On the Burer-Monteiro method for general semidefinite programs

Consider a semidefinite program (SDP) involving an $n\times n$ positive semidefinite matrix $X$. The Burer-Monteiro method uses the substitution $X=Y Y^T$ to obtain a nonconvex optimization problem in terms of an $n\times p$ matrix $Y$. Boumal et al. showed that this nonconvex method provably solves equality-constrained SDPs with a generic cost matrix when $p \gtrsim \sqrt{2m}$, where $m$ is the number of constraints. In this note we extend their result to arbitrary SDPs, possibly involving inequalities or multiple semidefinite constraints. We derive similar guarantees for a fixed cost matrix and generic constraints. We illustrate applications to matrix sensing and integer quadratic minimization.Comment: 10 page

On the degree-chromatic polynomial of a tree

The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree chromatic polynomial of a tree.Comment: 3 page

Banking Concentration: Implications for Systemic Risk and Safety Net Design

This paper explores the impact of banking concentration on safety net design –in particular, deposit insurance– and on systemic risk. The paper focuses on a system characterized by high concentration and low total number of banks. Each issue is addressed separately. The first section discusses best practices in deposit insurance design and derives conclusions for the case we are interested in. One is that in this context deposit insurance cannot be thought of as a stand-alone instrument, but rather must be understood as an element of the intervention and resolution policy. The second part of the paper studies systemic risk in such a system, using the Eisenberg and Noe (2001) approach to model and study risk in a network of banks. A working metric of the “too big to fail” situation can be derived in the model. More importantly, this section shows how the risk of idiosyncratic shocks spreading through the system are substantially higher in concentrated systems than in decentralized ones. Finally, the paper proposes and evaluates a specific regulatory measure that successfully contains systemic risk.

Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gr\"obner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gr\"obner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization and differential equations.Comment: 40 pages, 5 figure

Tax Incentives for Retirement Savings: Macro and Welfare Effects in an OLG-GE Model with Liquidity Constraints and Heterogeneous Consumers.

This paper uses an Overlapping Generations-General Equilibrium model to study the impact of the introduction of tax incentives to voluntary savings for retirement in Chile. The paper analyzes the macro impact of the reform, driven mainly by its effect on savings and capital accumulation, and its effect on welfare. A setting with heterogeneous consumers is considered where agents differ in their income levels, and therefore on the relevance that tax-incentives have for them. Both the transition and the final steady state are analyzed. The heterogeneity modeled allows unveiling important distributive effects of the reform, in particular during the transition to the new steady state.

Modelling distributed lag effects in mortality and air pollution studies: the case of Santiago

Most of the epidemiological literature on air pollution and mortality deals only with single or dual pollutant models whose results are hard to interpret and of questionable value from the policy perspective. In addition, much of the existing literature deals only with the very short-term effects of air pollution whereas policy makers need to know when, whether and to what extent pollution-induced increases in mortality counts are reversed. This involves modelling the infinite distributed lag effects of air pollution. Borrowing from econometrics this paper presents a method by which the infinite distributed lag effects can be estimated parsimoniously but plausibly estimated. The paper presents a time series study into the relationship between ambient levels of air pollution and daily mortality counts for Santiago employing this technique which confirms that the infinite lag effects are highly significant. It is also shown that day to day variations in NO2 concentrations and in the concentrations of both fine and coarse particulates are associated with short-term variations in death rates. These findings are made in the context of a model that simultaneously includes six different pollutants. Evidence is found pointing to the operation of a very short term harvesting effect