### 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.

Original language | English |
---|---|

Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | Journal of the Operational Research Society |

DOIs | |

Publication status | Accepted/In press - Jun 13 2017 |

### Fingerprint

### Keywords

- deterministic equivalent problem
- probability multimeasure
- set-valued optimization
- stochastic linear optimization

### ASJC Scopus subject areas

- Management Information Systems
- Strategy and Management
- Management Science and Operations Research
- Marketing

### Cite this

*Journal of the Operational Research Society*, 1-8. https://doi.org/10.1057/s41274-017-0249-9

**Stochastic linear optimization under partial uncertainty and incomplete information using the notion of probability multimeasure.** / la Torre, Davide; Mendivil, Franklin.

Research output: Contribution to journal › Article

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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

PY - 2017/6/13

Y1 - 2017/6/13

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 - deterministic equivalent problem

KW - probability multimeasure

KW - set-valued optimization

KW - stochastic linear optimization

UR - http://www.scopus.com/inward/record.url?scp=85020741569&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020741569&partnerID=8YFLogxK

U2 - 10.1057/s41274-017-0249-9

DO - 10.1057/s41274-017-0249-9

M3 - Article

SP - 1

EP - 8

JO - Journal of the Operational Research Society

JF - Journal of the Operational Research Society

SN - 0160-5682

ER -