Properties and Methods for Distributionally Robust Optimization with Decision Dependent Uncertainty
Abstract
The overarching goal of the proposed research is to study properties of stochastic optimization models, and develop algorithms for s""uch problems, where the probability distributions are am-biguous and decision dependent (D3RO). Such models are useful whenever the"" decision in~uences scenarios, moments, or the probability distribution by the decision; and the information of this in-~uence is im""precise. The proposed research builds upon PI~s previous research on distributionally robust optimization, where the ambiguity set i"s 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 spec-ifying an optimization problem. It com"bines the concept of stochastic optimization, where the distribution (or its sample average approximation) is assumed to be known ex""actly; and that of ro-bust 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, bu""t at the same time the model bene~ts from the statistical knowledge from the available data. Therefore, it provides a decision model""ing framework that is less conservative than the classical RO, but has risk aversion to ambiguity in knowledge of parameter distribu""tion. Studies in DRO use di~erent ways to de~ne the ambiguity set, and consider refor-mulation of the underlying models towards unde""rstanding 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, an""d hence the uncertainty of the system response is in~uenced by the decisions. For example, in a news-vendor model used in product pr""ocurement, the uncertainty in the demand of a product can depend on the selling price. In a security problem, the knowledge of a dec""ision by the adversary may result in him shifting his movement, hence resulting in a new ambiguity set describing uncertainty. Secon""d, decision dependent uncertainty also arises from data-driven sequential decision making. For instance, consider a sequential decis""ion making problem (e.g., dynamic programming, or multi-stage stochastic programming). The uncertainty of the model parameters can b""e reduced over time when more information on the uncertain parameters is observed to predict future parameters, and the decision bei"ng made in~uences the reduction in the ambiguity in specifying the probability distribution.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jan 23, 2018
- Source ID
- N000141812097
Entities
People
- Sanjay Mehrotra
Organizations
- Northwestern University
- Office of Naval Research
- United States Navy