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

Davide La Torre and Franklin Mendivil

Journal of the Operational Research Society, 2018, vol. 69, issue 10, 1549-1556

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.

Date: 2018
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DOI: 10.1057/s41274-017-0249-9

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