Size matters: cardinality-constrained clustering and outlier detection via conic optimization

Plain vanilla K-means clustering has proven to be successful in practice, yet it suffers from outlier sensitivity and may produce highly unbalanced clusters. To mitigate both shortcomings, we formulate a joint outlier detection and clustering problem, which assigns a prescribed number of datapoints to an auxiliary outlier cluster and performs cardinality-constrainedK-means clustering on the residual dataset, treating the cluster cardinalities as a given input. We cast this problem as a mixed-integer linear program (MILP) that admits tractable semidefinite and linear programming relaxations. We propose deterministic rounding schemes thattransform the relaxed solutions to feasible solutions for the MILP. We also prove that these solutions areoptimal in the MILP if a cluster separation condition holds

Chebychev inequalities for products of random variables

We derive sharp probability bounds on the tails of a product of symmetric non-negative random variables using only information about their first two moments. If the covariance matrix of the random variables is known exactly, these bounds can be computed numerically using semidefinite programming. If only an upper bound on the covariance matrix is avail- able, the probability bounds on the right tails can be evaluated analytically. The bounds under precise and imprecise covariance information coincide for all left tails as well as for all right tails corresponding to quantiles that are either sufficiently small or sufficiently large. We also prove that all left probability bounds reduce to the trivial bound 1 if the number of random variables in the product exceeds an explicit threshold. Thus, in the worst case, the weak-sense geometric random walk defined through the running product of the random variables is absorbed at 0 with certainty as soon as time exceeds the given threshold. The techniques devised for constructing Chebyshev bounds for products can also be used to de- rive Chebyshev bounds for sums, maxima and minima of non-negative random variables

Optimal Financial Decision Making under Uncertainty

We use a fairly general framework to analyze a rich variety of financial optimization models presented in the literature, with emphasis on contributions included in this volume and a related special issue of OR Spectrum. We do not aim at providing readers with an exhaustive survey, rather we focus on a limited but significant set of modeling and methodological issues. The framework is based on a benchmark discrete-time stochastic control optimization framework, and a benchmark financial problem, asset--liability management, whose generality is considered in this chapter. A wide set of financial problems, ranging from asset allocation to financial engineering problems, is outlined, in terms of objectives, risk models, solution methods, and model users. We pay special attention to the interplay between alternative uncertainty representations and solution methods, which have an impact on the kind of solution which is obtained. Finally, we outline relevant directions for further research and optimization paradigms integration