35,901 research outputs found
Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions
Two-stage stochastic optimization is a framework for modeling uncertainty,
where we have a probability distribution over possible realizations of the
data, called scenarios, and decisions are taken in two stages: we make
first-stage decisions knowing only the underlying distribution and before a
scenario is realized, and may take additional second-stage recourse actions
after a scenario is realized. The goal is typically to minimize the total
expected cost. A criticism of this model is that the underlying probability
distribution is itself often imprecise! To address this, a versatile approach
that has been proposed is the {\em distributionally robust 2-stage model}:
given a collection of probability distributions, our goal now is to minimize
the maximum expected total cost with respect to a distribution in this
collection.
We provide a framework for designing approximation algorithms in such
settings when the collection is a ball around a central distribution and the
central distribution is accessed {\em only via a sampling black box}.
We first show that one can utilize the {\em sample average approximation}
(SAA) method to reduce the problem to the case where the central distribution
has {\em polynomial-size} support. We then show how to approximately solve a
fractional relaxation of the SAA (i.e., polynomial-scenario
central-distribution) problem. By complementing this via LP-rounding algorithms
that provide {\em local} (i.e., per-scenario) approximation guarantees, we
obtain the {\em first} approximation algorithms for the distributionally robust
versions of a variety of discrete-optimization problems including set cover,
vertex cover, edge cover, facility location, and Steiner tree, with guarantees
that are, except for set cover, within -factors of the guarantees known
for the deterministic version of the problem
Robust Optimal Power Flow with Wind Integration Using Conditional Value-at-Risk
Integrating renewable energy into the power grid requires intelligent
risk-aware dispatch accounting for the stochastic availability of renewables.
Toward achieving this goal, a robust DC optimal flow problem is developed in
the present paper for power systems with a high penetration of wind energy. The
optimal dispatch is obtained as the solution to a convex program with a
suitable regularizer, which is able to mitigate the potentially high risk of
inadequate wind power. The regularizer is constructed based on the energy
transaction cost using conditional value-at-risk (CVaR). Bypassing the
prohibitive high-dimensional integral, the distribution-free sample average
approximation method is efficiently utilized for solving the resulting
optimization problem. Case studies are reported to corroborate the efficacy of
the novel model and approach tested on the IEEE 30-bus benchmark system with
real operation data from seven wind farms.Comment: To Appear in Proc. of the 4th Intl. Conf. on Smart Grid
Communication
Multi-Period Asset Allocation: An Application of Discrete Stochastic Programming
The issue of modeling farm financial decisions in a dynamic framework is addressed in this paper. Discrete stochastic programming is used to model the farm portfolio over the planning period. One of the main issues of discrete stochastic programming is representing the uncertainty of the data. The development of financial scenario generation routines provides a method to model the stochastic nature of the model. In this paper, two approaches are presented for generating scenarios for a farm portfolio problem. The approaches are based on copulas and optimization. The copula method provides an alternative to the multivariate normal assumption. The optimization method generates a number of discrete outcomes which satisfy specified statistical properties by solving a non-linear optimization model. The application of these different scenario generation methods is then applied to the topic of geographical diversification. The scenarios model the stochastic nature of crop returns and land prices in three separate geographic regions. The results indicate that the optimal diversification strategy is sensitive to both scenario generation method and initial acreage assumptions. The optimal diversification results are presented using both scenario generation methods.Agribusiness, Agricultural Finance, Farm Management,
- …