arXiv subjects

Juan Vera

Publications and source records attributed to Juan Vera.

9 recordsLinked to original sources

arXiv2025

Revisiting the convergence rate of the Lasserre hierarchy for polynomial optimization over the hypercube

We revisit the problem of minimizing a given polynomial $f$ on the hypercube $[-1,1]^n$. Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds $\{f_{(r)}\}_{r \in \mathbb N}$ on the minimum value $f^*$, where $r$ refers to the allowed degrees in the sum-of-squares hierarchy. A natural question is how fast the hierarchy converges as a function of the parameter $r$. The current state-of-the-art is due to Baldi and Slot [SIAM J. on Applied Algebraic Geometry, 2024] and roughly shows a convergence rate of order $1/r$. Here we obtain closely related results via a different approach: the polynomial kernel method. We also discuss limitations of the polynomial kernel method, suggesting a lower bound of order $1/r^2$ for our approach.

math.OC↗

arXiv2021

The maximum $k$-colorable subgraph problem and related problems

The maximum $k$-colorable subgraph (M$k$CS) problem is to find an induced $k$-colorable subgraph with maximum cardinality in a given graph. This paper is an in-depth analysis of the M$k$CS problem that considers various semidefinite programming relaxations including their theoretical and numerical comparisons. To simplify these relaxations we exploit the symmetry arising from permuting the colors, as well as the symmetry of the given graphs when applicable. We also show how to exploit invariance under permutations of the subsets for other partition problems and how to use the M$k$CS problem to derive bounds on the chromatic number of a graph. Our numerical results verify that the proposed relaxations provide strong bounds for the M$k$CS problem, and that those outperform existing bounds for most of the test instances.

math.OC↗

arXiv2020

New characterizations of Hoffman constants for systems of linear constraints

We give a characterization of the Hoffman constant of a system of linear constraints in $\R^n$ {\em relative} to a {\em reference polyhedron} $R\subseteq\R^n$. The reference polyhedron $R$ represents constraints that are easy to satisfy such as box constraints. In the special case $R = \R^n$, we obtain a novel characterization of the classical Hoffman constant. More precisely, suppose $R\subseteq \mathbb{R}^n$ is a reference polyhedron, $A\in \R^{m\times n},$ and $A(R):=\{Ax: x\in R\}$. We characterize the sharpest constant $H(A|R)$ such that for all $b \in A(R) + \R^m_+$ and $u\in R$ \[ \dist(u, P_{A}(b)\cap R) \le H(A|R) \cdot \|(Au-b)_+\|, \] where $P_A(b) = \{x\in \R^n:Ax\le b\}$. Our characterization is stated in terms of the largest of a canonical collection of easily computable Hoffman constants. Our characterization in turn suggests new algorithmic procedures to compute Hoffman constants.

math.OC↗

arXiv2018

Globally solving Non-Convex Quadratic Programs via Linear Integer Programming techniques

Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (NLP) solvers, and CPLEX's non-convex QP solver, on a large variety of QP test instances. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small- to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on box-constrained QP instances. Also, unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.

math.OC↗

arXiv2018

An algorithm to compute the Hoffman constant of a system of linear constraints

We propose a combinatorial algorithm to compute the Hoffman constant of a system of linear equations and inequalities. The algorithm is based on a characterization of the Hoffman constant as the largest of a finite canonical collection of easy-to-compute Hoffman constants. Our algorithm and characterization extend to the more general context where some of the constraints are easy to satisfy as in the case of box constraints. We highlight some natural connections between our characterizations of the Hoffman constant and Renegar's distance to ill-posedness for systems of linear constraints.

math.OC↗

arXiv2016

Computing semiparametric bounds on the expected payments of insurance instruments via column generation

It has been recently shown that numerical semiparametric bounds on the expected payoff of fi- nancial or actuarial instruments can be computed using semidefinite programming. However, this approach has practical limitations. Here we use column generation, a classical optimization technique, to address these limitations. From column generation, it follows that practical univari- ate semiparametric bounds can be found by solving a series of linear programs. In addition to moment information, the column generation approach allows the inclusion of extra information about the random variable; for instance, unimodality and continuity, as well as the construction of corresponding worst/best-case distributions in a simple way.

q-fin.PR↗

arXiv2009

Reconstruction for Colorings on Trees

Consider $k$-colorings of the complete tree of depth $\ell$ and branching factor $Δ$. If we fix the coloring of the leaves, as $\ell$ tends to $\infty$, for what range of $k$ is the root uniformly distributed over all $k$ colors? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which ``freeze'' the entire tree when $k\leΔ+1$. For $k\geqΔ+2$, Jonasson proved the root is ``unbiased'' for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for a {\em typical} coloring of the leaves? When the leaves have a non-vanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Non-reconstruction is equivalent to extremality of the free-boundary Gibbs measure. When $k<Δ/\lnΔ$, it is straightforward to show that reconstruction is possible and hence the measure is not extremal. We prove that for $C>1$ and $k =CΔ/\lnΔ$, that the Gibbs measure is extremal in a strong sense: with high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. Closely related results were also proven recently by Sly. The above strong form of extremality implies that a local Markov chain that updates constant sized blocks has inverse linear entropy constant and hence $O(N\log N)$ mixing time where $N$ is the number of vertices of the tree.

math.PR↗

arXiv2009

Logconcave Random Graphs

We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by picking a random point X from F and defining a graph on n vertices whose edges are pairs ij for which X_{ij} \le p. The standard Erdős-Rényi model is the special case when F is uniform on the 0-1 unit cube. We examine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the X_{ij} are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight.

math.CO↗

arXiv2007

Bias reduction in traceroute sampling: towards a more accurate map of the Internet

Traceroute sampling is an important technique in exploring the internet router graph and the autonomous system graph. Although it is one of the primary techniques used in calculating statistics about the internet, it can introduce bias that corrupts these estimates. This paper reports on a theoretical and experimental investigation of a new technique to reduce the bias of traceroute sampling when estimating the degree distribution. We develop a new estimator for the degree of a node in a traceroute-sampled graph; validate the estimator theoretically in Erdos-Renyi graphs and, through computer experiments, for a wider range of graphs; and apply it to produce a new picture of the degree distribution of the autonomous system graph.

physics.soc-ph↗