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Community Detection in Random Networks
, 2013
"... We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. Under the null hypothesis, the graph is a realization of an ErdösRényi graph with prob ..."
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We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. Under the null hypothesis, the graph is a realization of an ErdösRényi graph with probability p0. Under the (composite) alternative, there is a subgraph of n nodes where the probability of connection is p1> p0. We derive a detection lower bound for detecting such a subgraph in terms of N,n,p0,p1 and exhibit a test that achieves that lower bound. We do this both when p0 is known and unknown. We also consider the problem of testing in polynomialtime. As an aside, we consider the problem of detecting a clique, which is intimately related to the planted clique problem. Our focus in this paper is in the quasinormal regime where np0 is either bounded away from zero, or tends to zero slowly.
Efficient Cardinality/MeanVariance Portfolios
, 2012
"... A number of variants of the classical Markowitz meanvariance optimization model for portfolio selection have been investigated to render it more realistic. Recently, it has been studied the imposition of a cardinality constraint, setting an upper bound on the number of active positions taken in the ..."
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A number of variants of the classical Markowitz meanvariance optimization model for portfolio selection have been investigated to render it more realistic. Recently, it has been studied the imposition of a cardinality constraint, setting an upper bound on the number of active positions taken in the portfolio, in an attempt to improve its performance and reduce transactions costs. However, one can regard cardinality as an objective function itself, thus adding another goal to those two traditionally considered (the variance and the mean of the return). In this paper, we suggest a new approach to directly compute sparse portfolios by reformulating the cardinality constrained Markowitz meanvariance optimization model as a biobjective optimization problem, allowing the investor to analyze the efficient tradeoff between meanvariance and cardinality, in a general scenario where shortselling is allowed. Since cardinality is a nonsmooth objective function, one has chosen a derivativefree algorithm (based on direct multisearch) for the solution of the biobjective optimization problem. For the several data sets obtained from the FTSE 100 index and the Fama/French benchmark collection, direct multisearch was capable of quickly determining (insample) the efficient frontier for the biobjective cardinality/meanvariance problem. Our results showed that a number of efficient cardinality/meanvariance portfolios (with values of cardinality not high) overcome the naive strategy in terms of outofsample performance measured by the Sharpe ratio, which is known to be extremely difficult.
Recent Advances in Mathematical Programming with Semicontinuous Variables and Cardinality Constraint
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Efficient Cardinality/MeanVariance Portfolios
"... Abstract. We propose a novel approach to handle cardinality in portfolio selection, by means of a biobjective cardinality/meanvariance problem, allowing the investor to analyze the efficient tradeoff between returnrisk and number of active positions. Recent progress in multiobjective optimizati ..."
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Abstract. We propose a novel approach to handle cardinality in portfolio selection, by means of a biobjective cardinality/meanvariance problem, allowing the investor to analyze the efficient tradeoff between returnrisk and number of active positions. Recent progress in multiobjective optimization without derivatives allow us to robustly compute (insample) the whole cardinality/meanvariance efficient frontier, for a variety of data sets and meanvariance models. Our results show that a significant number of efficient cardinality/meanvariance portfolios can overcome (outofsample) the naive strategy, while keeping transaction costs relatively low.