We consider a scalar stochastic linear optimization problem subject to linear constraints. We introduce the notion of deterministic equivalent formulation when the underlying probability space is equipped with a probability multimeasure. The initial problem is then transformed into a set-valued optimization problem with linear constraints. We also provide a method for estimating the expected value with respect to a probability multimeasure and prove extensions of the classical strong law of large numbers, the Glivenko–Cantelli theorem, and the central limit theorem to this setting. The notion of sampling with respect to a probability multimeasure and the definition of cumulative distribution multifunction are also discussed. Finally, we show some properties of the deterministic equivalent problem.
Stochastic linear optimization under partial uncertainty and incomplete information using the notion of probability multimeasure / D. la Torre, F. Mendivil. - In: JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY. - ISSN 0160-5682. - (2017 Jun 13), pp. 1-8. [Epub ahead of print] [10.1057/s41274-017-0249-9]
Stochastic linear optimization under partial uncertainty and incomplete information using the notion of probability multimeasure
D. la TorrePrimo
;
2017
Abstract
We consider a scalar stochastic linear optimization problem subject to linear constraints. We introduce the notion of deterministic equivalent formulation when the underlying probability space is equipped with a probability multimeasure. The initial problem is then transformed into a set-valued optimization problem with linear constraints. We also provide a method for estimating the expected value with respect to a probability multimeasure and prove extensions of the classical strong law of large numbers, the Glivenko–Cantelli theorem, and the central limit theorem to this setting. The notion of sampling with respect to a probability multimeasure and the definition of cumulative distribution multifunction are also discussed. Finally, we show some properties of the deterministic equivalent problem.File | Dimensione | Formato | |
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