Title
Risk-averse Decision Making with Contextual Information: Model, Sample Average Approximation, and Kernelization
Abstract
We consider risk-averse contextual optimization problems where the decision maker (DM) faces two types of uncertainties: problem data uncertainty (PDU) and contextual uncertainty (CU) associated with PDU, the DM makes an optimal decision by minimizing the risk arising from PDU based on the present observation of CU and then assesses the risk of the optimal policy against the CU. A natural question arises as to whether the nested risk minimization/assessment process is equivalent to joint risk minimization/assessment against CU and PDU simultaneously. First, we demonstrate that the equivalence can be established by appropriate choices of the risk measures and give counter examples where such equivalence may fail. One of the interesting findings is that the optimal policies are independent of the choice of the risk measure against the CU under certain conditions. Second, by using the equivalence, we propose computational method for solving the risk-averse contextual optimization problem by solving a one-stage risk minimization problem. The latter is particularly helpful in data-driven environments. We consider a number of risk measures/metrics to characterize the DM’s risk preference for PDU and discuss the computational tractability for the resulting risk-averse contextual optimization problem. Third, when the risk-averse contextual optimization problem is defined in the reproducing kernel Hilbert space, we show consistency of the optimal values obtained from solving sample average approximation problems. Some numerical tests, in newsvendor problem and portfolio selection problem, are performed to validate the theoretical results.
This is a joint work with Yuan Tao (CUHK) and Erick Delage (HEC Montreal)
Bio
Huifu Xu is a Professor of the Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong. Prior to joining CUHK in 2019, he was a professor of operational research in the School of Mathematical Sciences, University of Southampton. Huifu Xu’s research is mainly on optimal decision making under uncertainty including stochastic mathematical programs with equilibrium constraints (SMPEC), stochastic generalized equations and distributionally robust optimization with applications in energy markets. More recently, he is actively working on preference robust optimization and statistical robustness in data-driven problems.