17,469 research outputs found
Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces
Optimal values and solutions of empirical approximations of stochastic
optimization problems can be viewed as statistical estimators of their true
values. From this perspective, it is important to understand the asymptotic
behavior of these estimators as the sample size goes to infinity. This area of
study has a long tradition in stochastic programming. However, the literature
is lacking consistency analysis for problems in which the decision variables
are taken from an infinite dimensional space, which arise in optimal control,
scientific machine learning, and statistical estimation. By exploiting the
typical problem structures found in these applications that give rise to hidden
norm compactness properties for solution sets, we prove consistency results for
nonconvex risk-averse stochastic optimization problems formulated in infinite
dimensional space. The proof is based on several crucial results from the
theory of variational convergence. The theoretical results are demonstrated for
several important problem classes arising in the literature.Comment: 24 page
Non-asymptotic confidence bounds for the optimal value of a stochastic program
We discuss a general approach to building non-asymptotic confidence bounds
for stochastic optimization problems. Our principal contribution is the
observation that a Sample Average Approximation of a problem supplies upper and
lower bounds for the optimal value of the problem which are essentially better
than the quality of the corresponding optimal solutions. At the same time, such
bounds are more reliable than "standard" confidence bounds obtained through the
asymptotic approach. We also discuss bounding the optimal value of MinMax
Stochastic Optimization and stochastically constrained problems. We conclude
with a simulation study illustrating the numerical behavior of the proposed
bounds
Bounding Optimality Gap in Stochastic Optimization via Bagging: Statistical Efficiency and Stability
We study a statistical method to estimate the optimal value, and the
optimality gap of a given solution for stochastic optimization as an assessment
of the solution quality. Our approach is based on bootstrap aggregating, or
bagging, resampled sample average approximation (SAA). We show how this
approach leads to valid statistical confidence bounds for non-smooth
optimization. We also demonstrate its statistical efficiency and stability that
are especially desirable in limited-data situations, and compare these
properties with some existing methods. We present our theory that views SAA as
a kernel in an infinite-order symmetric statistic, which can be approximated
via bagging. We substantiate our theoretical findings with numerical results
Accuracy of numerical solutions using the eulers equation residuals
In this paper we derive sorne asymptotic properties on the accuracy of numerical solutions. We sIlow tIlat the approximation error of the policy function is of the same order of magnitude as the size of the Euler equation residuals. Moreover, for bounding this approximation error tIle most relevant parameters are the discount factor and the curvature of the return function. These findings provide theoretical foundations for the construction of tests that can assess the performance of alternative computational methods
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