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

On Parallel Processors Design for Solving Stochastics Programs

A design based on parallel processing is laid out for solving (multistage) stochastic programs. Because of the very special nature of the decomposition used here, one could rely on hard-wired micro-processors that would be extremely simple in design and fabrication, and would reduce the time required to solving stochastic programs to that needed for solving deterministic linear programs of the same size (ignoring the time required to design the parallel decomposition)

Modeling and Solution Strategies for Unconstrained Stochastic Optimization Problems

We review some modeling alternatives for handling risk in decision making processes for unconstrained stochastic optimization problems. Solution strategies are discussed and compared

Constrained Estimation: Consistency and Asymptotics

We review some of the recent results obtained for constrained estimation, involving possibly nondifferentiable criterion functions. New tools are required to push consistency and asymptotic results beyond those that can be reached by classical means

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

On the Continuity of the Value of a Linear Program

Results about the continuity of the value of a linear program are reviewed. Particular attention is paid to the interconnection between various sufficient conditions

Quantitative Stability of Variational Systems: I. The Epigraphical Distance

This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems)

Existence Results and Finite Horizon Approximates for Infinite Horizon Optimization Problems

The paper deals with infinite horizon optimization problems. The existence of optimal solutions is obtained as a consequence of an asymptotic growth condition. We also exhibit finite horizon approximates that yield upper and lower bounds for the optimal values and whose optimal solutions converge to the long-term optimal trajectories