimensions

For the hard-core lattice gas model defined on independent sets weighted by an activity $λ$, we study the critical activity $λ_c(\mathbb{Z}^2)$ for the uniqueness/non-uniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$. The conjectured value of the critical activity is approximately $3.796$. Until recently, the best lower bound followed from algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree $Δ$ when $λ<λ_c(\mathbb{T}_Δ)$ where $\mathbb{T}_Δ$ is the infinite, regular tree of degree $Δ$. His result established a certain decay of correlations property called strong spatial mixing (SSM) on $\mathbb{Z}^2$ by proving that SSM holds on its self-avoiding walk tree $T_{\mathrm{saw}}^σ(\mathbb{Z}^2)$ where $σ=(σ_v)_{v\in \mathbb{Z}^2}$ and $σ_v$ is an ordering on the neighbors of vertex $v$. As a consequence he obtained that $λ_c(\mathbb{Z}^2)\geqλ_c( \mathbb{T}_4) = 1.675$. Restrepo et al. (2011) improved Weitz's approach for the particular case of $\mathbb{Z}^2$ and obtained that $λ_c(\mathbb{Z}^2)>2.388$. In this paper, we establish an upper bound for this approach, by showing that, for all $σ$, SSM does not hold on $T_{\mathrm{saw}}^σ(\mathbb{Z}^2)$ when $λ>3.4$. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to $λ_c(\mathbb{Z}^2)>2.48$.

cs.DM↗

arXiv2012

Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

We prove that the mixing time of the Glauber dynamics for random k-colorings of the complete tree with branching factor b undergoes a phase transition at $k=b(1+o_b(1))/\ln{b}$. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for $k=Cb/\ln{b}$ colors with constant C. For $C\geq1$ we prove the mixing time is $O(n^{1+o_b(1)}\ln{n})$. On the other side, for $C<1$ the mixing time experiences a slowing down; in particular, we prove it is $O(n^{1/C+o_b(1)}\ln{n})$ and $Ω(n^{1/C-o_b(1)})$. The critical point C=1 is interesting since it coincides (at least up to first order) with the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.

math.PR↗

arXiv2011

Randomly coloring planar graphs with fewer colors than the maximum degree

We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $Δ$. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar graphs when $k=Ω(Δ/\logΔ)$. Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most $Δ^{1-\eps}$, for fixed $\eps > 0$. The main challenge when $k \le Δ+ 1$ is the possibility of "frozen" vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when $Δ= O(1)$, even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using "local uniformity" properties.

math.PR↗

arXiv2010

Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees

We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $λ$, called the activity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $ω$ which is the positive solution to $λ=ω(1+ω)^b$, and vertices are occupied with probability $ω/(1+ω)$ when their parent is unoccupied. This broadcasting process undergoes a phase transition between the so-called reconstruction and non-reconstruction regions at $ω_r\approx \ln{b}/b$. Reconstruction has been of considerable interest recently since it appears to be intimately connected to the efficiency of local algorithms on locally tree-like graphs, such as sparse random graphs. In this paper we show that the relaxation time of the Glauber dynamics on regular $b$-ary trees $T_h$ of height $h$ and $n$ vertices, undergoes a phase transition around the reconstruction threshold. In particular, we construct a boundary condition for which the relaxation time slows down at the reconstruction threshold. More precisely, for any $ω\le \ln{b}/b$, for $T_h$ with any boundary condition, the relaxation time is $Ω(n)$ and $O(n^{1+o_b(1)})$. In contrast, above the reconstruction threshold we show that for every $δ>0$, for $ω=(1+δ)\ln{b}/b$, the relaxation time on $T_h$ with any boundary condition is $O(n^{1+δ+ o_b(1)})$, and we construct a boundary condition where the relaxation time is $Ω(n^{1+δ/2 - o_b(1)})$.

math.PR↗