TY - JOUR

T1 - Stochastic linear optimization under partial uncertainty and incomplete information using the notion of probability multimeasure

AU - La Torre, Davide

AU - Mendivil, Franklin

N1 - Funding Information:
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).

PY - 2018/10/3

Y1 - 2018/10/3

N2 - 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.

AB - 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.

KW - Stochastic linear optimization

KW - deterministic equivalent problem

KW - probability multimeasure

KW - set-valued optimization

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U2 - 10.1057/s41274-017-0249-9

DO - 10.1057/s41274-017-0249-9

M3 - Article

AN - SCOPUS:85020741569

VL - 69

SP - 1549

EP - 1556

JO - Journal of the Operational Research Society

JF - Journal of the Operational Research Society

SN - 0160-5682

IS - 10

ER -