# On Valid Inequalities for Mixed Integer *p*-Order Cone Programming

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## Abstract

We discuss two families of valid inequalities for linear mixed integer programming problems with cone constraints of arbitrary order, which arise in the context of stochastic optimization with downside risk measures. In particular, we extend the results of Atamtürk and Narayanan (Math. Program., 122:1–20, 2010, Math. Program., 126:351–363, 2011), who developed mixed integer rounding cuts and lifted cuts for mixed integer programming problems with second-order cone constraints. Numerical experiments conducted on randomly generated problems and portfolio optimization problems with historical data demonstrate the effectiveness of the proposed methods.

## Keywords

Valid inequalities Nonlinear cuts Mixed integer*p*-order cone programming Stochastic optimization Risk measures

## Notes

### Acknowledgements

This work was supported in part by AFOSR grant FA9550-12-1-0142 and NSF grant EPS1101284.

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