A Composite Risk Measure Framework for Decision Making Under Uncertainty
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A Composite Risk Measure Framework for Decision Making Under Uncertainty Peng-Yu Qian1 Zai-Wen Wen3
· Zi-Zhuo Wang2 ·
Received: 25 September 2017 / Revised: 16 March 2018 / Accepted: 8 June 2018 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In this paper, we present a unified framework for decision making under uncertainty. Our framework is based on the composite of two risk measures, where the inner risk measure accounts for the risk of decision if the exact distribution of uncertain model parameters were given, and the outer risk measure quantifies the risk that occurs when estimating the parameters of distribution. We show that the model is tractable under mild conditions. The framework is a generalization of several existing models, including stochastic programming, robust optimization, distributionally robust optimization. Using this framework, we study a few new models which imply probabilistic guarantees for solutions and yield less conservative results compared to traditional models. Numerical experiments are performed on portfolio selection problems to demonstrate the strength of our models. Keywords Risk management · Stochastic programming · Portfolio management
This paper is dedicated to Professor Yin-Yu Ye in celebration of his 70th birthday.
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Peng-Yu Qian [email protected] Zi-Zhuo Wang [email protected] Zai-Wen Wen [email protected]
1
Graduate School of Business, Columbia University, New York, NY 10027, USA
2
Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN 55455, USA
3
Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China
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P.-Y. Qian et al.
Mathematics Subject Classification 49N99
1 Introduction In this paper, we consider a decision maker who wants to minimize an objective function H (x, ξ ), where x ∈ Rn is the decision variable and ξ ∈ Rs is an uncertain/unknown parameter related to the model. For example, in a newsvendor problem, x is the order amount of newspapers by a newsvendor, and ξ is the uncertain future demand. Similarly, in an investment problem, x is the portfolio chosen by a portfolio manager, and ξ is the unknown future returns of the instruments. The existence of the uncertain parameters distinguishes the problem from ordinary optimization problems and has led to several decision making paradigms. One of the earliest attempts to deal with such decision making problems under uncertainty was proposed by [1], where it was assumed that the distribution of ξ is known exactly and the decision is chosen to minimize the expectation of H (x, ξ ). Such an approach is called stochastic programming. Another approach named robust optimization initiated by [2] assumes that all possible values of ξ lie within an uncertainty set, and the decision should be made to minimize the worst-case value of H (x, ξ ). Stochastic programming and robust optimization models can be viewed
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