  Preference robust choice models concern decision-making problems where the
decision maker's (DM) utility/risk preferences are ambiguous and the evaluation
is based on the worst-case utility function/risk measure from a set of
plausible utility functions/risk measures. The current preference robust choice
models are mostly built upon von Neumann-Morgenstern expected utility theory,
the theory of convex risk measures, or Yaari's dual theory of choice, all of
which assume the DM's preferences satisfy some specified axioms. In this paper,
we extend the preference robust approach to a broader class of choice functions
which satisfy monotonicity and quasi-concavity in the space of multi-attribute
random prospects. While our new model is non-parametric and significantly
extends the coverage of decision-making problems, it also brings new
computational challenges due to the non-convexity of the optimization
formulations, which arises from the non-concavity of the class of quasi-concave
choice functions. To tackle these challenges, we develop a sorting-based
algorithm that efficiently evaluates the robust choice function (RCF) by
solving a sequence of linear programming problems. Then, we show how to solve
preference robust optimization (PRO) problems by solving a sequence of convex
optimization problems. We test our robust choice model and computational scheme
on a single-attribute portfolio optimization problem and a multi-attribute
capital allocation problem.
