The Continuous Mixing Polyhedron

We analyze the polyhedral structure of the setsPCMIX = {( s, r, z) ?R XR+nXZ n|s +r j+z j=f j ,j = 1,..., n} andP+CMIX =PCMIXn{ s = 0}. The setP+CMIX is a natural generalization of the mixing set studied by Pochet and Wolsey [15, 16] and GA¼nlA¼k and Pochet [8] and recently has been introduced by Miller and Wolsey [12]. We introduce a new class of valid inequalities that has proven to be sufficient for describing conv( PCMIX). We give an extended formulation of sizeO( n) XO( n2) variables and constraints and indicate how to separate over conv( PCMIX) inO( n3) time. Finally, we show how the mixed integer rounding (MIR) inequalities of Nemhauser and Wolsey [14] and the mixing inequalities of GA¼nlA¼k and Pochet [8] constitute special cases of the cycle inequalities.