Let $V$ be an Euclidean Jordan algebra and $\Omega$ be a cone of invertible squares in $V$. Suppose that $g:\mathbb{R}_{+} \to \mathbb{R}$ is a matrix monotone function on the positive semiaxis which naturally induces a function $\tilde{g}: \Omega \to V$. We show that $-\tilde{g}$ is compatible (in the sense of Nesterov-Nemirovski) with the standard self-concordant … Read more
Consider a semidefinite program (SDP) involving an $n\times n$ positive semidefinite matrix $X$. The Burer-Monteiro method consists in solving a nonconvex program in $Y$, where $Y$ is an $n\times p$ matrix such that $X = Y Y^T$. Despite nonconvexity, Boumal et al. showed that the method provably solves generic equality-constrained SDP’s when $p > \sqrt{2m}$, … Read more
The Bandwidth Problem asks for a simultaneous permutation of the rows and columns of the adjacency matrix of a graph such that all nonzero entries are as close as possible to the main diagonal. This work focuses on investigating novel approaches to obtain lower bounds for the bandwidth problem. In particular, we use vertex partitions … Read more
We present several solution techniques for the noisy single source localization problem, i.e.,~the Euclidean distance matrix completion problem with a single missing node to locate under noisy data. For the case that the sensor locations are fixed, we show that this problem is implicitly convex, and we provide a purification algorithm along with the SDP … Read more
Several logarithmic-barrier interior-point methods are now available for solving two-stage stochastic optimization problems with recourse in the finite-dimensional setting. However, despite the genuine need for studying such methods in general spaces, there are no infinite-dimensional analogs of these methods. Inspired by this evident gap in the literature, in this paper, we propose logarithmic-barrier decomposition-based interior-point … Read more
We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy … Read more
In this work we present an Augmented Lagrangian algorithm for nonlinear semidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinear programming. This method works with two levels of constraints; one that is penalized and other that is kept within the subproblems. This is done in order to allow exploiting the … Read more
Computational speed and global optimality are key needs for practical algorithms for the optimal power flow problem. Two convex relaxations offer a favorable trade-off between the standard second-order cone and the standard semidefinite relaxations for large-scale meshed networks in terms of optimality gap and computation time: the tight-and-cheap relaxation (TCR) and the quadratic convex relaxation … Read more
We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregated and correlative sparsity of the data matrices of the CPP relaxation is represented by a block-clique graph $G$. By exploiting the correlative sparsity, we decompose the CPP relaxation problem into a clique-tree structured family of smaller … Read more
Sum-of-norms clustering is a method for assigning $n$ points in $\R^d$ to $K$ clusters, $1\le K\le n$, using convex optimization. Recently, Panahi et al.\ proved that sum-of-norms clustering is guaranteed to recover a mixture of Gaussians under the restriction that the number of samples is not too large. The purpose of this note is to … Read more