On the Complexity of Nonoverlapping Multivariate Marginal Bounds for Probabilistic Combinatorial Optimization Problems

Abstract

Given a combinatorial optimization problem with an arbitrary partition of the set of random objective coefficients, we evaluate the tightest possible bound on the expected optimal value for joint distributions consistent with the given multivariate marginals of the subsets in the partition.For univariate marginals, this bound was first proposed by Meilijson and Nadas (Journal of Applied Probability, 1979). We generalize the bound to non-overlapping multivariate marginals using multiple choice integer programming. For discrete distributions, new instances of polynomial time computable multivariate marginal bounds are identified. For the problem of selecting up to M items out a set of N items of maximum total weight, the bound is shown to be computable in polynomial time, when the size of each subset in the partition is O(log N ). For an activity-on-arc PERT network, the partition is naturally defined by subsets of incoming arcs into nodes. The worst-case expected project duration is shown to be computable in time polynomial in the maximum number of scenarios for any subset and the size of the network. An instance of a polynomial time solvable two stage stochastic program arising from project crashing is identified. An important feature of the bound is that it is exactly achievable by a joint distribution, unlike many of the existing bounds.

A distributionally robust perspective on uncertainty quantification and chance constrained programming

“…Ambiguity sets of special interest include the Markov ambiguity set containing all distributions with known mean and support [48], the Chebyshev ambiguity set containing all distributions with known bounds on the first and second-order moments [12,14,22,31,39,46,49,51,52], the Gauss ambiguity set containing all unimodal distributions from within the Chebyshev ambiguity set [38,41], various generalized Chebyshev ambiguity sets that specify asymmetric moments [12,13,35], higher-order moments [7,30,45] or marginal moments [17,18], the median-absolute deviation ambiguity set containing all symmetric distributions with known median and mean absolute deviation [24], the Huber ambiguity set containing all distributions with known upper bound on the expected Huber loss function [15,48], the Wasserstein ambiguity set containing all distributions that are close to the empirical distribution with respect to the Wasserstein metric [19,34,40], the KullbackLeibler divergence ambiguity set and likelihood ratio ambiguity set [10,26,27,31,47] containing all distributions that are sufficiently likely to have generated a given data set, the Hoeffding ambiguity set containing all component-wise independent distributions with a box support [3,8,10], the Bernstein ambiguity set containing all distributions from within the Hoeffding ambiguity set subject to marginal moment bounds [36], several φ-divergence-based ambiguity sets [2,…”

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

“…This distribution maximizes the Shannon entropy among all the measures θ ∈ Θ E (see Jiroušek We conclude this section by showing that the result in Doan and Natarajan (2012) for general partitions can be derived from the result of Theorem 1 for general covers. By assigning dual…”

“…Several models have been proposed to capture the uncertainty (or ambiguity) in distributions. This includes the class of distributions with information on the first and second moments (see El Ghaoui et al (2003), Delage and Ye (2010), Natarajan et al (2009a), Zymler et al (2013)), the Fréchet class of distributions with information on the univariate marginal distributions (see Meilijson and Nadas 3 (1979), Denuit et al (1999)) and the Fréchet class of distributions with information on the multivariate marginals of non-overlapping subsets of asset returns (see Doan and Natarajan (2012), Garlappi et al (2007), Rüschendorf (1991)). In this paper, we adopt a more general representation of distributional uncertainty where the multivariate marginals possibly overlap with each other.…”

Distributionally robust optimization with polynomial densities: theory, models and algorithms

“…Quantifying the worst-case copula amounts to solving a so-called Fréchet problem. In distributionally robust optimization, Fréchet problems with discrete marginals or approximate marginal matching conditions have been studied in [13,12,43].…”