We discuss two families of valid inequalities for linear mixed integer programming problems with p-order cone constraints that arise in the context of stochastic optimization with downside risk measures. In particular, we extend the results of Atamtürk and Narayanan (2010, 2011) who developed mixed integer rounding cuts and lifted cuts for for mixed integer programming problems with second-order cone constraints (MISOCP). Numerical experiments conducted on randomly generated problems and

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... io optimization problems with historical data demonstrate the efficiency of the proposed methods.