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 -