A Semidefinite Approach to the KiK_i Cover Problem

We apply theta body relaxations to the $K_i$-cover problem and show polynomial time solvability for certain classes of graphs. In particular, we give an effective relaxation where all $K_i$-$p$-hole facets are valid, and study its relation to an open question of Conforti et al. For the triangle free problem, we show for $K_n$ that the theta body relaxations do not converge by $(n-2)/4$ steps; we also prove for all $G$ an integrality gap of 2 for the second theta body

Convex Hulls of Algebraic Sets

This article describes a method to compute successive convex approximations of the convex hull of a set of points in R^n that are the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of polynomials and the dual theory of moment matrices. The main feature of the technique is that all computations are done modulo the ideal generated by the polynomials defining the set to the convexified. This work was motivated by questions raised by Lov\'asz concerning extensions of the theta body of a graph to arbitrary real algebraic varieties, and hence the relaxations described here are called theta bodies. The convexification process can be seen as an incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic set in R^n. When the defining ideal is real radical the results become especially nice. We provide several examples of the method and discuss convergence issues. Finite convergence, especially after the first step of the method, can be described explicitly for finite point sets.Comment: This article was written for the "Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications

Competências: moda ou inevitabilidade?

conceito de competência é aqui exaustivamente trabalhado, desde os seus contextos de emergência, no mundo do trabalho, até às diferentes acepções e definições com que nos surge, designadamente, como sucedâneo mal compreendido e apreendido das noções de objectivos, em Ciências daEducação. Importa colocar alguma ordem no universo de referenciação do conceito, desde logo, para que ele possa operacionalizar-se em práticas educativas intencionalmente concebidas e aplicadas, à luz dos novos paradigmas da sociedade do conhecimento.The concept of competence is here worked about, having into consideration not only its emergence contexts at the world of labor but also its different assumptions and definitions, even as a misunderstood concept confused with the objectives notions in the Sciences of Education. It is important to reorder the definition of this concept, so that it can be taken into account in educational practices, which shall be conceived and applied, according to the new paradigms of the society of knowledge

Theta Bodies for Polynomial Ideals

Inspired by a question of Lov\'asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lov\'asz's theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre's relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals.Comment: 26 pages, 3 figure

Approximate cone factorizations and lifts of polytopes

In this paper we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix. This provides a robust generalization of the famous result of Yannakakis that polyhedral lifts of a polytope are controlled by (exact) nonnegative factorizations of its slack matrix. Our approximations behave well under polarity and have efficient representations using second order cones. We establish a direct relationship between the quality of the factorization and the quality of the approximations, and our results extend to generalized slack matrices that arise from a polytope contained in a polyhedron

Polytopes of Minimum Positive Semidefinite Rank

The positive semidefinite (psd) rank of a polytope is the smallest $k$ for which the cone of $k \times k$ real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three

Risk vs return: A comparative analysis between a developed and an emerging stock market

Emerging stock markets have presented several opportunities for international investors over the last decades. This thesis addresses the risk-adjusted returns of an emerging stock market by comparing its returns and volatility with a developed stock market. Using the United States and China as the representative markets, the dissertation explores the opportunities offered by the emerging market for international investors. Most of the previous literature focuses on the diversification benefits of emerging stock markets. In this study, I innovate by looking at these markets as an alternative investment rather than the diversification potential. To compare the risk-adjusted returns of the two markets I calculate the weekly Sharpe ratio for the S&P 500 and the SSE Composite for 18 years. I perform multiple linear regression, for both the emerging and the developed markets, to analyze the factors that affect the Sharpe ratio calculated. The empirical results confirm that the financial market characteristics, the macroeconomic factors, and the correlation/contagion impact the performance of both the emerging and the developed stock markets. Regarding the risk-adjusted returns, the results show that in the period studied, the index representative of the US market presents higher returns, lower volatility, and consequently a higher Sharpe ratio than the one for the Chinese market index. Such results suggest that the developed stock market offers higher and more sustained risk-adjusted returns in comparison to the emerging stock market.Os mercados bolsistas emergentes têm apresentado diversas oportunidades para os investidores internacionais ao longo das últimas décadas. Esta tese aborda o retorno ajustado ao risco de um mercado emergente ao comparar os retornos e volatilidade deste mercado com os de um mercado desenvolvido. Usando os Estados Unidos e China como os mercados representativos, a dissertação explora as oportunidades oferecidas pelo mercado emergente para os investidores internacionais. A maioria da literatura desenvolvida anteriormente destaca os benefícios da diversificação conseguida através dos mercados bolsistas emergentes. Neste estudo, inovo ao analisar estes mercados como um investimento alternativo, em vez de focar nos benefícios de diversificação. Para comparar o retorno ajustado ao risco dos dois mercados, calculo o Sharpe ratio semanal para o S&P 500 e para o SSE Composite durante um período de 18 anos. Realizo uma regressão linear múltipla, para os mercados emergente e desenvolvido, a fim de analisar os factores que afectam o Sharpe ratio calculado. Os resultados empíricos confirmam que as características dos mercados financeiros, os factores macroeconómicos, e a correlação/contágio têm um impacto no desempenho de ambos os mercados. No que diz respeito aos retornos ajustados ao risco, os resultados indicam que o índice representativo do mercado dos Estados Unidos apresenta um retorno superior, uma volatilidade menor, e consequentemente um Sharpe ratio mais elevado do que o calculado para o índice de mercado chinês. Estes resultados permitem concluir que o mercado bolsista desenvolvido apresenta um maior e mais sustentado retorno ajustado ao risco em comparação com o mercado emergente