Robust decision making over a set of random targets or risk-averse utilities with an application to portfolio optimization

Abstract

In many situations, decision-makers need to exceed a random target or make decisions using expected utilities. These two situations are equivalent when a decision-maker’s utility function is increasing and bounded. This article focuses on the problem where the random target has a concave cumulative distribution function (cdf) or a risk-averse decision-maker’s utility is concave (alternatively, the probability density function (pdf) of the random target or the decision-maker’ marginal utility is decreasing) and the concave cdf or utility can only be specified by an uncertainty set. Specifically, a robust (maximin) framework is studied to facilitate decision making in such situations. Functional bounds on the random target’s cdf and pdf are used. Additional general auxiliary requirements may also be used to describe the uncertainty set. It is shown that a discretized version of the problem may be formulated as a linear program. A result showing the convergence of discretized models for uncertainty sets specified using continuous functions is also proved. A portfolio investment decision problem is used to illustrate the construction and usefulness of the proposed decision-making framework.

Keywords:

• Random target
• robust optimization
• utility function
• marginal utility function
• expected utility maximization
• portfolio optimization

Appendix

The proof of Lemma 9. The uniform convergence of π_L(x) to π(x) means that, for any ε > 0, there exists such that, for all , which implies that |y_L − y*| ≤ ε. It follows that y_L → y* as L → ∞.

Arguing with a contradiction, we suppose that . There is such that for some δ > 0. For the compactness of , we can assume that x_L converges to a point . It follows that and hence π(x*) < y*. Moreover, y_L = π_L(x_L) and whose first item |π_L(x_L) − π(x_L)| → 0 by the uniform convergence of π_L to π under Assumption (iii) and second item |π(x_L) − π(x*)| → 0 by the continuity of π under Assumption (ii). That is, we have a contradiction that y_L > y*. ▪

Additional information

Notes on contributors

Jian Hu

Jian Hu is an Assistant Professor in the Industrial and Manufacturing Systems Engineering Department at University of Michigan Dearborn. He earned his Ph.D. from the Industrial Engineering and Management Sciences Department at Northwestern University. His research interests include stochastic, robust, and risk-adjusted optimization and its applications. His work has been published in IIE Transactions, Mathematical Programming, Operations Research, European Journal of Operations Research, and other journals.

Sanjay Mehrotra

Sanjay Mehrotra is a professor in Industrial Engineering and Management Sciences Department within McCormick School of Engineering and Applied Sciences, Northwestern University. He is widely known for his methodology research in optimization that has spanned topics in linear, convex, mixed integer, stochastic, multi-objective, distributionally robust, and risk-adjusted optimization. His applied research in healthcare research has included topics in bioinformatics, predictive modeling, budgeting, hospital operations, and policy modeling using modern operations research tools. He is the current chair of the Institute for Operations Research and Management Science’s Optimization Society. He is the director of the Center for Engineering and Health at Northwestern University’s Institute for Public Health and Medicine. He has been a vice-president of (Chapter/Fora) at the Institute for Operations Research and Management Sciences. He is the current editor for the Healthcare Department of IIE Transactions. He also held the role of department editor for the Optimization Department for the same journal.