Multistage Utility Preference Robust Optimization

Abstract

In this paper, we consider a multistage expected utility maximization problem where the decision maker's utility function at each stage depends on historical data and the information on the true utility function is incomplete. To mitigate adverse impact arising from ambiguity of the true utility, we propose a maximin robust model where the optimal policy is based on the worst-case sequence of utility functions from an ambiguity set constructed with partially available information about the decision maker's preferences. We then show that the multistage maximin problem is time consistent when the utility functions are state-dependent and demonstrate with a counter example that the time consistency may not be retained when the utility functions are state-independent. With the time consistency, we show the maximin problem can be solved by a recursive formula whereby a one-stage maximin problem is solved at each stage beginning from the last stage. Moreover, we propose two approaches to construct the ambiguity set: a pairwise comparison approach and a zeta-ball approach where a ball of utility functions centered at a nominal utility function under zeta-metric is considered. To overcome the difficulty arising from solving the infinite dimensional optimization problem in computation of the worst-case expected utility value, we propose piecewise linear approximation of the utility functions and derive error bound for the approximation under moderate conditions. Finally, we use the SDDP method and the nested Benders' decomposition method to solve the multistage state-dependent preference robust problem and the scenario tree method to solve the state-independent problem, and carry out comparative analysis on the efficiency of the computational schemes as well as out-of-sample performances of the state-dependent and state-independent models.