83 research outputs found
Variational Analysis of Constrained M-Estimators
We propose a unified framework for establishing existence of nonparametric
M-estimators, computing the corresponding estimates, and proving their strong
consistency when the class of functions is exceptionally rich. In particular,
the framework addresses situations where the class of functions is complex
involving information and assumptions about shape, pointwise bounds, location
of modes, height at modes, location of level-sets, values of moments, size of
subgradients, continuity, distance to a "prior" function, multivariate total
positivity, and any combination of the above. The class might be engineered to
perform well in a specific setting even in the presence of little data. The
framework views the class of functions as a subset of a particular metric space
of upper semicontinuous functions under the Attouch-Wets distance. In addition
to allowing a systematic treatment of numerous M-estimators, the framework
yields consistency of plug-in estimators of modes of densities, maximizers of
regression functions, level-sets of classifiers, and related quantities, and
also enables computation by means of approximating parametric classes. We
establish consistency through a one-sided law of large numbers, here extended
to sieves, that relaxes assumptions of uniform laws, while ensuring global
approximations even under model misspecification
Log-Concave Duality in Estimation and Control
In this paper we generalize the estimation-control duality that exists in the
linear-quadratic-Gaussian setting. We extend this duality to maximum a
posteriori estimation of the system's state, where the measurement and
dynamical system noise are independent log-concave random variables. More
generally, we show that a problem which induces a convex penalty on noise terms
will have a dual control problem. We provide conditions for strong duality to
hold, and then prove relaxed conditions for the piecewise linear-quadratic
case. The results have applications in estimation problems with nonsmooth
densities, such as log-concave maximum likelihood densities. We conclude with
an example reconstructing optimal estimates from solutions to the dual control
problem, which has implications for sharing solution methods between the two
types of problems
Solving equilibrium problems in economies with financial markets, home production, and retention
We propose a new methodology to compute equilibria for general equilibrium
problems on exchange economies with real financial markets, home-production,
and retention. We demonstrate that equilibrium prices can be determined by
solving a related maxinf-optimization problem. We incorporate the non-arbitrage
condition for financial markets into the equilibrium formulation and establish
the equivalence between solutions to both problems. This reduces the complexity
of the original by eliminating the need to directly compute financial contract
prices, allowing us to calculate equilibria even in cases of incomplete
financial markets.
We also introduce a Walrasian bifunction that captures the imbalances and
show that maxinf-points of this function correspond to equilibrium points.
Moreover, we demonstrate that every equilibrium point can be approximated by a
limit of maxinf points for a family of perturbed problems, by relying on the
notion of lopsided convergence.
Finally, we propose an augmented Walrasian algorithm and present numerical
examples to illustrate the effectiveness of this approach. Our methodology
allows for efficient calculation of equilibria in a variety of exchange
economies and has potential applications in finance and economics
Fusion of Hard and Soft Information in Nonparametric Density Estimation
This article discusses univariate density estimation in situations when the sample (hard
information) is supplemented by “soft” information about the random phenomenon. These situations
arise broadly in operations research and management science where practical and computational reasons
severely limit the sample size, but problem structure and past experiences could be brought in. In
particular, density estimation is needed for generation of input densities to simulation and stochastic
optimization models, in analysis of simulation output, and when instantiating probability models. We
adopt a constrained maximum likelihood estimator that incorporates any, possibly random, soft information
through an arbitrary collection of constraints. We illustrate the breadth of possibilities by
discussing soft information about shape, support, continuity, smoothness, slope, location of modes,
symmetry, density values, neighborhood of known density, moments, and distribution functions. The
maximization takes place over spaces of extended real-valued semicontinuous functions and therefore
allows us to consider essentially any conceivable density as well as convenient exponential transformations.
