325 research outputs found
Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations
We consider stochastic programs where the distribution of the uncertain
parameters is only observable through a finite training dataset. Using the
Wasserstein metric, we construct a ball in the space of (multivariate and
non-discrete) probability distributions centered at the uniform distribution on
the training samples, and we seek decisions that perform best in view of the
worst-case distribution within this Wasserstein ball. The state-of-the-art
methods for solving the resulting distributionally robust optimization problems
rely on global optimization techniques, which quickly become computationally
excruciating. In this paper we demonstrate that, under mild assumptions, the
distributionally robust optimization problems over Wasserstein balls can in
fact be reformulated as finite convex programs---in many interesting cases even
as tractable linear programs. Leveraging recent measure concentration results,
we also show that their solutions enjoy powerful finite-sample performance
guarantees. Our theoretical results are exemplified in mean-risk portfolio
optimization as well as uncertainty quantification.Comment: 42 pages, 10 figure
Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions
Two-stage stochastic optimization is a framework for modeling uncertainty,
where we have a probability distribution over possible realizations of the
data, called scenarios, and decisions are taken in two stages: we make
first-stage decisions knowing only the underlying distribution and before a
scenario is realized, and may take additional second-stage recourse actions
after a scenario is realized. The goal is typically to minimize the total
expected cost. A criticism of this model is that the underlying probability
distribution is itself often imprecise! To address this, a versatile approach
that has been proposed is the {\em distributionally robust 2-stage model}:
given a collection of probability distributions, our goal now is to minimize
the maximum expected total cost with respect to a distribution in this
collection.
We provide a framework for designing approximation algorithms in such
settings when the collection is a ball around a central distribution and the
central distribution is accessed {\em only via a sampling black box}.
We first show that one can utilize the {\em sample average approximation}
(SAA) method to reduce the problem to the case where the central distribution
has {\em polynomial-size} support. We then show how to approximately solve a
fractional relaxation of the SAA (i.e., polynomial-scenario
central-distribution) problem. By complementing this via LP-rounding algorithms
that provide {\em local} (i.e., per-scenario) approximation guarantees, we
obtain the {\em first} approximation algorithms for the distributionally robust
versions of a variety of discrete-optimization problems including set cover,
vertex cover, edge cover, facility location, and Steiner tree, with guarantees
that are, except for set cover, within -factors of the guarantees known
for the deterministic version of the problem
Distributionally Robust Optimization: A Review
The concepts of risk-aversion, chance-constrained optimization, and robust
optimization have developed significantly over the last decade. Statistical
learning community has also witnessed a rapid theoretical and applied growth by
relying on these concepts. A modeling framework, called distributionally robust
optimization (DRO), has recently received significant attention in both the
operations research and statistical learning communities. This paper surveys
main concepts and contributions to DRO, and its relationships with robust
optimization, risk-aversion, chance-constrained optimization, and function
regularization
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Approximation schemes for network, clustering and queueing models
In this dissertation, we consider important optimization problems that arise in three different domains, namely network models, clustering problems and queueing models. To be more specific, we focus on devising efficient traffic routing models, deriving exact convex reformulation to the well-known K-means clustering problem and studying the classical Naor’s observable queues under uncertain parameters. In the following chapters, we discuss these problems in detail, design efficient and tractable solution methodologies, and assess the quality of proposed solutions. In the first part of the dissertation, we analyze a limited-adaptability traffic routing model for the Austin road network. Routing a person through a traffic network presents a tension between selecting a fixed route that is easy to navigate and selecting an aggressively adaptive route that minimizes the expected travel time. We develop non-aggressive adaptive routes in the middle-ground seeking the best of both these extremes. Specifically, these routes still adapt to changing traffic condition, however we limit the total number of allowable adjustments. This improves the user experience, by providing a continuum of options between saving travel time and minimizing navigation. We design strategies to model single and multiple route adjustments, and investigate enumerative techniques to solve these models. We also develop tractable algorithms with easily computable lower and upper bounds to handle real-size traffic data. We finally present the numerical results highlighting the benefit of different levels of adaptability in terms of reducing the expected travel time. In the second part of the dissertation, we study the well-known classical K-means clustering problem. We show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractable semidefinite programming (SDP) relaxation that is tighter than the current SDP relaxation schemes in the literature. In contrast to the existing schemes, our proposed SDP formulation gives rise to solutions that can be leveraged to identify the clusters. We devise a new approximation algorithm for K-means clustering that utilizes the improved formulation and empirically illustrate its superiority over the state-of-the-art solution schemes. Finally, we study an extension of Naor’s analysis [74] on the joining or balking problem in observable M/M/1 queues, relaxing the principal assumption of deterministic arrival and service rates. While all the Markovian assumptions still hold, we assume the arrival and service rates are uncertain and study this problem under stochastic and distributionally robust settings. In the former setting, the exact rates are unknown but we assume the distribution of rates are known to all the decision makers. We derive the optimal joining threshold strategies from the perspective of an individual customer, a social optimizer and a revenue maximizer, such that expected profit rate is maximized. In the distributionally robust setting, we go a step further to assume the true distributions are unknown and the decision makers have access to only a finite set of training samples. Similar to the stochastic setting, we derive optimal thresholds such that the worst-case expected profit rates are maximized. Finally, we compare our observations, both theoretically and numerically, with Naor’s classical results.Operations Research and Industrial Engineerin
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