Optimal transport (OT) formulation and algorithms allow for optimal allocation of limited resources in a network consisting of sources and targets. The standard OT paradigm does not extend well over a large population of different types. In this paper, we formulate innovate OT problem and design algorithms to cope with resource allocation with a large and heterogeneous population of target nodes. The heterogeneity of targets is described by a type distribution function. We consider two instances in which the distribution is known and unknown to the sources, i.e., transport designer. For the former case, we propose a fully distributed algorithm to obtain the solution. For the latter case in which the targets' type distribution is not available to the sources, we develop a collaborative learning algorithm to compute the OT scheme efficiently. We evaluate the performance of the proposed learning algorithm using a case study.

Entropic risk measure is widely used to account for tail risks associated with an uncertain parameter. We study distributionally robust optimization problems with entropic risk measures and use cross-validation to tune the radius (hyperparameter) of the ambiguity set. However, with limited data, the empirical average of the entropic risk associated with scenarios in the validation set underestimates the true entropic risk. We provide two procedures that first learn Gaussian mixture models and then use bootstrapping to identify scaling parameters to correct the bias in the entropic risk estimated using the validation data. We show that one can achieve lower risk using our debiasing procedure in project selection and portfolio optimization problems.

Decision-makers are often assumed to possess fixed risk preferences. However, evidences have shown that individual risk preferences are unstable and are subject to manipulation. This work proposes a Stackelberg risk preference design (STRIPE) problem to capture a designer’s incentive to influence DMs’ risk preferences. STRIPE consists of two levels. In the lower level, individual DMs in a population, known as the followers, respond to uncertainties according to their risk preference types. In the upper level, the leader influences the distribution of the types to induce targeted decisions and steers the follower’s preferences to it. Our analysis centers around the solution concept of approximate Stackelberg equilibrium that yields suboptimal behaviors of the players. We show the existence of the equilibrium, derive a single-level reformulation of the design problem, and suggest two applications for the proposed framework.