### Description

The overarching goal of the proposed research is to study properties of stochastic optimization models, and develop algorithms for such problems, where the probability distributions are ambiguous and decision dependent (D3RO). Such models are useful whenever the decision influences scenarios, moments, or the probability distribution by the decision; and the information of this influence is imprecise. The proposed research builds upon PI's previous research on distributionally robust optimization, where the ambiguity set is assumed to be independent of the decisions. The research literature has not considered the proposed model framework thus far. In distributionally robust optimization (DRO) an ambiguity set of distributions is used in specifying

an optimization problem. It combines the concept of stochastic optimization, where the

distribution (or its sample average approximation) is assumed to be known exactly; and that of robust optimization (RO), where the uncertainty set is speci ed by a deterministic set of constraints. Therefore, it provides a decision framework where the probability distribution of underlying random parameters cannot be speci ed exactly, but at the same time the model bene ts from the statistical knowledge from the available data. Therefore, it provides a decision modeling framework that is less conservative than the classical RO, but has risk aversion to ambiguity in knowledge of parameter distribution. Studies in DRO use di erent ways to de ne the ambiguity set, and consider reformulation of the underlying models towards understanding their tractability, developing algorithms, probability guarantee of the constraint satisfaction by the true distribution, and applications of the

developed models. The motivations of considering decision dependent uncertainty set in robust optimization are many. First, when certain decisions are made to optimize the objective, the system of interest will respond to the decision, and hence the uncertainty of the system response is influenced by the decisions. For example, in a news-vendor model used in product procurement, the uncertainty in the demand of a product can depend on the selling price. In a security problem, the knowledge of a decision by the adversary may result in him shifting his movement, hence resulting in a new ambiguity set describing uncertainty. Second, decision dependent uncertainty also arises from datadriven

sequential decision making. For instance, consider a sequential decision making problem

(e.g., dynamic programming, or multi-stage stochastic programming). The uncertainty of the model parameters can be reduced over time when more information on the uncertain parameters is observed to predict future parameters, and the decision being made influences the reduction in the ambiguity in specifying the probability distribution.

an optimization problem. It combines the concept of stochastic optimization, where the

distribution (or its sample average approximation) is assumed to be known exactly; and that of robust optimization (RO), where the uncertainty set is speci ed by a deterministic set of constraints. Therefore, it provides a decision framework where the probability distribution of underlying random parameters cannot be speci ed exactly, but at the same time the model bene ts from the statistical knowledge from the available data. Therefore, it provides a decision modeling framework that is less conservative than the classical RO, but has risk aversion to ambiguity in knowledge of parameter distribution. Studies in DRO use di erent ways to de ne the ambiguity set, and consider reformulation of the underlying models towards understanding their tractability, developing algorithms, probability guarantee of the constraint satisfaction by the true distribution, and applications of the

developed models. The motivations of considering decision dependent uncertainty set in robust optimization are many. First, when certain decisions are made to optimize the objective, the system of interest will respond to the decision, and hence the uncertainty of the system response is influenced by the decisions. For example, in a news-vendor model used in product procurement, the uncertainty in the demand of a product can depend on the selling price. In a security problem, the knowledge of a decision by the adversary may result in him shifting his movement, hence resulting in a new ambiguity set describing uncertainty. Second, decision dependent uncertainty also arises from datadriven

sequential decision making. For instance, consider a sequential decision making problem

(e.g., dynamic programming, or multi-stage stochastic programming). The uncertainty of the model parameters can be reduced over time when more information on the uncertain parameters is observed to predict future parameters, and the decision being made influences the reduction in the ambiguity in specifying the probability distribution.

Status | Active |
---|---|

Effective start/end date | 2/1/18 → 1/31/21 |

### Funding

- Office of Naval Research (N00014-18-1-2097 P00001)

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Uncertainty

Robust optimization

Probability distribution

Stochastic optimization

Optimization problem

Risk aversion

Optimization model

Procurement

Sample average approximation

Constraint satisfaction

Dynamic programming

Random parameters

Multistage stochastic programming

Decision modeling

Guarantee

Newsvendor

Scenarios

Decision making

Sequential decision making