Central Limit Theory for Multivalued Mappings

This paper presents new fundamental results concerning central limit behavior of sequences of random closed sets, modeled as a multivalued mapping of an underlying asymptotically normal sequence of random variables. The main theorem generalizes the classical result for differentiable functions in a mathematically satisfying way by combining recent developments in convergence theory for random closed sets and recent work in pseudo-differentiability of multifunctions. Potential applications to the asymptotic analysis of solution sets for stochastic and ordinary parametric programs with incomplete information are indicated in the examples. This paper reports research that was partly performed in the Adaptation and Optimization Project of the System and Decision Sciences Program

Distribution Sensitivity for a Chance Constrained Model of Optimal Load Dispatch

Using results from parametric optimization we derive for chance constrained stochastic programs (quantitative) stability properties for (locally) optimal values and sets of (local) minimizers when the underlying probability distribution is subjected to perturbations. Emphasis is placed on verifiable sufficient conditions for the constraint-set-mapping to fulfill a Lipschitz property which is essential for the stability results. Both convex and non-convex problems are investigated. We present an optimal-load-dispatch model with considering the demand as a random vector and putting the equilibrium between total generation and demand as a probabilistic constraint. Since in optimal load dispatch the information on the probability distribution of the demand is often incomplete, we discuss consequences of our general results for the stability of optimal generation costs and optimal generation policies

Incorporating statistical model error into the calculation of acceptability prices of contingent claims

The determination of acceptability prices of contingent claims requires the choice of a stochastic model for the underlying asset price dynamics. Given this model, optimal bid and ask prices can be found by stochastic optimization. However, the model for the underlying asset price process is typically based on data and found by a statistical estimation procedure. We define a confidence set of possible estimated models by a nonparametric neighborhood of a baseline model. This neighborhood serves as ambiguity set for a multi-stage stochastic optimization problem under model uncertainty. We obtain distributionally robust solutions of the acceptability pricing problem and derive the dual problem formulation. Moreover, we prove a general large deviations result for the nested distance, which allows to relate the bid and ask prices under model ambiguity to the quality of the observed data.Comment: 27 pages, 2 figure

Map Generation from Large Scale Incomplete and Inaccurate Data Labels

Accurately and globally mapping human infrastructure is an important and challenging task with applications in routing, regulation compliance monitoring, and natural disaster response management etc.. In this paper we present progress in developing an algorithmic pipeline and distributed compute system that automates the process of map creation using high resolution aerial images. Unlike previous studies, most of which use datasets that are available only in a few cities across the world, we utilizes publicly available imagery and map data, both of which cover the contiguous United States (CONUS). We approach the technical challenge of inaccurate and incomplete training data adopting state-of-the-art convolutional neural network architectures such as the U-Net and the CycleGAN to incrementally generate maps with increasingly more accurate and more complete labels of man-made infrastructure such as roads and houses. Since scaling the mapping task to CONUS calls for parallelization, we then adopted an asynchronous distributed stochastic parallel gradient descent training scheme to distribute the computational workload onto a cluster of GPUs with nearly linear speed-up.Comment: This paper is accepted by KDD 202

Constrained Cost-Coupled Stochastic Games with Independent State Processes

We consider a non-cooperative constrained stochastic games with N players with the following special structure. With each player there is an associated controlled Markov chain. The transition probabilities of the i-th Markov chain depend only on the state and actions of controller i. The information structure that we consider is such that each player knows the state of its own MDP and its own actions. It does not know the states of, and the actions taken by other players. Finally, each player wishes to minimize a time-average cost function, and has constraints over other time-avrage cost functions. Both the cost that is minimized as well as those defining the constraints depend on the state and actions of all players. We study in this paper the existence of a Nash equilirium. Examples in power control in wireless communications are given.Comment: 7 pages, submitted in september 2006 to Operations Research Letter