Statistical robustness in utility preference robust optimization models
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Statistical robustness in utility preference robust optimization models Shaoyan Guo1 · Huifu Xu2 Received: 18 August 2019 / Accepted: 19 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2020
Abstract Utility preference robust optimization (PRO) concerns decision making problems where information on decision maker’s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited. A key assumption in the PRO models is that the true probability distribution is either known or can be recovered by real data generated by the true distribution. In data-driven optimization, this assumption may not be satisfied when perceived data differ from real data and consequently it raises a question as to whether statistical estimators of the PRO models based on perceived data are reliable. In this paper, we investigate the issue which is also known as qualitative robustness in the literature of statistics (Huber in Robust statistics, 3rd edn, Wiley, New York, 1981) and risk management (Krätschmer et al. in Finance Stoch 18:271–295, 2014). By utilizing the framework proposed by Krätschmer et al. (2014), we derive moderate sufficient conditions under which the optimal value and optimal solution of the PRO models are robust against perturbation of the exogenous uncertainty data, and examine how the tail behaviour of utility functions affects the robustness. Moreover, under some additional conditions on the Lipschitz continuity of the underlying functions with respect to random data, we establish quantitative robustness of the statistical estimators under the Kantorovich metric. Finally, we investigate uniform consistency of the optimal value and optimal solution of the PRO models. The results cover utility selection problems and stochastic optimization problems as special cases. Keywords PRO · Qualitative statistical robustness · Quantitative statistical robustness · Uniform consistency Mathematics Subject Classification 90C15 · 90C31 · 90C47
S. Guo: The work of this author is supported by the NSFC Grant No. 11801057 and the Fundamental Research Funds for the Central Universities under Project No. DUT19LK09. H. Xu: The work of this author is supported by a CUHK startup funding and GRC Grant No. 14500620. Extended author information available on the last page of the article
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S. Guo, H. Xu
1 Introduction We consider the following one-stage expected utility maximization problem max E P [u( f (x, ξ ))], x∈X
(1)
where u : IR → IR is a real-valued increasing utility function and f : IRn × IRk → IR is a continuous function, x is a decision vector which is restricted to taking values over a specified compact feasible set X ⊂ IRn , ξ : → IRk is a vector of random variables defined over probability space (, F, P), P := P ◦ ξ −1 is the probability measure on IRk induced by ξ . In practice, f may represent a financial position or the performance of an engineering design.
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