The continuous mixing polyhedron

We analyze the polyhedral structure of the sets P^CMIX={(s,r,z)∈R×R+ⁿ×Zⁿ | s+rj+zj=fj, j=1,…,n} and P+^CMIX=P^CMIX n{s = 0}. The set P+^CMIX is a natural generalization of the mixing set studied by Pochet and Wolsey and Günlük and Pochet and recently has been introduced by Miller and Wolsey. We introduce a new class of valid inequalities that has been proven to be sufficient for describing conv(P^CMIX). We give an extended formulation of size O(n) × O(n²) variables and constraints and indicate how to separate over conv(P^CMIX) in O(n³) time. Finally, we show how the mixed integer rounding (MIR) inequalities of Nemhauser and Wolsey and the mixing inequalities of Günlük and Pochet constitute special cases of the cycle inequalities.