Risk Minimization, Regret Minimization and Progressive Hedging Algorithms

A dual relationship between risk measures and regret measures is established. It helps to build a list of correspondences between useful coherent and averse risk and regret measures. Based on such dual representation of risk measures, the multistage risk minimization problem can be converted to a multistage regret minimization problem. A progressive hedging algorithm is proposed for solving the corresponding regret minimization problem. In case that Ex-Funal constraints arise in the risk and regret minimization problem, the progressive hedging algorithm can be modified to take advantage of the hidden decomposability of the problems. Numerical results are reported to demonstrate the efficiency of the progressive hedging algorithms for riskneutral and risk-averse practical or randomly generated problems.