Stochastic linear optimization under partial uncertainty and incomplete information using the notion of probability multimeasure

Please note this paper has been re-typeset by Taylor & Francis from the manuscript originally provided to the previous publisher.

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.

Keywords:

• Stochastic linear optimization
• deterministic equivalent problem
• set-valued optimization
• probability multimeasure

Acknowledgements

The second author (FM) was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of a Discovery Grant (238549-2012).

Notes

Please note this paper has been re-typeset by Taylor & Francis from the manuscript originally provided to the previous publisher.