Optimization models based on coherent and averse risk measures are of essential importance in financial management and business operations. This paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty. The relationship between risk envelopes and regret envelopes is established by using the Lagrangian duality theory. It is then pointed out that such a relationship opens a door to a decomposition scheme, called progressive hedging, for solving multistage risk minimization and regret minimization problems. In particular, the classical progressive hedging algorithm is modified in order to handle a new class of  constraints that arises from a reformulation of risk and regret minimization problems. Numerical results are provided to show the efficiency of the progressive hedging algorithms.
 This paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty. A relationship between risk envelopes and regret envelopes is established by using the Lagrangian duality theory. Such a relationship opens a door to a decomposition scheme, called progressive hedging, for solving multistage risk minimization and regret minimization problems. In particular, the classical progressive hedging algorithm is modified in order to handle a new class of linkage constraints that arises from reformulations and other applications of risk and regret minimization problems. Numerical results are provided to show the efficiency of the progressive hedging algorithms.
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Optimization models based on coherent and averse risk measures are of essential importance in financial management and business operations.
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It is then pointed out that such
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linkage
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a reformulation
reformulations and other applications