For many decision-making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces, we develop a cut generation algorithm, where each cut is obtained by solving a mixed-integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible and computationally tractable way of modeling preferences in stochastic multicriteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods.

Given a directed graph G = (V, A) with a non-negative weight (length) function on its arcs w : A → R + and two terminals s, t ∈ V , our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A. This is known as the short paths interdiction problem. We consider several versions of it, and in each case analyze two subcases: total limited interdiction, when a fixed number k of arcs can be removed, and node-wise limited interdiction, when for each node v ∈ V a fixed number k(v) of out-going arcs can be removed. Our results indicate that the latter subcase is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra's algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k, it is NP-hard to approximate within a factor c < 2 the maximum s−t distance d(s, t) obtainable by removing (at most) k arcs from G. Furthermore, given d, it is NP-hard to approximate within a factor c < 10 √ 5 − 21 ≈ 1.36 the minimum number of arcs which has to be removed to guarantee d(s, t) ≥ d. Finally, we also show that the same inapproximability bounds hold for non-directed graphs and/or node elimination.Keywords: approximation algorithm, Dijkstra's algorithm, most vital arcs problem, cyclic game, maxmin mean cycle, minimal vertex cover, network inhibition, network interdiction. Introduction Node-wise limited interdictionLet G = (V, A) be a directed graph (digraph) with given arc-weights w(e), e ∈ A. For each vertex v ∈ V , we are allowed to delete (remove, block, interdict) a subset X(v) of the arcs A(v) = {e ∈ A | e = (v, u)} leaving v. We assume that these arc-sets X(v) ⊆ A(v) are selected for all vertices v ∈ V independently, and we call the collection B(v) of all admissible arc-sets X(v) a blocking system at v. We also assume that for each v, the family B(v) forms an independence system, i.e., if X(v) ∈ B(v) is an admissible arc-set at v, then so is any subset of X(v). Hence, we could replace B(v) by the collection of all inclusion-wise maximal admissible arc-sets. In general, we will only assume that the blocking system B(v) is given by a membership oracle O : (B 0 ) Given a list X(v) of out-going arcs for some vertex v, the oracle can determine whether or not the arcs in the list belong to B(v) and hence can be simultaneously deleted.A similar formalization of blocking sets via membership oracles is used by Pisaruk [37]. We will also consider two special types of blocking systems:For each vertex v, we can delete any collection of (at most) k(v) arcs leaving v. The numbers k(v) define digraphs with prohibitions considered by Karzanov and Lebedev [30].(B 2 ) There are two types of vertices: control vertices, where we can select any out-going arc e ∈ A(v) and block all the remaining arcs in A(v), and regular vertices, where we can block no arc. This case, considered...

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BackgroundMethylation studies are a promising complement to genetic studies of DNA sequence. However, detailed prior biological knowledge is typically lacking, so methylome-wide association studies (MWAS) will be critical to detect disease relevant sites. A cost-effective approach involves the next-generation sequencing (NGS) of single-end libraries created from samples that are enriched for methylated DNA fragments. A limitation of single-end libraries is that the fragment size distribution is not observed. This hampers several aspects of the data analysis such as the calculation of enrichment measures that are based on the number of fragments covering the CpGs.ResultsWe developed a non-parametric method that uses isolated CpGs to estimate sample-specific fragment size distributions from the empirical sequencing data. Through simulations we show that our method is highly accurate. While the traditional (extended) read count methods resulted in severely biased coverage estimates and introduces artificial inter-individual differences, through the use of the estimated fragment size distributions we could remove these biases almost entirely. Furthermore, we found correlations of 0.999 between coverage estimates obtained using fragment size distributions that were estimated with our method versus those that were “observed” in paired-end sequencing data.ConclusionsWe propose a non-parametric method for estimating fragment size distributions that is highly precise and can improve the analysis of cost-effective MWAS studies that sequence single-end libraries created from samples that are enriched for methylated DNA fragments.

Linear stochastic programming problems with first order stochastic dominance (FSD) constraints are non-convex. For their mixed 0-1 linear programming formulation we present two convex relaxations based on second order stochastic dominance (SSD). We develop necessary and sufficient conditions for FSD, used to obtain a disjunctive programming formulation and to strengthen one of the SSD-based relaxations.