arXiv subjects

Sanjay Mehrotra

Publications and source records attributed to Sanjay Mehrotra.

24 records · Page 2Linked to original sources

arXiv2018

Distributionally Robust Optimization with Decision Dependent Ambiguity Sets

We study decision dependent distributionally robust optimization models, where the ambiguity sets of probability distributions can depend on the decision variables. These models arise in situations with endogenous uncertainty. The developed framework includes two-stage decision dependent distributionally robust stochastic programming as a special case. Decision dependent generalizations of five types of ambiguity sets are considered. These sets are based on bounds on moments, Wasserstein metric, $ϕ$-divergence and Kolmogorov-Smirnov test. For the finite support case, we use linear, conic or Lagrangian duality to give reformulations of the models with a finite number of constraints. These reformulations allow solutions of such problems using global optimization techniques. Certain reformulations give rise to non-convex semi-infinite programs. Techniques from global optimization and semi-infinite programming can be used to solve these reformulations.

math.OC↗

arXiv2018

A Concentration Result of Estimating Phi-Divergence using Data Dependent Partition

Estimation of the $ϕ$-divergence between two unknown probability distributions using empirical data is a fundamental problem in information theory and statistical learning. We consider a multi-variate generalization of the data dependent partitioning method for estimating divergence between the two unknown distributions. Under the assumption that the distribution satisfies a power law of decay, we provide a convergence rate result for this method on the number of samples and hyper-rectangles required to ensure the estimation error is bounded by a given level with a given probability.

math.PR↗

arXiv2017

Decomposition Algorithm for Distributionally Robust Optimization using Wasserstein Metric

We study distributionally robust optimization (DRO) problems where the ambiguity set is defined using the Wasserstein metric. We show that this class of DRO problems can be reformulated as semi-infinite programs. We give an exchange method to solve the reformulated problem for the general nonlinear model, and a central cutting-surface method for the convex case, assuming that we have a separation oracle. We used a distributionally robust generalization of the logistic regression model to test our algorithm. Numerical experiments on the distributionally robust logistic regression models show that the number of oracle calls are typically 20 ? 50 to achieve 5-digit precision. The solution found by the model is generally better in its ability to predict with a smaller standard error.

math.OC↗

arXiv2016

Generating nested quadrature formulas for general weight functions with known moments

We revisit the problem of extending quadrature formulas for general weight functions, and provide a generalization of Patterson's method for the constant weight function. The method can be used to compute a nested sequence of quadrature formulas for integration with respect to any continuous probability measure on the real line with finite moments. The advantages of the method include that it works directly with the moments of the underlying distribution, and that for distributions with rational moments the existence of the formulas can be verified by exact rational arithmetic.

math.NA↗

arXiv2015

Designed Sampling from Large Databases for Controlled Trials

The increasing prevalence of rich sources of data and the availability of electronic medical record databases and electronic registries opens tremendous opportunities for enhancing medical research. For example, controlled trials are ubiquitously used to investigate the effect of a medical treatment, perhaps dependent on a set of patient covariates, and traditional approaches have relied primarily on randomized patient sampling and allocation to treatment and control group. However, when covariate data for a large cohort group of patients have already been collected and are available in a database, one can potentially design a treatment/control sample and allocation that provides far better estimates of the covariate-dependent effects of the treatment. In this paper, we develop a new approach that uses optimal design of experiments (DOE) concepts to accomplish this objective. The approach selects the patients for the treatment and control samples upfront, based on their covariate values, in a manner that optimizes the information content in the data. For the optimal sample selection, we develop simple guidelines and an optimization algorithm that provides solutions that are substantially better than random sampling. Moreover, our approach causes no sampling bias in the estimated effects, for the same reason that DOE principles do not bias estimated effects. We test our method with a simulation study based on a testbed data set containing information on the effect of statins on low-density lipoprotein (LDL) cholesterol.

stat.ME↗

arXiv2014

A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization

We present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems, and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of arbitrary order, as well as non-polynomial moments can be included in the formulation. We show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk-aversion, with classic robust optimization at one end of the spectrum, and stochastic programming at the other. Although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty, the cutting surface method for general semi-infinite convex programs is also of independent interest. The proposed method is applicable to problems with non-differentiable semi-infinite constraints indexed by an infinite-dimensional index set. Examples comparing the cutting surface algorithm to the central cutting plane algorithm of Kortanek and No demonstrate the potential of our algorithm even in the solution of traditional semi-infinite convex programming problems whose constraints are differentiable and are indexed by an index set of low dimension. After the rate of convergence analysis of the cutting surface algorithm, we extend the authors' moment matching scenario generation algorithm to a probabilistic algorithm that finds optimal probability distributions subject to moment constraints. The combination of this distribution optimization method and the central cutting surface algorithm yields a solution to a family of distributionally robust optimization problems that are considerably more general than the ones proposed to date.

math.OC↗