Inverse Optimization of Convex Risk Functions

The theory of convex risk functions has now been well established as the basis for identifying the families of risk functions that should be used in risk-averse optimization problems. Despite its theoretical appeal, the implementation of a convex risk function remains difficult, because there is little guidance regarding how a convex risk function should be chosen so that it also well represents a decision maker’s subjective risk preference. In this paper, we address this issue through the lens of inverse optimization. Specifically, given solution data from some (forward) risk-averse optimization problem (i.e., a risk minimization problem with known constraints), we develop an inverse optimization framework that generates a risk function that renders the solutions optimal for the forward problem. The framework incorporates the well-known properties of convex risk functions—namely, monotonicity, convexity, translation invariance, and law invariance—as the general information about candidate risk functions, as well as feedback from individuals—which include an initial estimate of the risk function and pairwise comparisons among random losses—as the more specific information. Our framework is particularly novel in that unlike classical inverse optimization, it does not require making any parametric assumption about the risk function (i.e., it is nonparametric). We show how the resulting inverse optimization problems can be reformulated as convex programs and are polynomially solvable if the corresponding forward problems are polynomially solvable. We illustrate the imputed risk functions in a portfolio selection problem and demonstrate their practical value using real-life data.