196 research outputs found
A distributionally robust perspective on uncertainty quantification and chance constrained programming
The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones
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
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
Data-Driven Chance Constrained Programs over Wasserstein Balls
We provide an exact deterministic reformulation for data-driven chance
constrained programs over Wasserstein balls. For individual chance constraints
as well as joint chance constraints with right-hand side uncertainty, our
reformulation amounts to a mixed-integer conic program. In the special case of
a Wasserstein ball with the -norm or the -norm, the cone is the
nonnegative orthant, and the chance constrained program can be reformulated as
a mixed-integer linear program. Our reformulation compares favourably to
several state-of-the-art data-driven optimization schemes in our numerical
experiments.Comment: 25 pages, 9 figure
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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