Facets for continuous multi-mixing set with general coefficients and bounded integer variables

Bansal and Kianfar (2015) introduced continuous multi-mixing set where the coefficients satisfy the so-called n-step MIR conditions and developed facet-defining inequalities for this set. In this paper, we first generalize their inequalities for the continuous multi-mixing set with general coefficients (where no conditions are imposed on the coefficients) and show that they are facet-defining in many cases. Next, we further generalize the continuous multi-mixing set with general coefficients by incorporating upper bounds on the integer variables. We introduce a family of valid inequalities for this set through a unified generalization of the n-step cycle inequalities and the mingled n-step MIR inequalities. We indicate how to separate over these inequalities in polynomial time and present the conditions under which a subset of these inequalities are facet-defining.