14,677 research outputs found
DEA Problems under Geometrical or Probability Uncertainties of Sample Data
This paper discusses the theoretical and practical aspects of new methods for solving DEA problems under real-life geometrical uncertainty and probability uncertainty of sample data. The proposed minimax approach to solve problems with geometrical uncertainty of sample data involves an implementation of linear programming or minimax optimization, whereas the problems with probability uncertainty of sample data are solved through implementing of econometric and new stochastic optimization methods, using the stochastic frontier functions estimation.DEA, Sample data uncertainty, Linear programming, Minimax optimization, Stochastic optimization, Stochastic frontier functions
Minimax risks for sparse regressions: Ultra-high-dimensional phenomenons
Consider the standard Gaussian linear regression model ,
where is a response vector and is a design matrix.
Numerous work have been devoted to building efficient estimators of
when is much larger than . In such a situation, a classical approach
amounts to assume that is approximately sparse. This paper studies
the minimax risks of estimation and testing over classes of -sparse vectors
. These bounds shed light on the limitations due to
high-dimensionality. The results encompass the problem of prediction
(estimation of ), the inverse problem (estimation of ) and
linear testing (testing ). Interestingly, an elbow effect occurs
when the number of variables becomes large compared to .
Indeed, the minimax risks and hypothesis separation distances blow up in this
ultra-high dimensional setting. We also prove that even dimension reduction
techniques cannot provide satisfying results in an ultra-high dimensional
setting. Moreover, we compute the minimax risks when the variance of the noise
is unknown. The knowledge of this variance is shown to play a significant role
in the optimal rates of estimation and testing. All these minimax bounds
provide a characterization of statistical problems that are so difficult so
that no procedure can provide satisfying results
Interval-Parameter Robust Minimax-regret Programming and Its Application to Energy and Environmental Systems Planning
In this study, an interval-parameter robust minimax-regret programming method is developed and applied to the planning of energy and environmental systems. Methods of robust programming, interval-parameter programming, and minimax-regret analysis are incorporated within a general optimization framework to enhance the robustness of the optimization effort. The interval-parameter robust minimax-regret programming can deal with uncertainties expressed as discrete intervals, fuzzy sets, and random variables. It can also be used for analyzing multiple scenarios associated with different system costs and risk levels. In its solution process, the fuzzy decision space is delimited into a more robust one through dimensional enlargement of the original fuzzy constraints; moreover, an interval-element cost matrix can be transformed into an interval-element regret matrix, such that the decision makers can identify desired alternatives based on the inexact minimax regret criterion. The developed method has been applied to a case study of energy and environmental systems planning under uncertainty. The results indicate that reasonable solutions have been generated
Linear programming on the Stiefel manifold
Linear programming on the Stiefel manifold (LPS) is studied for the first
time. It aims at minimizing a linear objective function over the set of all
-tuples of orthonormal vectors in satisfying additional
linear constraints. Despite the classical polynomial-time solvable case ,
general (LPS) is NP-hard. According to the Shapiro-Barvinok-Pataki theorem,
(LPS) admits an exact semidefinite programming (SDP) relaxation when
, which is tight when . Surprisingly, we can greatly
strengthen this sufficient exactness condition to , which covers the
classical case and . Regarding (LPS) as a smooth nonlinear
programming problem, we reveal a nice property that under the linear
independence constraint qualification, the standard first- and second-order
{\it local} necessary optimality conditions are sufficient for {\it global}
optimality when
Bayesian nonparametric multivariate convex regression
In many applications, such as economics, operations research and
reinforcement learning, one often needs to estimate a multivariate regression
function f subject to a convexity constraint. For example, in sequential
decision processes the value of a state under optimal subsequent decisions may
be known to be convex or concave. We propose a new Bayesian nonparametric
multivariate approach based on characterizing the unknown regression function
as the max of a random collection of unknown hyperplanes. This specification
induces a prior with large support in a Kullback-Leibler sense on the space of
convex functions, while also leading to strong posterior consistency. Although
we assume that f is defined over R^p, we show that this model has a convergence
rate of log(n)^{-1} n^{-1/(d+2)} under the empirical L2 norm when f actually
maps a d dimensional linear subspace to R. We design an efficient reversible
jump MCMC algorithm for posterior computation and demonstrate the methods
through application to value function approximation
An Interactive Fuzzy Satisficing Method for Fuzzy Random Multiobjective 0-1 Programming Problems through Probability Maximization Using Possibility
In this paper, we focus on multiobjective 0-1 programming problems under the situation where stochastic uncertainty and vagueness exist at the same time. We formulate them as
fuzzy random multiobjective 0-1 programming problems where coefficients of objective functions are fuzzy random variables. For the formulated problem, we propose an interactive fuzzy satisficing method through probability maximization using of possibility
Recursive Concurrent Stochastic Games
We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent
analysis of recursive simple stochastic games to a concurrent setting where the
two players choose moves simultaneously and independently at each state. For
multi-exit games, our earlier work already showed undecidability for basic
questions like termination, thus we focus on the important case of single-exit
RCSGs (1-RCSGs).
We first characterize the value of a 1-RCSG termination game as the least
fixed point solution of a system of nonlinear minimax functional equations, and
use it to show PSPACE decidability for the quantitative termination problem. We
then give a strategy improvement technique, which we use to show that player 1
(maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM)
strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM
strategies. Thus, such games are r-SM-determined. These results mirror and
generalize in a strong sense the randomized memoryless determinacy results for
finite stochastic games, and extend the classic Hoffman-Karp strategy
improvement approach from the finite to an infinite state setting. The proofs
in our infinite-state setting are very different however, relying on subtle
analytic properties of certain power series that arise from studying 1-RCSGs.
We show that our upper bounds, even for qualitative (probability 1)
termination, can not be improved, even to NP, without a major breakthrough, by
giving two reductions: first a P-time reduction from the long-standing
square-root sum problem to the quantitative termination decision problem for
finite concurrent stochastic games, and then a P-time reduction from the latter
problem to the qualitative termination problem for 1-RCSGs.Comment: 21 pages, 2 figure
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