The infinite dimensionality of the optimization problem is overcome by approximating splines
tailored to these spaces. To facilitate the treatment of small samples, the construction of these splines
is decoupled from the sample. We discuss existence and uniqueness of the estimator, examine consistency
under increasing hard and soft information, and give rates of convergence. Numerical examples
illustrate the value of soft information, the ability to generate a family of diverse densities, and the
effect of misspecification of soft information.U.S. Army Research Laboratory and the U.S. Army Research Office grant 00101-80683U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-10-1-0246U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-12-1-0273U.S. Army Research Laboratory and the U.S. Army Research Office grant 00101-80683U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-10-1-0246U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-12-1-027
Sublinear upper bounds for stochastic programs with recourse
Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47918/1/10107_2005_Article_BF01582286.pd
Challenges in Stochastic Programming
Remarkable progress has been made in the development of algorithmic procedures and the availability of software for stochastic programming problems. However, some fundamental questions have remained unexplored. This paper identifies the more challenging open questions in the field of stochastic programming. Some are purely technical in nature, but many also go to the foundations of designing models for decision making under uncertainty. Key words: stochastic programming, decisions under uncertainty, chance-constraints, probabilistic constraints, distribution problem, Markowitz portfolio model iii iv Challenges in Stochastic Programming Roger J.-B.Wets Recent work in stochastic programming has mostly been aimed at the design of solution procedures and the development of accompanying software; an overly brief review of the present state-of-the-art is provided in x1. This effort should be continued and expanded, and should remain the central concern of the research in stochastic pr..
Stability of ε-approximate solutions to convex stochastic programs
An analysis of convex stochastic programs is provided if the underlying proba-bility distribution is subjected to (small) perturbations. It is shown, in particular,that ε-approximate solution sets of convex stochastic programs behave Lipschitzcontinuous with respect to certain distances of probability distributions that aregenerated by the relevant integrands. It is shown that these results apply tolinear two-stage stochastic programs with random recourse. Consequences arediscussed on associating Fortet-Mourier metrics to two-stage models and on theasymptotic behavior of empirical estimates of such models, respectively
Variational Theory for Optimization under Stochastic Ambiguity
This paper is in review.Stochastic ambiguity provides a rich class of uncertainty models that includes those in
stochastic, robust, risk-based, and semi-in nite optimization, and that accounts for both uncertainty
about parameter values as well as incompleteness of the description of uncertainty. We provide a novel,
unifying perspective on optimization under stochastic ambiguity that rests on two pillars. First, the
paper models ambiguity by decision-dependent collections of cumulative distribution functions viewed
as subsets of a metric space of upper semicontinuous functions. We derive a series of results for this set-
ting including estimates of the metric, the hypo-distance, and a new proof of the equivalence with weak
convergence. Second, we utilize the theory of lopsided convergence to establish existence, convergence,
and approximation of solutions of optimization problems with stochastic ambiguity. For the rst time,
we estimate the lop-distance between bifunctions and show that this leads to bounds on the solution
quality for problems with stochastic ambiguity. Among other consequences, these results facilitate the
study of the \price of robustness" and related quantities
Multivariate Epi-Splines and Evolving Function Identification Problems
Includes erratumThe broad class of extended real-valued lower semicontinuous (lsc) functions on IRn captures
nearly all functions of practical importance in equation solving, variational problems, fitting, and
estimation. The paper develops piecewise polynomial functions, called epi-splines, that approximate
any lsc function to an arbitrary level of accuracy. Epi-splines provide the foundation for the solution
of a rich class of function identification problems that incorporate general constraints on the function
to be identified including those derived from information about smoothness, shape, proximity to other
functions, and so on. As such extrinsic information as well as observed function and subgradient values
often evolve in applications, we establish conditions under which the computed epi-splines converge
to the function we seek to identify. Numerical examples in response surface building and probability
density estimation illustrate the framework.U. S. Army Research Laboratory and the U. S. Army Research Office grant 00101-80683U. S. Army Research Laboratory and the U. S. Army Research Office grant W911NF-10-1-0246U. S. Army Research Laboratory and the U. S. Army Research Office grant W911NF-12-1-0273U. S. Army Research Laboratory and the U. S. Army Research Office grant 00101-80683U. S. Army Research Laboratory and the U. S. Army Research Office grant W911NF-10-1-0246U. S. Army Research Laboratory and the U. S. Army Research Office grant W911NF-12-1-027
- …